Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x) \cos (x)+\sin (2 x) \sin (x)=1$

Answer: $x=\frac{\pi}{48}, \frac{11 \pi}{48}, \frac{13 \pi}{48}, \frac{23 \pi}{48}, \frac{25 \pi}{48}, \frac{35 \pi}{48}, \frac{37 \pi}{48}, \frac{47 \pi}{48}, \frac{49 \pi}{48}, \frac{59 \pi}{48}, \frac{61 \pi}{48}, \frac{71 \pi}{48}, \frac{73 \pi}{48}, \frac{83 \pi}{48}, \frac{85 \pi}{48}, \frac{95 \pi}{48}$


Question: Find the real solutions of the polynomial equation $x^{3}-7 x^{2}=7-x$.

Answer: $x=7$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\tan ^{2}(x) \geq 1$

Answer: $\left[-\frac{7 \pi}{4},-\frac{3 \pi}{2}\right) \cup\left(-\frac{3 \pi}{2},-\frac{5 \pi}{4}\right] \cup\left[-\frac{3 \pi}{4},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2},-\frac{\pi}{4}\right] \cup\left[\frac{\pi}{4}, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right] \cup\left[\frac{5 \pi}{4}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, \frac{7 \pi}{4}\right]$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\tan ^{2}(x)=3$

Answer: $x=\frac{\pi}{3}+\pi k$ or $x=\frac{2 \pi}{3}+\pi k ; x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3}$


Question:
The temperature $T$, in degrees Fahrenheit, $t$ hours after $6 \mathrm{AM}$ is given by:
$T(t)=-\frac{1}{2} t^{2}+8 t+32, \quad 0 \leq t \leq 12$
What is the warmest temperature of the day? When does this happen?

Answer: $64^{\circ}$ at 2 PM (8 hours after 6 AM.)


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cot (x) \leq 4$

Answer: $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$


Question: Rewrite the sum using summation notation: $2+\frac{3}{2}+\frac{4}{3}+\frac{5}{4}+\frac{6}{5}$

Answer: $\sum_{k=1}^{5} \frac{k+1}{k}$


Question: Convert the point from polar coordinates into rectangular coordinates: $(\pi, \arctan (\pi))$

Answer: $\left(\frac{\pi}{\sqrt{1+\pi^{2}}}, \frac{\pi^{2}}{\sqrt{1+\pi^{2}}}\right)$


Question: Write out the first four terms of the given sequence: $\{5 k-2\}_{k=1}^{\infty}$

Answer: $3,8,13,18$


Question: Use the properties of logarithms to write the expression as a single logarithm: $-\frac{1}{3} \ln (x)-\frac{1}{3} \ln (y)+\frac{1}{3} \ln (z)$

Answer: $\ln \left(\sqrt[3]{\frac{z}{x y}}\right)$


Question: Write the set using interval notation: $\{x \mid-3<x<3$ or $x=4\}$

Answer: $(-3,3) \cup\{4\}$


Question: Find the inverse of the matrix or state that the matrix is not invertible: $H=\left[\begin{array}{rrrr}1 & 0 & -3 & 0 \\ 2 & -2 & 8 & 7 \\ -5 & 0 & 16 & 0 \\ 1 & 0 & 4 & 1\end{array}\right]$

Answer: $H^{-1}=\left[\begin{array}{rrrr}16 & 0 & 3 & 0 \\ -90 & -\frac{1}{2} & -\frac{35}{2} & \frac{7}{2} \\ 5 & 0 & 1 & 0 \\ -36 & 0 & -7 & 1\end{array}\right]$


Question: Solve the system of nonlinear equations: $\left\{\begin{aligned} 2 y+2 z & =\lambda y z \\ 2 x+2 z & =\lambda x z \\ 2 y+2 x & =\lambda x y \\ x y z & =1000\end{aligned}\right.$

Answer: $x=10, y=10, z=10, \lambda=\frac{2}{5}$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\cos (x)>\frac{\sqrt{3}}{2}$

Answer: $\left[-\pi,-\frac{\pi}{2}\right) \cup\left[-\frac{\pi}{3}, \frac{\pi}{3}\right] \cup\left(\frac{\pi}{2}, \pi\right]$


Question:
Determine whether or not the relation represents $y$ as a function of $x$ and find the domain and range of those relations which are functions:
$\{(-3,0),(1,6),(2,-3),(4,2),(-5,6),(4,-9),(6,2)\}$

Answer: Not a function


Question: Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\left(\frac{1}{2}, 4\right),\left(\frac{3}{2},-1\right)$

Answer: $d=\sqrt{26}, M=\left(1, \frac{3}{2}\right)$


Question: Expand the given logarithm and simplify: $\log _{2}\left(\frac{128}{x^{2}+4}\right)$

Answer: $7-\log _{2}\left(x^{2}+4\right)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $3 \cos (2 x)=\sin (x)+2$

Answer: $x=\arctan (2), \pi+\arctan (2)$


Question: Find the inverse of the matrix or state that the matrix is not invertible: $C=\left[\begin{array}{rr}6 & 15 \\ 14 & 35\end{array}\right]$

Answer: $C$ is not invertible


Question: Evaluate the expression: $\log \left(e^{\ln (100)}\right)$

Answer: $\log \left(e^{\ln (100)}\right)=2$


Question: Solve the equation: $12 \operatorname{arccsc}\left(\frac{x}{3}\right)=2 \pi$

Answer: $x=\frac{1}{2}$


Question: Find the quadratic function passing through the points $(-2,1),(1,4),(3,-2)$

Answer: $f(x)=-\frac{4}{5} x^{2}+\frac{1}{5} x+\frac{23}{5}$


Question: A salesperson is paid $\$ 200$ per week plus $5 \%$ commission on her weekly sales of $x$ dollars. Find a linear function that represents her total weekly pay, $W$ (in dollars) in terms of $x$. What must her weekly sales be in order for her to earn $\$ 475.00$ for the week?

Answer: $W(x)=200+.05 x, x \geq 0$ She must make $\$ 5500$ in weekly sales.


Question: Approximate the given value to three decimal places: $\cos (e)$

Answer: $\cos (e) \approx-0.912$


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $45^{\circ}$

Answer: $\frac{\pi}{4}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (5 x)=-\cos (2 x)$

Answer: $x=\frac{\pi}{7}, \frac{\pi}{3}, \frac{3 \pi}{7}, \frac{5 \pi}{7}, \pi, \frac{9 \pi}{7}, \frac{11 \pi}{7}, \frac{5 \pi}{3}, \frac{13 \pi}{7}$


Question: Convert the point from polar coordinates into rectangular coordinates: $\left(11,-\frac{7 \pi}{6}\right)$

Answer: $\left(-\frac{11 \sqrt{3}}{2}, \frac{11}{2}\right)$


Question: Find the exact value of the cosine and sine of the given angle: $\theta=\frac{5 \pi}{4}$

Answer: $\cos \left(\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}, \sin \left(\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $2 x y=4$

Answer: Function


Question: Solve the following system: $\left\{\begin{aligned} x^{2}+\sqrt{y}+\log _{2}(z) & =6 \\ 3 x^{2}-2 \sqrt{y}+2 \log _{2}(z) & =5 \\ -5 x^{2}+3 \sqrt{y}+4 \log _{2}(z) & =13\end{aligned}\right.$

Answer: $(1,4,8),(-1,4,8)$


Question: Solve the rational equation: $\frac{x^{2}-2 x+1}{x^{3}+x^{2}-2 x}=1$

Answer: No solution


Question: Suppose $C(x)=x^{2}-10 x+27$ represents the costs, in hundreds, to produce $x$ thousand pens. How many pens should be produced to minimize the cost? What is this minimum cost?

Answer: 5000 pens should be produced for a cost of $\$ 200$.


Question: Convert the equation from rectangular coordinates into polar coordinates: $y=7$

Answer: $\left(7 \sqrt{2}, \frac{7 \pi}{4}\right)$


Question: Evaluate the expression: $\ln \left(\frac{1}{\sqrt{e}}\right)$

Answer: $\ln \left(\frac{1}{\sqrt{e}}\right)=-\frac{1}{2}$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $y=x^{3}-x$

Answer: Function


Question: Evaluate the expression: $\log _{2}(32)$

Answer: $\log _{2}(32)=5$


Question: Solve the equation analytically: $\log _{3}(7-2 x)=2$

Answer: $x=-1$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (6 x) \cos (x)=-\cos (6 x) \sin (x)$

Answer: $x=0, \frac{\pi}{7}, \frac{2 \pi}{7}, \frac{3 \pi}{7}, \frac{4 \pi}{7}, \frac{5 \pi}{7}, \frac{6 \pi}{7}, \pi, \frac{8 \pi}{7}, \frac{9 \pi}{7}, \frac{10 \pi}{7}, \frac{11 \pi}{7}, \frac{12 \pi}{7}, \frac{13 \pi}{7}$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cos ^{2}(x)>\frac{1}{2}$

Answer: $\left[0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right) \cup\left(\frac{7 \pi}{4}, 2 \pi\right]$


Question: Solve the equation analytically: $500\left(1-e^{2 x}\right)=250$

Answer: $x=\frac{1}{2} \ln \left(\frac{1}{2}\right)=-\frac{1}{2} \ln (2)$


Question: Solve the quadratic equation $x^{2}-m x=1$ for $x$.

Answer: $(-\infty, 1) \cup\left(2, \frac{3+\sqrt{17}}{2}\right)$


Question: Solve the following system of linear equations: $\left\{\begin{aligned} x_{1}-x_{3} & =-2 \\ 2 x_{2}-x_{4} & =0 \\ x_{1}-2 x_{2}+x_{3} & =0 \\ -x_{3}+x_{4} & =1\end{aligned}\right.$

Answer: $(1,2,3,4)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan (x)=\cos (x)$

Answer: $x=\arcsin \left(\frac{-1+\sqrt{5}}{2}\right) \approx 0.6662, \pi-\arcsin \left(\frac{-1+\sqrt{5}}{2}\right) \approx 2.4754$


Question: Solve the equation for $t$: $\sin (t)=-\frac{\sqrt{2}}{2}$

Answer: $\sin (t)=-\frac{\sqrt{2}}{2}$ when $t=\frac{5 \pi}{4}+2 \pi k$ or $t=\frac{7 \pi}{4}+2 \pi k$ for any integer $k$.


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\sin (x)>\frac{1}{3}$

Answer: $\left(\arcsin \left(\frac{1}{3}\right), \pi-\arcsin \left(\frac{1}{3}\right)\right)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $2 \tan (x)=1-\tan ^{2}(x)$

Answer: $x=\frac{\pi}{8}, \frac{5 \pi}{8}, \frac{9 \pi}{8}, \frac{13 \pi}{8}$


Question: Evaluate the expression: $\log (0.01)$

Answer: $\log (0.01)=-2$


Question: Solve the equation analytically: $\ln \left(8-x^{2}\right)=\ln (2-x)$

Answer: $x=-2$


Question: Find the exact value of the cosine and sine of the given angle: $\theta=-\frac{13 \pi}{2}$

Answer: $\cos \left(-\frac{13 \pi}{2}\right)=0, \sin \left(-\frac{13 \pi}{2}\right)=-1$


Question: Write out the first four terms of the given sequence: $s_{0}=1, s_{n+1}=x^{n+1}+s_{n}, n \geq 0$

Answer: $1, x+1, x^{2}+x+1, x^{3}+x^{2}+x+1$


Question: Convert the angle from radian measure into degree measure: $\frac{\pi}{2}$

Answer: $90^{\circ}$


Question: Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\left(-\frac{2}{3}, \frac{3}{2}\right),\left(\frac{7}{3}, 2\right)$

Answer: $d=\frac{\sqrt{37}}{2}, M=\left(\frac{5}{6}, \frac{7}{4}\right)$


Question: Evaluate: $\left(\begin{array}{c}n \\ n-2\end{array}\right), n \geq 2$

Answer: $\frac{n(n-1)}{2}$


Question: A small boat leaves the dock at Camp DuNuthin and heads across the Nessie River at 17 miles per hour (that is, with respect to the water) at a bearing of $S 68^{\circ} \mathrm{W}$. The river is flowing due east at 8 miles per hour. What is the boat's true speed and heading? Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree.

