Question: Find the derivative of the function: $y=f(x)=x^{2}-(1 / x)$.

Answer: $2 x+1 / x^{2}$

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Question: Find the derivative of the function: $-4 x^{5}+3 x^{2}-5 / x^{2} $

Answer: $-20 x^{4}+6 x+10 / x^{3}$

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Question: For the velocity function $v=\sin (\pi t / 3)-t$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 1$.

Answer: $(3-\pi) /(2 \pi),(18-12 \sqrt{3}+\pi) /(4 \pi)$

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Question: Find the derivative of the function: $(4-x)^{3} $

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

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Question: Evaluate the integral: $\int\left(t^{3 / 2}+47\right)^{3} \sqrt{t} d t $

Answer: $\frac{\left(t^{3 / 2}+47\right)^{4}}{6}+C$

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Question: Evaluate the integral: $\int \frac{1}{t\left(t^{2}-4\right)} d t $

Answer: $\frac{1}{8} \ln \left|1-4 / t^{2}\right|+C$

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Question: Find the derivative of the function: $\left(x^{2}-4 x+5\right) \sqrt{25-x^{2}} $

Answer: $(2 x-4) \sqrt{25-x^{2}}-$ $\left(x^{2}-4 x+5\right) x / \sqrt{25-x^{2}}$

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Question: Describe the concavity of the function: $y=\sin ^{3} x $

Answer: inflection points at $n \pi$, $\pm \arcsin (\sqrt{2 / 3})+n \pi$

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Question: Compute the value of the integral: $\int_{1}^{2} x^{5} d x $

Answer: $2^{6} / 6-1 / 6$

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Question: Find the derivative of $\arctan \left(e^{x}\right)$.

Answer: $\frac{e^{x}}{1+e^{2 x}}$

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Question: Approximate the fifth root of 7 , using $x_{0}=1.5$ as a first guess. Use Newton's method to find $x_{3}$ as your approximation.

Answer: $x_{3}=1.475773162$

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Question: Find the derivative of the function: $5\left(-3 x^{2}+5 x+1\right) $

Answer: $-30 x+25$

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Question: Find the antiderivative: $\int \frac{1}{x^{2}+3 x} d x $

Answer: $(1 / 3) \ln |x|-(1 / 3) \ln |x+3|+C$

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

Answer: $R=1, I=(-1,1)$

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Question: Find the antiderivative of the function: $7 s^{-1} $

Answer: $7 \ln s+C$

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Question: The observation deck on the 102nd floor of the Empire State Building is 1,224 feet above the ground. If a steel ball is dropped from the observation deck its velocity at time $t$ is approximately $v(t)=-32 t$ feet per second. Find the average speed between the time it is dropped and the time it hits the ground, and find its speed when it hits the ground.

Answer: $-4 \sqrt{1224} \mathrm{ft} / \mathrm{s} ;-8 \sqrt{1224} \mathrm{ft} / \mathrm{s}$

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Question: Find the derivative of the function: $\frac{2 x^{-1}-x^{-2}}{3 x^{-1}-4 x^{-2}} $

Answer: $-5 /(3 x-4)^{2}$

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Question: Let $f(x)=x^{2}-2 x+3$. Approximate the area under the curve between $x=1$ and $x=3$ using 4 rectangles.

Answer: $23 / 4$

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Question: Compute the limit: $\lim _{x \rightarrow \infty} x^{1 / x} . $

Answer: 1

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Question: Find the average height of $\sqrt{1-x^{2}}$ over the interval $[-1,1] . $

Answer: $\pi / 4$

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Question: Determine whether the sequence $\left\{\frac{n+47}{\sqrt{n^{2}+3 n}}\right\}_{n=1}^{\infty}$ converges or diverges. If it converges, compute the limit.

Answer: 1

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Question: Compute the limit: $\lim _{x \rightarrow 1}\left\{\begin{array}{ll}x-5 & x \neq 1, \\ 7 & x=1 .\end{array} \right.$. If a limit does not exist, explain why.