Answer: The current is moving at about 10 miles per hour bearing $\mathrm{N} 54.6^{\circ} \mathrm{W}$.


Question: Let $L$ be the line $y=2 x+1$. Find a function $D(x)$ which measures the distance squared from a point on $L$ to $(0,0)$. Use this to find the point on $L$ closest to $(0,0)$.

Answer: $(-\infty, \infty)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sec (x)=2 \csc (x)$

Answer: $x=\frac{\pi}{6}, \frac{\pi}{2}$


Question: Find the standard equation of the circle which satisfies the given criteria: center $(3,6)$, passes through $(-1,4)$

Answer: $(x-3)^{2}+(y-6)^{2}=20$


Question: Carbon-14 cannot be used to date inorganic material such as rocks, but there are many other methods of radiometric dating which estimate the age of rocks. One of them, RubidiumStrontium dating, uses Rubidium-87 which decays to Strontium-87 with a half-life of 50 billion years. Use Equation 6.5 to express the amount of Rubidium-87 left from an initial 2.3 micrograms as a function of time $t$ in billions of years. Research this and other radiometric techniques and discuss the margins of error for various methods with your classmates.

Answer: $A(t)=2.3 e^{-0.0138629 t}$


Question: Simplify the expression: $\frac{9 !}{4 ! 3 ! 2 !}$

Answer: 1260


Question: Find all of the points on the line $y=1-x$ which are 2 units from $(1,-1)$.

Answer: $\left(\frac{3-\sqrt{7}}{2}, \frac{-1+\sqrt{7}}{2}\right),\left(\frac{3+\sqrt{7}}{2}, \frac{-1-\sqrt{7}}{2}\right)$


Question: Solve the equation: $|x-4|=x-5$

Answer: no solution


Question: Find the domain of the function: $f(x)=\ln (4 x-20)$

Answer: $(5, \infty)$


Question: Solve the equation analytically: $-\log (x)=5.4$

Answer: $x=10^{-5.4}$


Question: Write the set using interval notation: $\{x \mid x \neq-3,4\}$

Answer: $(-\infty,-3) \cup(-3,4) \cup(4, \infty)$


Question: Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(\sqrt{2}, \sqrt{3}),(-\sqrt{8},-\sqrt{12})$

Answer: $d=3 \sqrt{5}, M=\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{3}}{2}\right)$


Question: Solve the given inequality: $3 \arccos (x) \leq \pi$

Answer: $\left[\frac{1}{2}, 1\right]$


Question: Solve the equation analytically: $\ln \left(x^{2}\right)=(\ln (x))^{2}$

Answer: $x=1, x=e^{2}$


Question: Evaluate the expression: $\log _{36}(216)$

Answer: $\log _{36}(216)=\frac{3}{2}$


Question: Solve the equation for $t$: $\cos (t)=0$

Answer: $\cos (t)=0$ when $t=\frac{\pi}{2}+\pi k$ for any integer $k$.


Question: Use the properties of logarithms to write the expression as a single logarithm: $\log _{3}(x)-2 \log _{3}(y)$

Answer: $\log _{3}\left(\frac{x}{y^{2}}\right)$


Question: Solve the equation analytically: $e^{2 x}-3 e^{x}-10=0$

Answer: $x=\ln (5)$


Question:
For each isotope:
- Find the decay constant $k$. Round your answer to four decimal places.
- Find a function which gives the amount of isotope $A$ which remains after time $t$. (Keep the units of $A$ and $t$ the same as the given data.)
- Determine how long it takes for $90 \%$ of the material to decay. Round your answer to two decimal places. (HINT: If $90 \%$ of the material decays, how much is left?)
14. Cobalt 60, used in food irradiation, initial amount 50 grams, half-life of 5.27 years.

Answer: $\bullet k=\frac{\ln (1 / 2)}{5.27} \approx-0.1315$


Question: Use the properties of logarithms to write the expression as a single logarithm: $\log _{2}(x)+\log _{4}(x-1)$

Answer: $\log _{2}(x \sqrt{x-1})$


Question: Solve the equation or inequality: $x^{\frac{2}{3}}=4$

Answer: $x= \pm 8$


Question: Evaluate the expression: $\log \left(\sqrt[9]{10^{11}}\right)$

Answer: $\log \left(\sqrt[9]{10^{11}}\right)=\frac{11}{9}$


Question: Solve the rational equation: $\frac{3 x-1}{x^{2}+1}=1$

Answer: $x=1, x=2$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\cot (x) \geq-1$

Answer: $\left(-\pi,-\frac{\pi}{4}\right] \cup\left(0, \frac{3 \pi}{4}\right]$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cos (2 x) \leq 0$

Answer: $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right] \cup\left[\frac{5 \pi}{4}, \frac{7 \pi}{4}\right]$


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $150^{\circ}$

Answer: $\frac{5 \pi}{6}$


Question: Find the exact value of the cosine and sine of the given angle: $\theta=\frac{\pi}{3}$

Answer: $\cos \left(\frac{\pi}{3}\right)=\frac{1}{2}, \sin \left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}$


Question: Find the domain of the function: $f(x)=\ln \left(-2 x^{3}-x^{2}+13 x-6\right)$

Answer: $(-\infty,-3) \cup\left(\frac{1}{2}, 2\right)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)+\csc ^{2}(x)=0$

Answer: $x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (5 x) \cos (3 x)-\sin (5 x) \sin (3 x)=\frac{\sqrt{3}}{2}$

Answer: $x=0, \frac{\pi}{2}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)-\sqrt{3} \sin (2 x)=\sqrt{2}$

Answer: $x=0, \pi, \frac{\pi}{3}, \frac{4 \pi}{3}$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\tan (x) \geq \sqrt{3}$

Answer: $x=\left[\frac{\pi}{3}, \frac{\pi}{2}\right) \cup\left[\frac{4 \pi}{3}, \frac{3 \pi}{2}\right)$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\sin \left(\frac{x}{3}\right)=\frac{\sqrt{2}}{2}$

Answer: $x=\frac{3 \pi}{4}+6 \pi k$ or $x=\frac{9 \pi}{4}+6 \pi k ; x=\frac{3 \pi}{4}$


Question: Find all of the points on the $y$-axis which are 5 units from the point $(-5,3)$.

Answer: $(0,3)$


Question: Solve the equation: $\pi-2 \arcsin (x)=2 \pi$

Answer: $x=-1$


Question: Solve the following system of linear equations: $\left\{\begin{aligned} x-y+z & =-4 \\ -3 x+2 y+4 z & =-5 \\ x-5 y+2 z & =-18\end{aligned}\right.$

Answer: $(1,3,-2)$


Question: Solve the equation analytically: $5^{2 x-1}=125$

Answer: $x=2$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sqrt{3} \sin (2 x)+\cos (2 x)=1$

Answer: $x=\frac{17 \pi}{24}, \frac{41 \pi}{24}, \frac{23 \pi}{24}, \frac{47 \pi}{24}$


Question: A yo-yo which is 2.25 inches in diameter spins at a rate of 4500 revolutions per minute. How fast is the edge of the yo-yo spinning in miles per hour? Round your answer to two decimal places.

Answer: About 30.12 miles per hour


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (x)+\cos (x)=1$

Answer: $x=0, \frac{\pi}{2}$


Question: Convert the equation from polar coordinates into rectangular coordinates: $r=1+\sin (\theta)$

Answer: $r=\sin (\theta)$


Question: A mobile plan charges a base monthly rate of $\$ 10$ for the first 500 minutes of air time plus a charge of $15 \notin$ for each additional minute. Write a piecewise-defined linear function which calculates the monthly cost $C$ (in dollars) for using $m$ minutes of air time.

Answer: $C(m)=\left\{\begin{array}{rll}10 & \text { if } & 0 \leq m \leq 500 \\ 10+0.15(m-500) & \text { if } & m>500\end{array}\right.$


Question:
Use one matrix inverse to solve the following system of linear equations:
$\left\{\begin{array}{r}3 x+7 y=26 \\ 5 x+12 y=39\end{array}\right.$

Answer: $\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]\left[\begin{array}{l}26 \\ 39\end{array}\right]=\left[\begin{array}{r}39 \\ -13\end{array}\right]$ So $x=39$ and $y=-13$.


Question:
Create a polynomial $f$ with the following characteristics:
- The leading term of $f(x)$ is $-2x^3$
- $c=2i$ is a zero
- $f(0)=-16$

Answer: $f(x)=-2(x-2 i)(x+2 i)(x+2)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=\sin (x)$

Answer: $x=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{2}$


Question: Evaluate the expression: $\log _{8}(4)$

Answer: $\log _{8}(4)=\frac{2}{3}$


Question: How much money needs to be invested now to obtain $\$ 5000$ in 10 years if the interest rate in a CD is $2.25 \%$, compounded monthly? Round your answer to the nearest cent.

Answer: $P=\frac{5000}{\left(1+\frac{0.025}{12}\right)^{12 \cdot 10}} \approx \$ 3993.42$


Question: Simplify the expression: $\frac{(n+1) !}{n !}, n \geq 0$.

Answer: $n+1$


Question: Convert the angle into the DMS system and round the answer to the nearest second: $63.75^{\circ}$

Answer: $63^{\circ} 45^{\prime}$


Question: Solve the equation analytically: $\log _{\frac{1}{2}}(2 x-1)=-3$

Answer: $x=\frac{9}{2}$


Question: Let $L$ be the line $y=2 x+1$. Find a function $D(x)$ which measures the distance squared from a point on $L$ to $(0,0)$. Use this to find the point on $L$ closest to $(0,0)$.

Answer: $D(x)=x^{2}+(2 x+1)^{2}=5 x^{2}+4 x+1, D$ is minimized when $x=-\frac{2}{5}$, so the point on $y=2 x+1$ closest to $(0,0)$ is $\left(-\frac{2}{5}, \frac{1}{5}\right)$


Question: Solve the equation analytically: $2000 e^{0.1 t}=4000$

Answer: $t=\frac{\ln (2)}{0.1}=10 \ln (2)$


Question: Solve the equation: $|4-x|-|x+2|=0$

Answer: $x=1$


Question: Solve the equation: $|4-x|=7$

Answer: $x=-3$ or $x=11$


Question: Solve the inequality analytically: $\frac{1-\ln (x)}{x^{2}}<0$

Answer: $(e, \infty)$


Question: Find the inverse of the matrix or state that the matrix is not invertible: $G=\left[\begin{array}{rrr}1 & 2 & 3 \\ 2 & 3 & 11 \\ 3 & 4 & 19\end{array}\right]$

Answer: $G$ is not invertible


Question: Solve the equation: $6 \operatorname{arccot}(2 x)-5 \pi=0$

Answer: $x=-\frac{\sqrt{3}}{2}$


Question: Solve the rational equation: $\frac{x}{5 x+4}=3$

Answer: $x=-\frac{6}{7}$


Question:
Use one matrix inverse to solve the following system of linear equations:
$\left\{\begin{aligned} 3 x+7 y & =0 \\ 5 x+12 y & =-1\end{aligned}\right.$

Answer: $\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]\left[\begin{array}{r}0 \\ -1\end{array}\right]=\left[\begin{array}{r}7 \\ -3\end{array}\right] \quad$ So $x=7$ and $y=-3$.


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\csc (x)=0$

Answer: No solution


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (3 x)=\cos (5 x)$

Answer: $x=0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \pi, \frac{5 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}$


Question: Simplify the given power of $i$: $i^{26}$

Answer: $i^{26}=\left(i^{4}\right)^{6} \cdot i^{2}=1 \cdot(-1)=-1$


Question: Solve the equation for $t$: $\cos (t)=\frac{1}{2}$

Answer: $\cos (t)=\frac{1}{2}$ when $t=\frac{\pi}{3}+2 \pi k$ or $t=\frac{5 \pi}{3}+2 \pi k$ for any integer $k$.


Question: Solve the equation: $\frac{2}{3}|5-2 x|-\frac{1}{2}=5$

Answer: $x=-\frac{13}{8}$ or $x=\frac{53}{8}$


Question: Simplify the given power of $i$: $i^6$

Answer: $i^{6}=i^{4} \cdot i^{2}=1 \cdot(-1)=-1$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (x)+\sqrt{3} \cos (x)=1$

Answer: $x=\frac{\pi}{2}, \frac{11 \pi}{6}$


Question: Solve the equation analytically: $e^{-5730 k}=\frac{1}{2}$

Answer: $k=\frac{\ln \left(\frac{1}{2}\right)}{-5730}=\frac{\ln (2)}{5730}$


Question: Solve the equation analytically: $e^{2 x}=2 e^{x}$

Answer: $x=\ln (2)$


Question: Express the domain of the function using the extended interval notation: $f(x)=\sqrt{\tan ^{2}(x)-1}$

Answer: $\bigcup_{k=-\infty}^{\infty}\left\{\left[\frac{(4 k+1) \pi}{4}, \frac{(2 k+1) \pi}{2}\right) \cup\left(\frac{(2 k+1) \pi}{2}, \frac{(4 k+3) \pi}{4}\right]\right\}$


Question: Find the standard equation of the circle which satisfies the given criteria: endpoints of a diameter: $\left(\frac{1}{2}, 4\right),\left(\frac{3}{2},-1\right)$

Answer: $(x-1)^{2}+\left(y-\frac{3}{2}\right)^{2}=\frac{13}{2}$


Question: Solve the given inequality: $\pi>2 \arctan (x)$

Answer: $(-\infty, \infty)$


Question: Find the domain of the function: $f(x)=\sqrt[4]{\log _{4}(x)}$

Answer: $[1, \infty)$


Question: Find the inverse of the matrix or state that the matrix is not invertible: $D=\left[\begin{array}{rr}2 & -1 \\ 16 & -9\end{array}\right]$

Answer: $D^{-1}=\left[\begin{array}{cc}\frac{9}{2} & -\frac{1}{2} \\ 8 & -1\end{array}\right]$


Question: The current $i$ measured in amps in a certain electronic circuit with a constant impressed voltage of 120 volts is given by $i(t)=2-2 e^{-10 t}$ where $t \geq 0$ is the number of seconds after the circuit is switched on. Determine the value of $i$ as $t \rightarrow \infty$. (This is called the steady state current.)

Answer: The steady state current is 2 amps.


Question: Solve the equation or inequality: $2(x-2)^{-\frac{1}{3}}-\frac{2}{3} x(x-2)^{-\frac{4}{3}} \leq 0$

Answer: $(-\infty, 2) \cup(2,3]$


Question: Solve the equation: $2|5 x+1|-3=0$

Answer: $x=-\frac{1}{2}$ or $x=\frac{1}{10}$


Question: Solve the equation or inequality: $\frac{2}{3}(x+4)^{\frac{3}{5}}(x-2)^{-\frac{1}{3}}+\frac{3}{5}(x+4)^{-\frac{2}{5}}(x-2)^{\frac{2}{3}} \geq 0$

Answer: $(-\infty,-4) \cup\left(-4,-\frac{22}{19}\right] \cup(2, \infty)$


Question: Solve the rational inequality and express your answer using interval notation: $\frac{-x^{3}+4 x}{x^{2}-9} \geq 4 x$

Answer: $(-\infty,-3) \cup[-2 \sqrt{2}, 0] \cup[2 \sqrt{2}, 3)$


Question: Approximate the given value to three decimal places: $\cos \left(207^{\circ}\right)$

Answer: $\cos \left(207^{\circ}\right) \approx-0.891$


Question: Find the inverse of $k(x)=\frac{2 x}{\sqrt{x^{2}-1}}$.

Answer: $k^{-1}(x)=\frac{x}{\sqrt{x^{2}-4}}$


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $0^{\circ}$

Answer: 0


Question: Write the set using interval notation: $\{x \mid x<3$ or $x \geq 2\}$

Answer: $(-\infty, \infty)$


Question: Evaluate the expression: $\log _{2}\left(3^{-\log _{3}(2)}\right)$

Answer: $\log _{2}\left(3^{-\log _{3}(2)}\right)=-1$


Question: Simplify the power of a complex number: $\left(\frac{\sqrt{3}}{2}+\frac{1}{2} i\right)^{3}$

Answer: $i$


Question: Solve the rational equation: $\frac{1}{x+3}+\frac{1}{x-3}=\frac{x^{2}-3}{x^{2}-9}$

Answer: $x=-1$


Question: Find the domain of the function: $f(x)=\ln (\sqrt{x-4}-3)$

Answer: $(13, \infty)$


Question: Convert the equation from polar coordinates into rectangular coordinates: $r=1-2 \cos (\theta)$

Answer: $r=6 \sin (\theta)$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $2 x+3 y=4$

Answer: Function


Question: Find the domain of the function: $f(x)=\frac{1}{3-\log _{5}(x)}$

Answer: $(0,125) \cup(125, \infty)$


Question: A plumber charges $\$ 50$ for a service call plus $\$ 80$ per hour. If she spends no longer than 8 hours a day at any one site, find a linear function that represents her total daily charges $C$ (in dollars) as a function of time $t$ (in hours) spent at any one given location.

Answer: $C(t)=80 t+50,0 \leq t \leq 8$.


Question: How much money needs to be invested now to obtain $\$ 2000$ in 3 years if the interest rate in a savings account is $0.25 \%$, compounded continuously? Round your answer to the nearest cent.

Answer: $P=\frac{2000}{e^{0.0025 \cdot 3}} \approx \$ 1985.06$


Question: If $\theta=37.5^{\circ}$ and the side opposite $\theta$ has length 306 , how long is the side adjacent to $\theta$ ?

Answer: The hypotenuse has length $c=\frac{306}{\sin \left(37.5^{\circ}\right)} \approx 502.660$, so the side adjacent to $\theta$ has length $\sqrt{c^{2}-306^{2}} \approx 398.797$.


Question: Solve the rational equation: $\frac{-x^{3}+4 x}{x^{2}-9}=4 x$

Answer: $x=0, x= \pm 2 \sqrt{2}$


Question: Solve the following system of linear equations: $\left\{\begin{aligned} 2 x-y+z & =1 \\ 2 x+2 y-z & =1 \\ 3 x+6 y+4 z & =9\end{aligned}\right.$

Answer: $\left(\frac{1}{3}, \frac{2}{3}, 1\right)$


Question: Use a calculator to approximate the five fifth roots of 1.

Answer: $w_{0}=1$


Question: Solve the following system of linear equations: $\left\{\begin{aligned}-5 x+y & =17 \\ x+y & =5\end{aligned}\right.$

Answer: $(-2,7)$


Question: Solve the rational inequality and express your answer using interval notation: $\frac{3 x-1}{x^{2}+1} \leq 1$

Answer: $(-\infty, 1] \cup[2, \infty)$


Question: If $\tan (\theta)=\frac{x}{7}$ for $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$, find an expression for $\sin (2 \theta)$ in terms of $x$.

Answer: $\frac{14 x}{x^{2}+49}$


Question: Write out the first four terms of the given sequence: $\left\{\frac{n^{2}+1}{n+1}\right\}_{n=0}^{\infty}$

Answer: $1,1, \frac{5}{3}, \frac{5}{2}$


Question: Convert the angle from radian measure into degree measure: $\pi$

Answer: $180^{\circ}$


Question: Solve the equation analytically: $5^{x}=-2$

Answer: No solution.


Question: The sounds we hear are made up of mechanical waves. The note ' $A$ ' above the note 'middle $\mathrm{C}^{\prime}$ is a sound wave with ordinary frequency $f=440 \mathrm{Hertz}=440 \frac{\mathrm{cycles}}{\text { second }}$. Find a sinusoid which models this note, assuming that the amplitude is 1 and the phase shift is 0 .

Answer: $S(t)=\sin (880 \pi t)$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $x=-6$

Answer: Not a function


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $-270^{\circ}$

Answer: $-\frac{3 \pi}{2}$


Question: Simplify the expression: $(3 !)^{2}$

Answer: 36


Question: Solve the equation analytically: $\log (3 x-1)=\log (4-x)$

Answer: $x=\frac{5}{4}$


Question: Convert the point from polar coordinates into rectangular coordinates: $(6, \arctan (2))$

Answer: $\left(\frac{6 \sqrt{5}}{5}, \frac{12 \sqrt{5}}{5}\right)$


Question: Find the domain of the function: $f(x)=\log _{9}(|x+3|-4)$

Answer: $(-\infty,-7) \cup(1, \infty)$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\sec (x) \leq 2$

Answer: $\left[-\frac{2 \pi}{3},-\frac{\pi}{3}\right) \cup\left(\frac{\pi}{3}, \frac{2 \pi}{3}\right)$


Question: Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference $d$; if it is geometric, find the common ratio $r$: $a_{n}=\frac{n !}{2}, n \geq 0$

Answer: neither


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $135^{\circ}$

Answer: $\frac{3 \pi}{4}$


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $240^{\circ}$

Answer: $\frac{4 \pi}{3}$


Question: If $\sin (\theta)=\frac{x}{2}$ for $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$, find an expression for $\cos (2 \theta)$ in terms of $x$.

Answer: $1-\frac{x^{2}}{2}$


Question: Convert the point from polar coordinates into rectangular coordinates: $(-20,3 \pi)$

Answer: $(20,0)$


Question: Solve the equation analytically: $\frac{5000}{1+2 e^{-3 t}}=2500$

Answer: $t=\frac{1}{3} \ln (2)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=5 \sin (x)-2$

Answer: $x=0, \frac{2 \pi}{3}, \frac{4 \pi}{3}$


Question: Solve the equation: $|x|=6$

Answer: $x=-6$ or $x=6$


Question: Simplify the quantity $\sqrt{-49}$

Answer: $7 i$


Question: Solve the polynomial inequality $-2 x^{3}+19 x^{2}-49 x+20>0$ and state your answer using interval notation.

Answer: $(-\infty,-1) \cup(-1,0) \cup(2, \infty)$


Question: Find the exact value of the cosine and sine of the given angle: $\theta=\frac{2 \pi}{3}$

Answer: $\cos \left(\frac{2 \pi}{3}\right)=-\frac{1}{2}, \sin \left(\frac{2 \pi}{3}\right)=\frac{\sqrt{3}}{2}$


Question: Solve the equation analytically: $3^{(x-1)}=29$

Answer: $x=\frac{\ln (29)+\ln (3)}{\ln (3)}$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\sin \left(2 x-\frac{\pi}{3}\right)=-\frac{1}{2}$

Answer: $x=\frac{3 \pi}{4}+\pi k$ or $x=\frac{13 \pi}{12}+\pi k ; x=\frac{\pi}{12}, \frac{3 \pi}{4}, \frac{13 \pi}{12}, \frac{7 \pi}{4}$


Question: Expand the given logarithm and simplify: $\ln \left(\frac{\sqrt{z}}{x y}\right)$

Answer: $\frac{1}{2} \ln (z)-\ln (x)-\ln (y)$


Question: Expand the binomial: $\left(\frac{1}{3} x+y^{2}\right)^{3}$

Answer: $\left(\frac{1}{3} x+y^{2}\right)^{3}=\frac{1}{27} x^{3}+\frac{1}{3} x^{2} y^{2}+x y^{4}+y^{6}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=\cos (x)$

Answer: $x=0, \frac{2 \pi}{3}, \frac{4 \pi}{3}$


Question: What is the largest rectangular area one can enclose with 14 inches of string?