Answer: -4

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Question: Find the antiderivative of the function: $\int(1-t)^{9} d t $

Answer: $-(1-t)^{10 / 10+C}$

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Question: A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point $5 \mathrm{ft}$ higher than the front of the boat. The rope is being pulled through the ring at a rate of $0.6 \mathrm{ft} / \mathrm{sec}$. Find the rate at which the boat is approaching the dock when $13 \mathrm{ft}$ of rope are out.

Answer: $13 / 20 \mathrm{ft} / \mathrm{s}$

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Question: You are designing a poster to contain a fixed amount $A$ of printing (measured in square centimeters) and have margins of $a$ centimeters at the top and bottom and $b$ centimeters at the sides. Find the ratio of vertical dimension to horizontal dimension of the printed area on the poster if you want to minimize the amount of posterboard needed.

Answer: $a / b$

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Question: An object moves so that its velocity at time $t$ is $v(t)=1+2 \sin t \mathrm{~m} / \mathrm{s}$. Find the net distance traveled by the object between $t=0$ and $t=2 \pi$, and find the total distance traveled during the same period.

Answer: net: $2 \pi$, total: $2 \pi / 3+4 \sqrt{3}$

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Question: Find the area bounded by the curves: $x=3 y-y^{2}$ and $x+y=3 $

Answer: $4 / 3$

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

Answer: $y=24 x-48-\pi^{3}$

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

Answer: $x^{\sin (x)}(\cos (x) \ln (x)+\sin (x) / x)$

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

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

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Question: Find the antiderivative of the function: $\frac{2}{x \sqrt{x}} $

Answer: $-4 / \sqrt{x}+C$

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Question:
Evaluate the three integrals:
$$
A=\int_{0}^{3}-x^{2}+9 d x \quad B=\int_{0}^{4}-x^{2}+9 d x \quad C=\int_{4}^{3}-x^{2}+9 d x,
$$
and verify that $A=B+C$. $$

Answer: $A=18, B=44 / 3, C=10 / 3$

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Question: Find the area of the largest rectangle that fits inside a semicircle of radius 10 (one side of the rectangle is along the diameter of the semicircle).

Answer: 100

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Question: Does the improper integral $\int_{1}^{\infty} 1 / \sqrt{x} d x$ converge or diverge? If it converges, find the value.

Answer: diverges

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

Answer: $56 x^{6}+72 x^{5}+110 x^{4}+100 x^{3}+$ $60 x^{2}+28 x+6$

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

Answer: $\bar{x}=0, \bar{y}=28 /(9 \pi)$

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Question: Find the domain of the function: $h(x)=\left\{\begin{array}{ll}\left(x^{2}-9\right) /(x-3) & x \neq 3 \\ 6 & \text { if } x=3 .\end{array} \right.$

Answer: $\mathbb{R}$

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Question: Find the antiderivative of the function: $\int \tan x d x $

Answer: $-\ln |\cos x|+C$

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

Answer: converges

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Question: Find the antiderivative of the function: $8 \sqrt{x} $

Answer: $(16 / 3) x^{3 / 2}+C$

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Question: Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{\sin ^{3}(n)}{n^{2}} $

Answer: converges

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Question: Find the antiderivative of the function: $4 / \sqrt{x} $

Answer: $8 \sqrt{x}+C$

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Question: Find the average height of $\cos x$ over the intervals $[0, \pi / 2],[-\pi / 2, \pi / 2]$, and $[0,2 \pi] . $

Answer: $2 / \pi ; 2 / \pi ; 0$

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Question: Describe the concavity of the function: $y=x^{5}-x $

Answer: concave up on $(0, \infty)$

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Question: For the velocity function $v=-9.8 t+49$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 10$.

Answer: 0,245

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Question: Find the derivative of the function: $\ln (\cos (x)) $

Answer: $-\tan (x)$

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

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

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Question: Find the derivative of the function: $\frac{-3}{4 x^{2}-2 x+1} $

Answer: $3(8 x-2) /\left(4 x^{2}-2 x+1\right)^{2}$

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Question: Find an equation for the tangent line to $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$ at a point $\left(x_{1}, y_{1}\right)$ on the curve, with $x_{1} \neq 0$ and $y_{1} \neq 0$. (This curve is an astroid.)