Answer: The largest rectangle has area 12.25 square inches.


Question: Evaluate the expression: $\log _{36}(36)$

Answer: $\log _{36}(36)=1$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\sec ^{2}(x) \leq 4$

Answer: $\left(-\infty, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right]$


Question: Find the standard equation of the circle which satisfies the given criteria: endpoints of a diameter: $(3,6)$ and $(-1,4)$

Answer: $(x-1)^{2}+(y-5)^{2}=5$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}+y^{2}=4$

Answer: Not a function


Question: Rewrite the sum using summation notation: $1+2+4+\cdots+2^{29}$

Answer: $\sum_{k=1}^{30} 2^{k-1}$


Question: Simplify the expression: $\frac{(k-1) !}{(k+2) !}, k \geq 1$.

Answer: $\frac{1}{k(k+1)(k+2)}$


Question: Solve the rational inequality and express your answer using interval notation: $\frac{x^{3}+2 x^{2}+x}{x^{2}-x-2} \geq 0$

Answer: $(-1,0] \cup(2, \infty)$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}-y^{2}=4$

Answer: Not a function


Question: Write the set using interval notation: $\{x \mid x \neq 0,2\}$

Answer: $(-\infty, 0) \cup(0,2) \cup(2, \infty)$


Question: Solve the equation analytically: $\log \left(x^{2}-3 x\right)=1$

Answer: $x=-2,5$


Question: Evaluate the expression: $\log _{13}(\sqrt{13})$

Answer: $\log _{13}(\sqrt{13})=\frac{1}{2}$


Question: Find an equation for the parabola which fits the given criteria: Vertex $(7,0)$, focus $(0,0)$

Answer: $y^{2}=-28(x-7)$


Question: Solve the equation analytically: $4^{x}+2^{x}=12$

Answer: $x=\frac{\ln (3)}{\ln (2)}$


Question: Simplify the quantity $\sqrt{-(-9)}$

Answer: 3


Question: Write the set using interval notation: $\{x \mid x>2$ or $x= \pm 1\}$

Answer: $\{-1\} \cup\{1\} \cup(2, \infty)$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cot ^{2}(x) \geq \frac{1}{3}$

Answer: $\left(0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \pi\right) \cup\left(\pi, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right)$


Question: Solve the inequality analytically: $25\left(\frac{4}{5}\right)^{x} \geq 10$

Answer: $\left(-\infty, \frac{\ln \left(\frac{2}{5}\right)}{\ln \left(\frac{4}{5}\right)}\right]=\left(-\infty, \frac{\ln (2)-\ln (5)}{\ln (4)-\ln (5)}\right]$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cos ^{2}(x)=\frac{1}{2}$

Answer: $x=\frac{\pi}{4}+\frac{\pi k}{2} ; x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$


Question: Solve the equation analytically: $\log (x)-\log (2)=\log (x+8)-\log (x+2)$

Answer: $x=4$


Question: Convert the angle from radian measure into degree measure: $-\frac{\pi}{6}$

Answer: $-30^{\circ}$


Question: A 600 pound Sasquatch statue is suspended by two cables from a gymnasium ceiling. If each cable makes a $60^{\circ}$ angle with the ceiling, find the tension on each cable. Round your answer to the nearest pound.

Answer: The resultant force is only about 296 pounds so the couch doesn't budge. Even if it did move, the stronger force on the third rope would have made the couch drift slightly to the south as it traveled down the street.


Question: The height of an object dropped from the roof of an eight story building is modeled by $h(t)=-16 t^{2}+64,0 \leq t \leq 2$. Here, $h$ is the height of the object off the ground, in feet, $t$ seconds after the object is dropped. How long before the object hits the ground?

Answer: 2 seconds.


Question: Convert the equation from polar coordinates into rectangular coordinates: $r=-\csc (\theta) \cot (\theta)$

Answer: $r=7 \sin (\theta)$


Question: Use the properties of logarithms to write the expression as a single logarithm: $\ln (x)+\frac{1}{2}$

Answer: $\ln (x \sqrt{e})$


Question: Solve the equation or inequality: $5-(4-2 x)^{\frac{2}{3}}=1$

Answer: $x=-2,6$


Question: Evaluate the expression: $\log _{\frac{1}{5}}(625)$

Answer: $\log _{\frac{1}{5}}(625)=-4$


Question: Solve the equation analytically: $\log \left(\frac{x}{10^{-3}}\right)=4.7$

Answer: $x=10^{1.7}$


Question: Rewrite the sum using summation notation: $8+11+14+17+20$

Answer: $\sum_{k=1}^{5}(3 k+5)$


Question: Find the domain of the function: $f(x)=\log _{7}(4 x+8)$

Answer: $(-2, \infty)$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\tan (6 x)=1$

Answer: $x=\frac{\pi}{24}+\frac{\pi k}{6} ; x=\frac{\pi}{24}, \frac{5 \pi}{24}, \frac{3 \pi}{8}, \frac{13 \pi}{24}, \frac{17 \pi}{24}, \frac{7 \pi}{8}, \frac{25 \pi}{24}, \frac{29 \pi}{24}, \frac{11 \pi}{8}, \frac{37 \pi}{24}, \frac{41 \pi}{24}, \frac{15 \pi}{8}$


Question: Evaluate the expression: $7^{\log _{7}(3)}$

Answer: $7^{\log _{7}(3)}=3$


Question: Use the properties of logarithms to write the expression as a single logarithm: $\log _{5}(x)-3$

Answer: $\log _{5}\left(\frac{x}{125}\right)$


Question: Solve the equation or inequality using your calculator: $\ln (x)=e^{-x}$

Answer: $x \approx 1.3098$


Question: Solve the inequality analytically: $70+90 e^{-0.1 t} \leq 75$

Answer: $\left[\frac{\ln \left(\frac{1}{18}\right)}{-0.1}, \infty\right)=[10 \ln (18), \infty)$


Question: Simplify the given power of $i$: $i^{304}$

Answer: $i^{304}=\left(i^{4}\right)^{76}=1^{76}=1$


Question: Solve the equation analytically: $e^{x}+15 e^{-x}=8$

Answer: $x=\ln (3), \ln (5)$


Question: Solve the equation analytically: $\ln (\ln (x))=3$

Answer: $x=e^{e^{3}}$


Question: Approximate the given value to three decimal places: $\sin (392.994)$

Answer: $\sin (392.994) \approx-0.291$


Question: Simplify the power of a complex number: $(-1+i \sqrt{3})^{3}$

Answer: 8


Question: Solve the equation: $|3 x-1|=10$

Answer: $x=-3$ or $x=\frac{11}{3}$


Question: Evaluate the expression: $\log _{3}(27)$

Answer: $\log _{3}(27)=3$


Question: Find the exact value of the cosine and sine of the given angle: $\theta=0$

Answer: $\cos (0)=1, \sin (0)=0$


Question: A rock got stuck in the tread of my tire and when I was driving 70 miles per hour, the rock came loose and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling when it came out of the tread? (The tire has a diameter of 23 inches.)

Answer: 70 miles per hour


Question: Solve the quadratic equation $y^{2}-4 y=x^{2}-4$ for $y$.

Answer: $y=2 \pm x$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $y=\sqrt{4-x^{2}}$

Answer: Function


Question: Solve the equation: $\frac{5-|x|}{2}=1$

Answer: $x=-3$ or $x=3$


Question: Find the domain of the function: $f(x)=\ln (4 x-20)+\ln \left(x^{2}+9 x+18\right)$

Answer: $(5, \infty)$


Question: Convert the angle into the DMS system and round the answer to the nearest second: $179.999^{\circ}$

Answer: $179^{\circ} 59^{\prime} 56^{\prime \prime}$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}=y^{2}$

Answer: Not a function


Question: Solve the equation analytically: $\ln (3-y)-\ln (y)=2 x+\ln (5)$ for $y$.

Answer: $y=\frac{3}{5 e^{2 x}+1}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $3 \cos (2 x)+\cos (x)+2=0$

Answer: $x=\frac{7 \pi}{6}, \frac{11 \pi}{6}, \arcsin \left(\frac{1}{3}\right), \pi-\arcsin \left(\frac{1}{3}\right)$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}-y^{2}=1$

Answer: Not a function


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=2-5 \cos (x)$

Answer: $x=\frac{2 \pi}{3}, \frac{4 \pi}{3}, \arccos \left(\frac{1}{3}\right), 2 \pi-\arccos \left(\frac{1}{3}\right)$


Question: Solve the equation: $\arccos (2 x)=\pi$

Answer: $x=-\frac{1}{2}$


Question: Solve the equation analytically: $\log _{125}\left(\frac{3 x-2}{2 x+3}\right)=\frac{1}{3}$

Answer: $x=-\frac{17}{7}$


Question: Solve the equation analytically: $30-6 e^{-0.1 x}=20$

Answer: $x=-10 \ln \left(\frac{5}{3}\right)=10 \ln \left(\frac{3}{5}\right)$


Question: Write the set using interval notation: $\{x \mid x \leq 5$ or $x=6\}$

Answer: $(-\infty, 5] \cup\{6\}$


Question: At a local buffet, 22 diners (5 of whom were children) feasted for $\$162.25$, before taxes. If the kids buffet is $\$4.50$, the basic buffet is $\$7.50$, and the deluxe buffet (with crab legs) is $\$9.25$, find out how many diners chose the deluxe buffet.

Answer: This time, 7 diners chose the deluxe buffet.


Question: Solve the equation: $8 \operatorname{arccot}^{2}(x)+3 \pi^{2}=10 \pi \operatorname{arccot}(x)$

Answer: $x=-1,0$


Question: Simplify the given power of $i$: $i^8$

Answer: $i^{8}=i^{4} \cdot i^{4}=\left(i^{4}\right)^{2}=(1)^{2}=1$


Question: Write the set using interval notation: $\{x \mid x \neq 5\}$

Answer: $(-\infty, 5) \cup(5, \infty)$


Question: Solve the quadratic equation $y^{2}-3 y=4 x$ for $y$.

Answer: $y=\frac{3 \pm \sqrt{16 x+9}}{2}$


Question: Use the properties of logarithms to write the expression as a single logarithm: $2 \ln (x)-3 \ln (y)-4 \ln (z)$

Answer: $\ln \left(\frac{x^{2}}{y^{3} z^{4}}\right)$


Question: Solve the following system of linear equations: $\left\{\begin{aligned} 2 x-3 y+z & =-1 \\ 4 x-4 y+4 z & =-13 \\ 6 x-5 y+7 z & =-25\end{aligned}\right.$

Answer: $\left(-2 t-\frac{35}{4},-t-\frac{11}{2}, t\right)$ for all real numbers $t$


Question: Approximate the given value to three decimal places: $\sin \left(78.95^{\circ}\right)$

Answer: $\sin \left(78.95^{\circ}\right) \approx 0.981$


Question: Solve the equation analytically: $9 \cdot 3^{7 x}=\left(\frac{1}{9}\right)^{2 x}$

Answer: $x=-\frac{2}{11}$


Question: Solve the given inequality: $\arcsin (2 x)>0$

Answer: $\left(0, \frac{1}{2}\right]$


Question: Use your calculator to help you solve the inequality: $3^{(x-1)}<2^{x}$

Answer: $\approx(-\infty, 2.7095)$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\sin (-2 x)=\frac{\sqrt{3}}{2}$

Answer: $x=\frac{2 \pi}{3}+\pi k$ or $x=\frac{5 \pi}{6}+\pi k ; x=\frac{2 \pi}{3}, \frac{5 \pi}{6}, \frac{5 \pi}{3}, \frac{11 \pi}{6}$


Question:
Create a polynomial $f$ with real number coefficients that has the following characteristics:
- The zeros of $f$ are $c=\pm 1$ and $c=\pm i$
- The leading term of $f(x)$ is $42x^4$

Answer: $f(x)=42(x-1)(x+1)(x-i)(x+i)$


Question: Solve the equation analytically: $2^{\left(x^{3}-x\right)}=1$

Answer: $x=-1,0,1$


Question: Solve the following system of linear equations: $\left\{\begin{aligned} 2 x-4 y+z & =-7 \\ x-2 y+2 z & =-2 \\ -x+4 y-2 z & =3\end{aligned}\right.$

Answer: $\left(-3, \frac{1}{2}, 1\right)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (4 x)=\cos (2 x)$

Answer: $x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}$


Question: Solve the rational inequality and express your answer using interval notation: $\frac{1}{x+2} \geq 0$

Answer: $(-2, \infty)$


Question: Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $-225^{\circ}$

Answer: $-\frac{5 \pi}{4}$


Question: Solve the equation: $3|x-1|=2|x+1|$

Answer: $x=\frac{1}{5}$ or $x=5$


Question: If $\theta=5^{\circ}$ and the hypotenuse has length 10 , how long is the side opposite $\theta$ ?

Answer: The side opposite $\theta$ has length $10 \sin \left(5^{\circ}\right) \approx 0.872$.


Question: Expand the given logarithm and simplify: $\log _{6}\left(\frac{216}{x^{3} y}\right)^{4}$

Answer: $12-12 \log _{6}(x)-4 \log _{6}(y)$


Question: Solve the equation: $\left|x^{2}-1\right|=3$

Answer: $x=-2$ or $x=2$


Question: Solve the inequality analytically: $x \ln (x)-x>0$

Answer: $(e, \infty)$


Question: Find the domain of the function: $f(x)=\log \left(\frac{x+2}{x^{2}-1}\right)$

Answer: $(-2,-1) \cup(1, \infty)$


Question: Solve the equation analytically: $7^{3+7 x}=3^{4-2 x}$

Answer: $x=\frac{4 \ln (3)-3 \ln (7)}{7 \ln (7)+2 \ln (3)}$


Question: Write the set using interval notation: $\{x \mid x \neq-1\}$

Answer: $(-\infty,-1) \cup(-1, \infty)$


Question:
Find the work done pushing a 200 pound barrel 10 feet up a $12.5^{\circ}$ incline. Ignore all forces acting on the barrel except gravity, which acts downwards. Round your answer to two decimal places.

HINT: Since you are working to overcome gravity only, the force being applied acts directly upwards. This means that the angle between the applied force in this case and the motion of the object is not the $12.5^{\circ}$ of the incline!

Answer: (1500 pounds) $\left(300\right.$ feet) $\cos \left(0^{\circ}\right)=450,000$ foot-pounds


Question: Write out the first four terms of the given sequence: $d_{j}=(-1)^{\frac{j(j+1)}{2}}, j \geq 1$

Answer: $-1,-1,1,1$


Question: Convert the equation from polar coordinates into rectangular coordinates: $r=3 \sin (\theta)$

Answer: $r=\frac{19}{4 \cos (\theta)-\sin (\theta)}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $3 \sqrt{3} \sin (3 x)-3 \cos (3 x)=3 \sqrt{3}$

Answer: $x=\frac{\pi}{6}, \frac{5 \pi}{18}, \frac{5 \pi}{6}, \frac{17 \pi}{18}, \frac{3 \pi}{2}, \frac{29 \pi}{18}$


Question: Convert the angle from radian measure into degree measure: $\frac{7 \pi}{6}$

Answer: $210^{\circ}$


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $2 \cos (x) \geq 1$

Answer: $\left[0, \frac{\pi}{2}\right) \cup\left(\frac{11 \pi}{6}, 2 \pi\right]$


Question: A faucet can fill a sink in 5 minutes while a drain will empty the same sink in 8 minutes. If the faucet is turned on and the drain is left open, how long will it take to fill the sink?

Answer: $\frac{40}{3} \approx 13.33$ minutes


Question: A parabolic arch is constructed which is 6 feet wide at the base and 9 feet tall in the middle. Find the height of the arch exactly 1 foot in from the base of the arch.

Answer: The arch can be modeled by $x^{2}=-(y-9)$ or $y=9-x^{2}$. One foot in from the base of the arch corresponds to either $x= \pm 2$, so the height is $y=9-( \pm 2)^{2}=5$ feet.


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos ^{3}(x)=-\cos (x)$

Answer: $x=\frac{\pi}{2}, \frac{3 \pi}{2}$


Question: Expand the given logarithm and simplify: $\log _{5}\left(x^{2}-25\right)$

Answer: $\log _{5}(x-5)+\log _{5}(x+5)$


Question: The International Silver Strings Submarine Band holds a bake sale each year to fund their trip to the National Sasquatch Convention. It has been determined that the cost in dollars of baking $x$ cookies is $C(x)=0.1 x+25$ and that the demand function for their cookies is $p=10-.01 x$. How many cookies should they bake in order to maximize their profit?

Answer: $\left(1, \frac{5}{3}\right)$


Question: Evaluate the expression: $\log _{4}(8)$

Answer: $\log _{4}(8)=\frac{3}{2}$


Question: Convert the angle from radian measure into degree measure: $\frac{11 \pi}{6}$

Answer: $330^{\circ}$


Question: Find the domain of the function: $f(x)=\ln (7-x)+\ln (x-4)$

Answer: $(4,7)$


Question: Use your calculator to help you solve the equation: $2^{x}=x^{2}$

Answer: $x \approx-0.76666, x=2, x=4$


Question: Write the set using interval notation: $\{x \mid x \neq 0, \pm 4\}$

Answer: $(-\infty,-4) \cup(-4,0) \cup(0,4) \cup(4, \infty)$


Question: Find the real solutions of the polynomial equation $9 x^{2}+5 x^{3}=6 x^{4}$.

Answer: $x=0, \frac{5 \pm \sqrt{241}}{12}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cot ^{2}(x)=3 \csc (x)-3$

Answer: $x=\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{6}, \frac{11 \pi}{6}$


Question: Write out the first four terms of the given sequence: $a_{1}=3, a_{n+1}=a_{n}-1, n \geq 1$

Answer: $3,2,1,0$


Question: The Topology Taxi Company charges $\$ 2.50$ for the first fifth of a mile and $\$ 0.45$ for each additional fifth of a mile. Find a linear function which models the taxi fare $F$ as a function of the number of miles driven, $m$. Interpret the slope of the linear function and find and interpret $F(0)$.

Answer: $F(m)=2.25 m+2.05$ The slope 2.25 means it costs an additional $\$ 2.25$ for each mile beyond the first 0.2 miles. $F(0)=2.05$, so according to the model, it would cost $\$ 2.05$ for a trip of 0 miles. Would this ever really happen? Depends on the driver and the passenger, we suppose.


Question: Solve the inequality analytically: $\frac{150}{1+29 e^{-0.8 t}} \leq 130$

Answer: $\left(-\infty, \frac{\ln \left(\frac{2}{377}\right)}{-0.8}\right]=\left(-\infty, \frac{5}{4} \ln \left(\frac{377}{2}\right)\right]$


Question: Solve the given inequality: $2 \arcsin (x)^{2}>\pi \arcsin (x)$

Answer: $[-1,0)$


Question: Express the domain of the function using the extended interval notation: $f(x)=\frac{1}{\cos (x)-1}$

Answer: $\bigcup_{k=-\infty}^{\infty}(2 k \pi,(2 k+2) \pi)$


Question: Evaluate: $\left(\begin{array}{l}8 \\ 3\end{array}\right)$

Answer: 56


Question: Simplify the given power of $i$: $i^7$

Answer: $i^{7}=i^{4} \cdot i^{3}=1 \cdot(-i)=-i$


Question: Convert the equation from rectangular coordinates into polar coordinates: $x^{2}+y^{2}-2 y=0$

Answer: $(20, \pi-\arctan (3))$


Question: Rewrite the sum using summation notation: $x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}$

Answer: $\sum_{k=1}^{4}(-1)^{k-1} \frac{x^{2 k-1}}{2 k-1}$


Question: Use the properties of logarithms to write the expression as a single logarithm: $\log _{2}(x)+\log _{4}(x)$

Answer: $\log _{2}\left(x^{3 / 2}\right)$


Question: Find the inverse of the given matrix: $F=\left[\begin{array}{rrr}4 & 6 & -3 \\ 3 & 4 & -3 \\ 1 & 2 & 6\end{array}\right]$

Answer: $F^{-1}=\left[\begin{array}{rrr}-\frac{5}{2} & \frac{7}{2} & \frac{1}{2} \\ \frac{7}{4} & -\frac{9}{4} & -\frac{1}{4} \\ -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right]$


Question: Solve the rational inequality and express your answer using interval notation: $\frac{x}{x^{2}-1}>0$

Answer: $(-1,0) \cup(1, \infty)$


Question: Solve the equation for $t$: $\sin (t)=-\frac{1}{2}$

Answer: $\sin (t)=-\frac{1}{2}$ when $t=\frac{7 \pi}{6}+2 \pi k$ or $t=\frac{11 \pi}{6}+2 \pi k$ for any integer $k$.


Question: Convert the angle from radian measure into degree measure: $-\frac{2 \pi}{3}$

Answer: $-120^{\circ}$


Question: Solve the equation analytically: $3^{x}+25 \cdot 3^{-x}=10$

Answer: $x=\frac{\ln (5)}{\ln (3)}$


Question:
Determine whether or not the relation represents $y$ as a function of $x$ and find the domain and range of those relations which are functions:
$\{(x, y) \mid x$ is an odd integer, and $y$ is an even integer $\}$

Answer: Not a function


Question: Expand the given logarithm and simplify: $\log _{\frac{1}{3}}\left(9 x\left(y^{3}-8\right)\right)$

Answer: $-2+\log _{\frac{1}{3}}(x)+\log _{\frac{1}{3}}(y-2)+\log _{\frac{1}{3}}\left(y^{2}+2 y+4\right)$


Question: A certain bacteria culture follows the Law of Uninbited Growth, Equation 6.4. After 10 minutes, there are 10,000 bacteria. Five minutes later, there are 14,000 bacteria. How many bacteria were present initially? How long before there are 50,000 bacteria?

Answer: Initially, there are $\frac{250000}{49} \approx 5102$ bacteria. It will take $\frac{5 \ln (49 / 5)}{\ln (7 / 5)} \approx 33.92$ minutes for the colony to grow to 50,000 bacteria.


Question: Solve the equation: $4-|x|=2 x+1$

Answer: $x=1$


Question: The International Silver Strings Submarine Band holds a bake sale each year to fund their trip to the National Sasquatch Convention. It has been determined that the cost in dollars of baking $x$ cookies is $C(x)=0.1 x+25$ and that the demand function for their cookies is $p=10-.01 x$. How many cookies should they bake in order to maximize their profit?