Answer: $y=\left(-y_{1}^{1 / 3} x+y_{1}^{1 / 3} x_{1}+x_{1}^{1 / 3} y_{1}\right) / x_{1}^{1 / 3}$

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Question: Find the antiderivative: $\int \tan ^{3} x \sec x d x $

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

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Question: Evaluate the integral: $\int t^{3} e^{t} d t $

Answer: $\left(t^{3}-3 t^{2}+6 t-6\right) e^{t}+C$

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Question: Find the derivative of the function: $\sqrt{x \tan x} $

Answer: $\frac{\tan x+x \sec ^{2} x}{2 \sqrt{x \tan x}}$

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Question: Find the derivative of $\arcsin \left(x^{2}\right) . $

Answer: $\frac{2 x}{\sqrt{1-x^{4}}}$

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

Answer: $\max$ at $\pi / 2+2 n \pi$, min at $3 \pi / 2+2 n \pi$

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

Answer: converges

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Question: Let $f(x)=x^{4}$. If $a=1$ and $d x=\Delta x=1 / 2$, what are $\Delta y$ and $d y$ ?

Answer: $\Delta y=65 / 16, d y=2$

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

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

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Question: You have $l$ feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area?

Answer: $l^{2} / 8$ square feet

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Question: Determine the domain of the composition $(g \circ f)(x)$ if $f(x)=3 x-9$ and $g(x)=\sqrt{x}$. What is the domain of $(f \circ g)(x) ? $

Answer: $\{x \mid x \geq 3\},\{x \mid x \geq 0\}$

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Question: You want to make cylindrical containers to hold 1 liter using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $2 r$, so that $2(2 r)^{2}=8 r^{2}$ of material is needed (rather than $2 \pi r^{2}$, which is the total area of the top and bottom). Find the dimensions of the container using the least amount of material, and also find the ratio of height to radius for this container.

Answer: $r=5, h=40 / \pi, h / r=8 / \pi$

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Question: Find the derivative of the function: $\sin (\cos (6 x)) $

Answer: $-6 \cos (\cos (6 x)) \sin (6 x)$

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

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

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Question: The sun is rising at a rate of $1 / 4 \mathrm{deg} / \mathrm{min}$ and appears to be climbing into the sky perpendicular to the horizon. Find the rate at which the shadow of a 200 meter building is shrinking at the moment when the shadow is 500 meters long.

Answer: $145 \pi / 72 \mathrm{~m} / \mathrm{s}$

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Question: Describe the concavity of the function: $y=x^{2}-x $

Answer: concave up everywhere

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Question: Does the improper integral $\int_{-\infty}^{\infty} \sin x d x$ 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: Determine whether the series converges: $\sum_{n=1}^{\infty} \frac{1}{n \sqrt{n}} $

Answer: converges

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Question: Explain why the series $\sum_{n=1}^{\infty} \frac{5}{2^{1 / n}+14}$ diverges.

Answer: $\lim _{n \rightarrow \infty} 5 /\left(2^{1 / n}+14\right)=1 / 3$

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Question: Find the derivative of the function: $x^{3} \sin \left(23 x^{2}\right) $

Answer: $3 x^{2} \sin \left(23 x^{2}\right)+46 x^{4} \cos \left(23 x^{2}\right)$

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Question: Find an equation for the tangent line to $\frac{\left(x^{2}+x+1\right)}{(1-x)}$ at $(2,-7)$.

Answer: $y=2 x-11$

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Question: Find the area bounded by the curves: $x=y^{3}$ and $x=y^{2} $

Answer: $1 / 12$

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Question: Describe the concavity of the function: $y=x^{2}+1 / x $

Answer: concave up on $(-\infty,-1)$ and $(0, \infty)$

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Question: Find the derivative of the function: $\sin x \cos x $

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

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Question: Find the antiderivative: $\int \frac{1}{x^{2}+10 x+25} d x $

Answer: $-1 /(x+5)+C$

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

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

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

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

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Question: Find the arc length of $f(x)=(1 / 3)\left(x^{2}+2\right)^{3 / 2}$ on the interval $[0, a]$.