Answer: 495 cookies


Question: Find the exact value of the cosine and sine of the given angle: $\theta=\frac{3 \pi}{2}$

Answer: $\cos \left(\frac{3 \pi}{2}\right)=0, \sin \left(\frac{3 \pi}{2}\right)=-1$


Question: Solve the following system of linear equations: $\left\{\begin{aligned} x-2 y+3 z & =7 \\ -3 x+y+2 z & =-5 \\ 2 x+2 y+z & =3\end{aligned}\right.$

Answer: Inconsistent


Question: Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\csc (x)>1$

Answer: $\left(-2 \pi,-\frac{3 \pi}{2}\right) \cup\left(-\frac{3 \pi}{2},-\pi\right) \cup\left(0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right)$


Question: Express the domain of the function using the extended interval notation: $f(x)=\ln (|\cos (x)|)$

Answer: $\bigcup_{k=-\infty}^{\infty}\left(\frac{(2 k-1) \pi}{2}, \frac{(2 k+1) \pi}{2}\right)$


Question: Simplify the quantity $\sqrt{(-25)(-4)}$

Answer: 10


Question: Express the repeating decimal as a fraction of integers: $10 . \overline{159}$

Answer: $\frac{3383}{333}$


Question: Convert the equation from rectangular coordinates into polar coordinates: $4 x^{2}+4\left(y-\frac{1}{2}\right)^{2}=1$

Answer: $(\sqrt{13}, \pi-\arctan (2))$


Question: Simplify the quantity $\sqrt{-9}\sqrt{-16}$

Answer: -12


Question: Expand the given logarithm and simplify: $\log \left(1.23 \times 10^{37}\right)$

Answer: $\log (1.23)+37$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan ^{3}(x)=3 \tan (x)$

Answer: $x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}$


Question: The temperature $T$, in degrees Fahrenheit, $t$ hours after $6 \mathrm{AM}$ is given by $T(t)=-\frac{1}{2} t^{2}+8 t+32, \quad 0 \leq t \leq 12$. What is the warmest temperature of the day? When does this happen?

Answer: $(-\infty,-3] \cup[1, \infty)$


Question: Find the real solutions of the polynomial equation $2 x^{3}=19 x^{2}-49 x+20$.

Answer: $x=\frac{1}{2}, 4,5$


Question: Expand the given logarithm and simplify: $\log _{5}\left(\frac{z}{25}\right)^{3}$

Answer: $3 \log _{5}(z)-6$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $2 \sec ^{2}(x)=3-\tan (x)$

Answer: $x=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{\pi}{2}$


Question: Solve the system of nonlinear equations after making the appropriate substitutions: $\left\{\begin{array}{l}4 \ln (x)+3 y^{2}=1 \\ 3 \ln (x)+2 y^{2}=-1\end{array}\right.$

Answer: $\left(e^{-5}, \pm \sqrt{7}\right)$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan ^{2}(x)=\frac{3}{2} \sec (x)$

Answer: $x=\frac{\pi}{2}, \frac{3 \pi}{2}$


Question: Simplify the expression: $\frac{10 !}{7 !}$

Answer: 720


Question: Jeff can walk comfortably at 3 miles per hour. Find a linear function $d$ that represents the total distance Jeff can walk in $t$ hours, assuming he doesn't take any breaks.

Answer: $d(t)=3 t, t \geq 0$.


Question: Solve the equation or inequality: $x+1=\sqrt{3 x+7}$

Answer: $x=3$


Question: Solve the equation analytically: $\log _{5}(2 x+1)+\log _{5}(x+2)=1$

Answer: $x=\frac{1}{2}$


Question: Use your calculator to help you solve the inequality: $e^{x}<x^{3}-x$

Answer: $\approx(2.3217,4.3717)$


Question: Express the domain of the function using the extended interval notation: $f(x)=\frac{\sin (x)}{2+\cos (x)}$

Answer: $(-\infty, \infty)$


Question: Solve the equation: $|x|=12-x^{2}$

Answer: $x=-3$ or $x=3$


Question:
The period of a pendulum in seconds is given by
$$
T=2 \pi \sqrt{\frac{L}{g}}
$$
(for small displacements) where $L$ is the length of the pendulum in meters and $g=9.8$ meters per second per second is the acceleration due to gravity. My Seth-Thomas antique schoolhouse clock needs $T=\frac{1}{2}$ second and I can adjust the length of the pendulum via a small dial on the bottom of the bob. At what length should I set the pendulum?

Answer: $9.8\left(\frac{1}{4 \pi}\right)^{2} \approx 0.062$ meters or 6.2 centimeters


Question: A finance company offers a promotion on $\$ 5000$ loans. The borrower does not have to make any payments for the first three years, however interest will continue to be charged to the loan at $29.9 \%$ compounded continuously. What amount will be due at the end of the three-year period, assuming no payments are made? If the promotion is extended an additional three years, and no payments are made, what amount would be due?

Answer: $A(3)=5000 e^{0.299 \cdot 3} \approx \$ 12,226.18, A(6)=5000 e^{0.299 \cdot 6} \approx \$ 30,067.29$


Question: Find the standard equation of the circle which satisfies the given criteria: center $(3,5)$, passes through $(-1,-2)$

Answer: $(x-3)^{2}+(y-5)^{2}=65$


Question: Write out the first four terms of the given sequence: $b_{1}=2, b_{k+1}=3 b_{k}+1, k \geq 1$

Answer: $2,7,22,67$


Question: Use your calculator to help you solve the equation: $e^{x}=\ln (x)+5$

Answer: $x \approx 0.01866, x \approx 1.7115$


Question: Find the real solutions of the polynomial equation $14 x^{2}+5=3 x^{4}$.

Answer: $\{-2\} \cup[1,3]$


Question: Solve the inequality analytically: $\ln \left(x^{2}\right) \leq(\ln (x))^{2}$

Answer: $(0,1] \cup\left[e^{2}, \infty\right)$


Question: Solve the equation analytically: $\log _{3}(x-4)+\log _{3}(x+4)=2$

Answer: $x=5$


Question: Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cos \left(x+\frac{5 \pi}{6}\right)=0$

Answer: $x=-\frac{\pi}{3}+\pi k ; x=\frac{2 \pi}{3}, \frac{5 \pi}{3}$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $y=x^{2}+4$

Answer: Function


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan (2 x)-2 \cos (x)=0$

Answer: $x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \frac{3 \pi}{2}$


Question: Express the domain of the function using the extended interval notation: $f(x)=\arcsin (\tan (x))$

Answer: $\bigcup_{k=-\infty}^{\infty}\left[\frac{(4 k-1) \pi}{4}, \frac{(4 k+1) \pi}{4}\right]$


Question: Solve the equation: $4 \operatorname{arcsec}\left(\frac{x}{2}\right)=\pi$

Answer: $x=2 \sqrt{2}$


Question: Solve the equation analytically: $8^{x}=\frac{1}{128}$

Answer: $x=-\frac{7}{3}$


Question: Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(3,-10),(-1,2)$

Answer: $d=4 \sqrt{10}, M=(1,-4)$


Question: Solve the equation analytically: $(1.005)^{12 x}=3$

Answer: $x=\frac{\ln (3)}{12 \ln (1.005)}$


Question: Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (x)=\cos (x)$

Answer: $x=\frac{\pi}{4}, \frac{5 \pi}{4}$


Question: Determine whether or not the equation represents $y$ as a function of $x$: $x^{3} y=-4$

Answer: Function


Question: Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{3 n^{2}+4}{2 n^{2}+3 n+5} $ converges absolutely, converges conditionally, or diverges.

Answer: diverges

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Question: Find the antiderivative: $\int \sin ^{2} x d x $

Answer: $x / 2-\sin (2 x) / 4+C$

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Question: An object moves along a straight line with acceleration given by $a(t)=-\cos (t)$, and $s(0)=1$ and $v(0)=0$. Find the maximum distance the object travels from zero, and find its maximum speed. Describe the motion of the object.

Answer: $s(t)=\cos t, v(t)=-\sin t$, maximum distance is 1 , maximum speed is 1

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Question: Find all local maximum and minimum points of the function: $y=(x+1) / \sqrt{5 x^{2}+35} $

Answer: $\max$ at $x=7$

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Question: Find the domain of the function: $y=f(x)=\sqrt[3]{x} $

Answer: $\{x \mid x \in \mathbb{R}\}$, i.e., all $x$

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Question: Compute the limit of $\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin x} $.

Answer: 0

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Question: Find the antiderivative: $\int x \sin x \cos x d x $

Answer: $x / 4-\left(x \cos ^{2} x\right) / 2+(\cos x \sin x) / 4+$ C

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Question: Find the derivative of the function: $(1+3 x)^{2} $

Answer: $6+18 x$

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Question: Find the derivative of $\operatorname{arccot} x$, the inverse cotangent.

Answer: $-1 /\left(1+x^{2}\right)$

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Question: Find the derivative of the function: $5 x^{3}+12 x^{2}-15 $

Answer: $15 x^{2}+24 x$

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Question: A thin plate lies in the region contained by $y=4-x^{2}$ and the $x$-axis. Find the centroid.

Answer: $\bar{x}=0, \bar{y}=8 / 5$

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Question: Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{1}{\sqrt{n^{2}+4}} $

Answer: diverges

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Question: A hemispheric bowl of radius $r$ contains water to a depth $h$. Find the volume of water in the bowl.

Answer: $\pi h^{2}(3 r-h) / 3$

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Question: Describe all functions with derivative $x^{2}+47 x-5$.

Answer: $x^{3} / 3+47 x^{2} / 2-5 x+k$

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Question: Find the derivative of the function: $x^{3}-2 x^{2}+4 \sqrt{x} $

Answer: $3 x^{2}-4 x+2 / \sqrt{x}$

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Question: Determine whether the series $\sum_{n=2}^{\infty}(-1)^{n} \frac{1}{\ln n} $ converges absolutely, converges conditionally, or diverges.

Answer: converges conditionally

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Question: Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n !}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} $

Answer: converges

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Question: Determine whether the series converges: $1-\frac{3}{4}+\frac{5}{8}-\frac{7}{12}+\frac{9}{16}+\cdots $

Answer: diverges

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Question: Evaluate the integral: $\int t\left(t^{2}-9\right)^{3 / 2} d t $

Answer: $\frac{\left(t^{2}-9\right)^{5 / 2}}{5}+C$

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Question: Determine whether the series $\sum_{n=1}^{\infty} \frac{n !}{n^{n}} $ converges.

Answer: converges

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Question: Find the antiderivative: $\int x \sqrt{1-x^{2}} d x $

Answer: $-\left(1-x^{2}\right)^{3 / 2} / 3+C$

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Question: Find the domain of the function: $y=f(x)=\sqrt[4]{x} $

Answer: $\{x \mid x \geq 0\}$

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Question: Find the volume of the solid obtained by revolving the region bounded by $y=\sqrt{\sin x}$, the $y$-axis, and the lines $y=1$ and $x=\pi / 2$ around the $x$-axis.

Answer: $\pi(\pi / 2-1)$

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Question: Find the derivative of the function: $3^{x^{2}} $

Answer: $2 \ln (3) x 3^{x^{2}}$

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Question: Find the derivative of the function: $G(x)=\int_{1}^{x^{2}} t^{2}-3 t d t $

Answer: $2 x\left(x^{4}-3 x^{2}\right)$

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Question: Find the derivative of the function: $x^{3 / 4} $

Answer: $(3 / 4) x^{-1 / 4}$

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Question: A box with square base and no top is to hold a volume $V$. Find (in terms of $V$ ) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve $V$.)

Answer: $w=l=2^{1 / 3} V^{1 / 3}, h=V^{1 / 3} / 2^{2 / 3}$, $h / w=1 / 2$

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Question: Does the improper integral $\int_{-\infty}^{\infty} \frac{x^{2}}{4+x^{6}} d x$ converge or diverge? If it converges, find the value.

Answer: $\pi / 6$

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Question: A water tank has the shape of the bottom half of a sphere with radius $r=1$ meter. If the tank is full, how much work is required to pump all the water out the top of the tank?

Answer: $2450 \pi \mathrm{N}-\mathrm{m}$

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Question: Find the derivative of the function: $\sqrt[3]{x+x^{3}} $

Answer: $\frac{1+3 x^{2}}{3\left(x+x^{3}\right)^{2 / 3}}$

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Question: Find the arc length of $f(x)=\ln (\sin x)$ on the interval $[\pi / 4, \pi / 3] . $

Answer: $\ln ((\sqrt{2}+1) / \sqrt{3})$

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Question: Find the derivative of the function: $e^{4 x} / x $

Answer: $e^{4 x}(4 x-1) / x^{2}$

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Question: Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{2 n^{2}-3 n-5} $ converges or diverges.

Answer: converges

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Question: Find a series representation for the function: $\sum_{n=0}^{\infty} \frac{(x-1)^{n}}{n !} $

Answer: $(-\infty, \infty)$

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Question: Find the derivative of the function: $\left(3 x^{2}+1\right)(2 x-4)^{3} $

Answer: $120 x^{4}-576 x^{3}+888 x^{2}-480 x+96$

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Question: A beam 10 meters long has density $\sigma(x)=x^{2}$ at distance $x$ from the left end of the beam. Find the center of mass $\bar{x}$.

Answer: $15 / 2$

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Question: Find the area bounded by the curves: $y=\sin (\pi x / 3)$ and $y=x$ (in the first quadrant)

Answer: $3 / \pi-3 \sqrt{3} /(2 \pi)-1 / 8$

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Question: Find the antiderivative: $\int \csc x d x $

Answer: $-\ln |\csc x+\cot x|+C$

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Question: For all $x \geq 0,4 x-9 \leq f(x) \leq x^{2}-4 x+7$. Find $\lim _{x \rightarrow 4} f(x)$.

Answer: 7

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Question: Let $f(x)=\sqrt{x}$. If $a=1$ and $d x=\Delta x=1 / 10$, what are $\Delta y$ and $d y$ ?

Answer: $\Delta y=\sqrt{11 / 10}-1, d y=0.05$

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Question: Determine whether the series converges or diverges: $\sum_{n=2}^{\infty} \frac{1}{n \ln n} $

Answer: diverges

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Question: A cylindrical tank standing upright (with one circular base on the ground) has radius 20 $\mathrm{cm}$. Find the rate at which the water level in the tank drops when the water is being drained at 25 $\mathrm{cm}^{3} / \mathrm{sec}$.

Answer: $1 /(16 \pi) \mathrm{cm} / \mathrm{s}$

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Question: Find the derivative of the function: $f(x)+g(x)$, where $f(x)=x^{2}-3 x+2$ and $g(x)=2 x^{3}-5 x $

Answer: $6 x^{2}+2 x-8$

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Question: Determine whether the area under the curve $y=1 / x$ from 1 to infinity is finite or infinite. If it is finite, compute the area.

Answer: $\infty$

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Question: Find the antiderivative of the function: $|2 t-4| $

Answer: $4 t-t^{2}+C, t<2 ; t^{2}-4 t+8+C$, $t \geq 2$

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Question: Find the derivative of the function: $\frac{1}{\sin x} $

Answer: $-\frac{\cos x}{\sin ^{2} x}$

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Question: Find the antiderivative: $\int x \arctan x d x $

Answer: $\left(x^{2} \arctan x+\arctan x-x\right) / 2+C$

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Question: Find the antiderivative: $\int \arctan (\sqrt{x}) d x $

Answer: $x \arctan (\sqrt{x})+\arctan (\sqrt{x})-\sqrt{x}+C$

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Question: Find a power series representation for $2 /(1-x)^{3}$.

Answer: $\sum_{n=0}^{\infty}(n+1)(n+2) x^{n}$

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Question: Determine whether the series converges: $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{(2 n) !} $

Answer: converges

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Question: Find all critical points of the function $f(x)=x^{3} /(x+1) $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\min$ at $x=-3 / 2$, neither at $x=0$

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Question: Find all local maximum and minimum points of the function: $y=2+3 x-x^{3} $

Answer: $\min$ at $x=-1, \max$ at $x=1$

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Question: Find all of the solutions of $2 \sin (t)-1-\sin ^{2}(t)=0$ in the interval $[0,2 \pi]$.

Answer: $t=\pi / 2$

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Question: Determine whether the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{n}}{2^{n}+3^{n}} $ converges absolutely, converges conditionally, or diverges.

Answer: diverges

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Question: Find the antiderivative: $\int \frac{1}{\sqrt{1+x^{2}}} d x $

Answer: $\ln \left|x+\sqrt{1+x^{2}}\right|+C$

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Question: Find the derivative of the function: $\left(x^{2}+1\right)(5-2 x) / 2 $

Answer: $-3 x^{2}+5 x-1$

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Question: Evaluate the integral: $\int \sin ^{3} t \cos ^{4} t d t $

Answer: $\frac{\cos ^{7} t}{7}-\frac{\cos ^{5} t}{5}+C$

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Question: Find the derivative of the function: $\csc x $

Answer: $-\csc x \cot x$

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Question: Marketing tells you that if you set the price of an item at $\$ 10$ then you will be unable to sell it, but that you can sell 500 items for each dollar below $\$ 10$ that you set the price. Suppose your fixed costs total $\$ 3000$, and your marginal cost is $\$ 2$ per item. What is the most profit you can make?

Answer: $\$ 5000$

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Question: Determine whether the sequence $\left\{\frac{2^{n}}{n !}\right\}_{n=0}^{\infty}$ converges or diverges.

Answer: 0

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Question: Explain why the series $\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}$ diverges.

Answer: $\lim _{n \rightarrow \infty} n^{2} /\left(2 n^{2}+1\right)=1 / 2$

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Question: What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere?

Answer: $1 / \sqrt{3} \approx 58 \%$

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Question: A water tank has the shape of a cylinder with radius $r=1$ meter and height 10 meters. If the depth of the water is 5 meters, how much work is required to pump all the water out the top of the tank?

Answer: $367,500 \pi \mathrm{N}-\mathrm{m}$

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Question: Find the area bounded by the curves: $y=\sin x \cos x$ and $y=\sin x, 0 \leq x \leq \pi $

Answer: 2

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Question: Determine whether the series $\sum_{n=2}^{\infty} \frac{1}{\ln n} $ converges or diverges.

Answer: diverges

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Question: An object moves along a straight line with acceleration given by $a(t)=1+\sin (\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t)$ and $v(t)$.

Answer: $s(t)=t^{2} / 2-\sin (\pi t) / \pi^{2}+t / \pi$, $v(t)=t-\cos (\pi t) / \pi+1 / \pi$

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Question: A thin plate lies in the region between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1$ in the first quadrant. Find the centroid.

Answer: $\bar{x}=\bar{y}=28 /(9 \pi)$

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Question: Find the derivative of the function: $\sqrt{1-\sin ^{2} x} $

Answer: $\frac{-\sin x \cos x}{\sqrt{1-\sin ^{2} x}}$

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Question: Evaluate the integral: $\int \frac{1}{t^{2}-6 t+9} d t $

Answer: $\frac{-1}{t-3}+C$

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Question: Find the derivative of the function: $100 /\left(100-x^{2}\right)^{3 / 2} $

Answer: $\frac{300 x}{\left(100-x^{2}\right)^{5 / 2}}$

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Question: Find the antiderivative: $\int \frac{x^{4}}{4+x^{2}} d x $

Answer: $-4 x+x^{3} / 3+8 \arctan (x / 2)+C$

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Question: Find the domain of the function: $y=f(x)=1 /(\sqrt{x}-1) $

Answer: $\{x \mid x \geq 0$ and $x \neq 1\}$

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Question: Describe the concavity of the function: $y=(x+1) / \sqrt{5 x^{2}+35} $

Answer: concave up on $(-\infty,(21-\sqrt{497}) / 4)$ and $(21+\sqrt{497}) / 4, \infty)$

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Question: Compute the limit: $\lim _{x \rightarrow-4} \frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why.

Answer: 0

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Question: Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{6^{n}}{n !} $

Answer: converges

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Question: Describe all functions with derivative $x^{3}-\frac{1}{x}$.

Answer: $x^{4} / 4-\ln x+k$

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Question: Find an equation for the tangent line to $x^{4}=y^{2}+x^{2}$ at $(2, \sqrt{12})$. (This curve is the kampyle of Eudoxus.)

Answer: $y=7 x / \sqrt{3}-8 / \sqrt{3}$

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Question: Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=t^{2}+2$, and that at $t=0$ the object is at position 5 . Find the position of the object at $t=2 . $

Answer: $35 / 3$

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Question: Find the radius and interval of convergence for the series: $\sum_{n=0}^{\infty} \frac{x^{n}}{n !} $

Answer: $R=\infty, I=(-\infty, \infty)$

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Question: For a cylinder with given surface area $S$, including the top and the bottom, find the ratio of height to base radius that maximizes the volume.

Answer: $h / r=2$

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Question: Determine whether the area under the curve $y=1 / x^{3}$ from 1 to infinity is finite or infinite. If it is finite, compute the area.

Answer: $1 / 2$

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Question: Find the derivative of the function: $\sin ^{2}(\sqrt{x}) $

Answer: $\sin (\sqrt{x}) \cos (\sqrt{x}) / \sqrt{x}$

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Question: Express the improper integral $\int_{0}^{1 / 2}(2 x-1)^{-3} d x$ as a limit and determine whether it converges or diverges. If it converges, find the value.

Answer: diverges

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Question: Find an $N$ such that $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}}=\sum_{n=2}^{N} \frac{1}{n(\ln n)^{2}} \pm 0.005$.

Answer: any integer greater than $e^{200}$

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Question: A police helicopter is flying at a speed of $150 \mathrm{~mph}$ at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and that this distance is decreasing at a rate of $190 \mathrm{~mph}$. Find the speed of the car.

Answer: $380 / \sqrt{3}-150 \approx 69.4 \mathrm{mph}$

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Question: Find the domain of the function: $y=f(x)=1 /(x+1) $

Answer: $\{x \mid x \neq-1\}$

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Question: Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{1}{n^{2}+1} $

Answer: converges

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Question: Compute the limit of $\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{3}} $.

Answer: $\infty$

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Question: Determine whether the series $\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}+5^{n}} $ converges or diverges.

Answer: converges

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Question: A light shines from the top of a pole $20 \mathrm{~m}$ high. An object is dropped from the same height from a point $10 \mathrm{~m}$ away, so that its height at time $t$ seconds is $h(t)=20-9.8 t^{2} / 2$. Find the rate at which the object's shadow is moving on the ground one second later.

Answer: $4000 / 49 \mathrm{~m} / \mathrm{s}$

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Question: Find the derivative of the function: $\frac{x^{2}+x}{\sin x} $

Answer: $\frac{(2 x+1) \sin x-\left(x^{2}+x\right) \cos x}{\sin ^{2} x}$

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Question: Find the antiderivative of the function: $2 / z^{2} $

Answer: $-2 / z+C$

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Question: Find the antiderivative of the function: $\int \frac{\sin x}{\cos ^{3} x} d x $

Answer: $1 /\left(2 \cos ^{2} x\right)=(1 / 2) \sec ^{2} x+C$

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Question: Find the derivative of the function: $5(x+1-1 / x) $

Answer: $5+5 / x^{2}$

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Question: Determine whether the series $\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}+3^{n}} $ converges or diverges.

Answer: diverges

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Question: Evaluate the definite integral: $\int_{3}^{4} \frac{1}{(3 x-7)^{2}} d x $

Answer: $1 / 10$

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Question: Find the antiderivative: $\int x e^{x^{2}} d x $

Answer: $(1 / 2) e^{x^{2}}+C$

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Question: Evaluate the integral: $\int \frac{1}{t^{2}+3 t} d t $

Answer: $\frac{\ln |t|}{3}-\frac{\ln |t+3|}{3}+C$

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Question: Find the derivative of the function: $\left(6-2 x^{2}\right)^{3} $

Answer: $-12 x\left(6-2 x^{2}\right)^{2}$

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Question: Find all local maximum and minimum points of the function: $y=x^{5}-x $

Answer: $\max$ at $-5^{-1 / 4}$, min at $5^{-1 / 4}$

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Question: Find the area bounded by the curves: $y=\sqrt{x}$ and $y=\sqrt{x+1}, 0 \leq x \leq 4 $

Answer: $10 \sqrt{5} / 3-6$

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Question: Find the antiderivative: $\int \sin ^{4} x d x $

Answer: $3 x / 8-(\sin 2 x) / 4+(\sin 4 x) / 32+C$

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Question: Find all critical points of the function $y=x^{3}-9 x^{2}+24 x $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\max$ at $x=2$, min at $x=4$

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Question: Compute the limit of $\lim _{x \rightarrow \infty} \frac{\ln x}{x} $.

Answer: 0

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Question: Find an equation for the tangent line to $\sin ^{2}(x)$ at $x=\pi / 3$.

Answer: $\sqrt{3} x / 2+3 / 4-\sqrt{3} \pi / 6$

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Question: Find the derivative of the function: $G(x)=\int_{1}^{x^{2}} e^{t^{2}} d t $

Answer: $2 x e^{x^{4}}$

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Question: Find the antiderivative of the function: $\int \frac{\sin (\tan x)}{\cos ^{2} x} d x $

Answer: $-\cos (\tan x)+C$

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Question: Find the radius and interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{n^{n}}(x-2)^{n} $

Answer: $R=0$, converges only when $x=2$

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Question: Find the derivative of the function: $(x+1)\left(x^{2}+2 x-3\right) $

Answer: $3 x^{2}+6 x-1$

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Question: The function $f(x)=\frac{x}{\sqrt{x^{2}+1}}$ has two horizontal asymptotes. Find them and give a rough sketch of $f$ with its horizontal asymptotes.

Answer: $y=1$ and $y=-1$

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Question: Find the domain of the function: $y=f(x)=1 / \sqrt{1-(3 x)^{2}} $

Answer: $\{x \mid-1 / 3<x<1 / 3\}$

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Question: Find the domain of the function: $y=f(x)=\sqrt{-1 / x} $

Answer: $\{x \mid x<0\}$

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Question: Find the derivative of the function: $e^{\sin x} $

Answer: $\cos (x) e^{\sin x}$

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Question: How much work is done in lifting a 100 kilogram weight from the surface of the earth to an orbit 35,786 kilometers above the surface of the earth?

Answer: $\approx 5,305,028,517 \mathrm{~N}-\mathrm{m}$

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Question: Find the derivative of the function: $x^{-9 / 7} $

Answer: $-(9 / 7) x^{-16 / 7}$

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Question: A cylindrical tank standing upright (with one circular base on the ground) has radius 1 meter. Find the rate at which the water level in the tank drops when the water is being drained at 3 liters per second.

Answer: $3 /(1000 \pi)$ meters $/$ second

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Question: Find an equation for the tangent line to $f(x)=\left(x^{2}-4\right) /(5-x)$ at $x=3$.

Answer: $y=17 x / 4-41 / 4$

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Question: Describe the concavity of the function: $y=\sin x+\cos x $

Answer: concave down on $((8 n-1) \pi / 4,(8 n+$ $3) \pi / 4)$, concave up on $((8 n+$ $3) \pi / 4,(8 n+7) \pi / 4)$, for integer $n$

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Question: Find the antiderivative of the function: $\int \frac{1}{\sqrt[3]{1-5 t}} d t $

Answer: $-3(1-5 t)^{2 / 3} / 10+C$

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Question: Find the antiderivative: $\int \frac{x^{3}}{4+x^{2}} d x $

Answer: $x^{2} / 2-2 \ln \left(4+x^{2}\right)+C$

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Question: Determine whether the volume of the solid obtained by rotating the curve $y=1 / x$ around the x-axis, from $x=1$ to infinity, is finite or infinite. If it is finite, compute the volume.

Answer: $\pi$

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Question: A baseball diamond is a square $90 \mathrm{ft}$ on a side. A player runs from first base to second base at a speed of $15 \mathrm{ft} / \mathrm{sec}$. At what rate is the player's distance from third base decreasing when she is halfway from first to second base?

Answer: $3 \sqrt{5} \mathrm{ft} / \mathrm{s}$

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Question: Find the antiderivative: $\int \sin (\sqrt{x}) d x $

Answer: $2 \sin (\sqrt{x})-2 \sqrt{x} \cos (\sqrt{x})+C$

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Question: Find the derivative of the function: $y=f(t)=80-4.9 t^{2}$.

Answer: $-9.8 t$

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Question: Compute the value of the integral: $\int_{1}^{10} \frac{1}{x} d x $

Answer: $\ln (10)$

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Question: Find the antiderivative: $\int \sec ^{2} x \csc ^{2} x d x $

Answer: $\sec x \csc x-2 \cot x+C$

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Question: Find the derivative of the function: $y=f(x)=\sqrt{169-x^{2}}$.

Answer: $-x / \sqrt{169-x^{2}}$

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Question: Find the value of $a$ so that the tangent line to $y=\ln (x)$ at $x=a$ is a line through the origin. Sketch the resulting situation.

Answer: $e$

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Question: Find all local maximum and minimum points of the function: $y=x^{2}-x $

Answer: $\min$ at $x=1 / 2$

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Question: Let $f(x)=x^{2}+3 x+2$. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles.

Answer: 4 rectangles: $41 / 4=10.25$, 8 rectangles: $183 / 16=11.4375$

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Question: Find the derivative of the function: $\frac{1}{1+1 / x} $

Answer: $1 /(x+1)^{2}$

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Question: Describe the concavity of the function: $y=6 x+\sin 3 x $

Answer: concave down on $(2 n \pi / 3,(2 n+$ 1) $\pi / 3)$

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Question: Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $1 / x^{2}, a=1 $

Answer: $\sum_{n=0}^{\infty}(-1)^{n}(n+1)(x-1)^{n}, R=1$

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Question: Find the antiderivative: $\int \csc ^{3} x d x $

Answer: $-\csc x \cot x / 2-(1 / 2) \ln \mid \csc x+$ $\cot x \mid+C$

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Question: If $f^{\prime}(4)=5, g^{\prime}(4)=12,(f g)(4)=f(4) g(4)=2$, and $g(4)=6$, compute $f(4)$ and $\frac{d}{d x} \frac{f}{g}$ at 4 .

Answer: $13 / 18$

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Question: Find an equation for the tangent line to $y=9 x^{-2}$ at $(3,1)$.

Answer: $y=3-2 x / 3$

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Question: Determine whether the series $\sum_{n=4}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n-3}} $ converges or diverges.

Answer: converges

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Question: Describe the concavity of the function: $y=(x+5)^{1 / 4} $

Answer: concave down everywhere

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Question: Evaluate the integral: $\int \frac{t^{3}}{\left(2-t^{2}\right)^{5 / 2}} d t $

Answer: $\frac{2}{3\left(2-t^{2}\right)^{3 / 2}}-\frac{1}{\left(2-t^{2}\right)^{1 / 2}}+C$

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Question: Find an $N$ such that $\sum_{n=0}^{\infty} \frac{1}{e^{n}}=\sum_{n=0}^{N} \frac{1}{e^{n}} \pm 10^{-4}$.

Answer: $N=10$

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Question: Determine whether the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{n}}{2^{n}+5^{n}} $ converges absolutely, converges conditionally, or diverges.

Answer: converges absolutely

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Question: Compute the value of the series $\sum_{n=0}^{\infty} \frac{3}{2^{n}}+\frac{4}{5^{n}}$.

Answer: 11

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Question: Find the derivative of the function: $\frac{x^{2}-1}{x^{2}+1} $

Answer: $4 x /\left(x^{2}+1\right)^{2}$

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Question: Does the improper integral $\int_{-\infty}^{\infty} x d x$ converge or diverge? If it converges, find the value. Also, find the Cauchy Principal Value, if it exists.

Answer: diverges, 0

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Question: Find all local maximum and minimum points of the function: $y=x^{3}-9 x^{2}+24 x $

Answer: $\max$ at $x=2$, min at $x=4$

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Question: Find the derivative of the function: $\sqrt{\frac{169}{x}-x} $

Answer: $\frac{1}{2}\left(\frac{-169}{x^{2}}-1\right) / \sqrt{\frac{169}{x}-x}$

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Question: Compute the limit: $\lim _{x \rightarrow 4} 3 x^{3}-5 x $.

Answer: 172

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Question: Find an equation for the tangent line to $\sec ^{2} x$ at $x=\pi / 3$.

Answer: $8 \sqrt{3} x+4-8 \sqrt{3} \pi / 3$

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Question: An object moves along a straight line with acceleration given by $a(t)=1-\sin (\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t)$ and $v(t)$.

Answer: $s(t)=t^{2} / 2+\sin (\pi t) / \pi^{2}-t / \pi$, $v(t)=t+\cos (\pi t) / \pi-1 / \pi$

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Question: Find the derivative of the function: $x^{4}-3 x^{3}+(1 / 2) x^{2}+7 x-\pi $

Answer: $4 x^{3}-9 x^{2}+x+7$

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Question: Find the derivative of the function: $x^{3} e^{x} $

Answer: $3 x^{2} e^{x}+x^{3} e^{x}$

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Question: Find the derivative of the function: $(x+8)^{5} $

Answer: $5(x+8)^{4}$

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Question: Find the derivative of the function: $\left(x^{2}+1\right)^{3} $

Answer: $6\left(x^{2}+1\right)^{2} x$

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Question: Find the standard equation of the circle passing through $(-2,1)$ and tangent to the line $3 x-2 y=6$ at the point $(4,3)$. Sketch. (Hint: The line through the center of the circle and the point of tangency is perpendicular to the tangent line.)

Answer: $(x+2 / 7)^{2}+(y-41 / 7)^{2}=1300 / 49$

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Question: Determine whether the sequence $\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}$ converges or diverges. If it converges, compute the limit.

Answer: 0

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Question: Find the derivative of the function: $\sqrt{x^{3}-x^{2}-(1 / x)} $

Answer: $\frac{3 x^{2}-2 x+1 / x^{2}}{2 \sqrt{x^{3}-x^{2}-(1 / x)}}$

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Question: Describe the concavity of the function: $y=\left(x^{2}-1\right) / x $

Answer: concave up when $x<0$, concave down when $x>0$

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Question: Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{3 n-2} $ converges or diverges.

Answer: diverges

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Question: Find the antiderivative: $\int x^{2} \cos x d x $

Answer: $x^{2} \sin x-2 \sin x+2 x \cos x+C$

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Question: Describe all functions with derivative $\sin (2 x)$.

Answer: $-\cos (2 x) / 2+k$

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Question: If you fit the cone with the largest possible surface area (lateral area plus area of base) into a sphere, what percent of the volume of the sphere is occupied by the cone?

Answer: The ratio of the volume of the sphere to the volume of the cone is $1033 / 4096+33 / 4096 \sqrt{17} \approx 0.2854$, so the cone occupies approximately $28.54 \%$ of the sphere.

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Question: A thin plate lies in the region contained by $\sqrt{x}+\sqrt{y}=1$ and the axes in the first quadrant. Find the centroid.

Answer: $\bar{x}=\bar{y}=1 / 5$

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Question: Use Newton's Method to approximate the cube root of 10 to two decimal places.

Answer: 2.15

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Question: An object moves with velocity $v(t)=-t^{2}+1$ feet per second between $t=0$ and $t=2$. Find the average velocity and the average speed of the object between $t=0$ and $t=2 . $

Answer: $-1 / 3,1$

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Question: Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{n}{e^{n}} $

Answer: converges

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Question: Use an angle sum identity to compute $\cos (\pi / 12)$.

Answer: $(\sqrt{2}+\sqrt{6}) / 2$

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