Answer: $a+a^{3} / 3$

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Question: Find the derivative of the function: $\frac{1}{x^{5}} $

Answer: $-5 x^{-6}$

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Question: Find the antiderivative of the function: $\int f(x) f^{\prime}(x) d x $

Answer: $f(x)^{2} / 2$

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Question: A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top. Suppose that the colored glass transmits only $k$ times as much light per unit area as the clear glass ( $k$ is between 0 and 1). If the distance from top to bottom (across both the rectangle and the semicircle) is a fixed distance $H$, find (in terms of $k$ ) the ratio of vertical side to horizontal side of the rectangle for which the window lets through the most light.

Answer: If $k \leq 2 / \pi$ the ratio is $(2-k \pi) / 4$; if $k \geq 2 / \pi$, the ratio is zero: the window should be semicircular with no rectangular part.

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Question: Find all critical points of the function $y=3 x^{4}-4 x^{3} $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\min$ at $x=1$

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Question: Find a power series representation for $\int \ln (1-x) d x$.

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

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Question: Determine whether the sequence $\left\{\frac{n^{2}+1}{(n+1)^{2}}\right\}_{n=0}^{\infty}$ converges or diverges. If it converges, compute the limit.

Answer: 1

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

Answer: $-x-\ln |x-2|+\ln |x+2|+C$

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Question: Does the improper integral $\int_{0}^{\infty} e^{-x} d x$ converge or diverge? If it converges, find the value.

Answer: 1

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Question: An object is shot upwards from ground level with an initial velocity of 3 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground.

Answer: $45 / 98$ meters, $30 / 49$ seconds

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Question: Compute the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why.

Answer: 5

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Question: Find the antiderivative: $\int \sqrt{9+4 x^{2}} d x $

Answer: $x \sqrt{9+4 x^{2}} / 2+$ $(9 / 4) \ln \left|2 x+\sqrt{9+4 x^{2}}\right|+C$

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Question: Find the antiderivative of the function: $\int \cos (\pi t) \cos (\sin (\pi t)) d t $

Answer: $\sin (\sin \pi t) / \pi+C$

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Question: A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top; the colored glass transmits only $1 / 2$ as much light per unit area as the the clear glass. If the distance from top to bottom (across both the rectangle and the semicircle) is 2 meters and the window may be no more than 1.5 meters wide, find the dimensions of the rectangular portion of the window that lets through the most light.

Answer: 1.5 meters wide by 1.25 meters tall

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

Answer: $\tan x-\cot x+C$

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Question: Find the points on the curve $y=x+2 \cos x$ that have a horizontal tangent line.

Answer: $\pi / 6+2 n \pi, 5 \pi / 6+2 n \pi$, any integer $n$

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Question: Find the antiderivative: $\int \frac{1}{2 x^{2}-x-3} d x $

Answer: $(1 / 5) \ln |2 x-3|-(1 / 5) \ln |1+x|+C$

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Question: Find the area bounded by the curves: $y=\cos (\pi x / 2)$ and $y=1-x^{2}$ (in the first quadrant)

Answer: $2 / 3-2 / \pi$

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Question: Compute the limit: $\lim _{x \rightarrow 2} 3 $.

Answer: 3

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Question: Find a series representation for the function: $\ln (1+x) $

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

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Question: Find the dimensions of the rectangle of largest area having fixed perimeter $100 . $

Answer: $25 \times 25$

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Question: Compute the limit: $\lim _{x \rightarrow \pi / 4} \frac{\sin x-\cos x}{\cos (2 x)} $

Answer: $-\sqrt{2} / 2$

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Question: Find a polynomial approximation for $\cos x$ on $[0, \pi]$, accurate to $\pm 10^{-3} $

Answer: $1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}+\cdots+\frac{x^{12}}{12 !}$

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Question: Find a series representation for the function: $\sum_{n=1}^{\infty} \frac{x^{n}}{n 3^{n}} $

Answer: $(-3,3)$

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Question: Does the improper integral $\int_{0}^{\infty} x^{2}+2 x-1 d x$ converge or diverge? If it converges, find the value.

Answer: diverges

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Question: Consider the function $f(x)=x^{2}-3 x+2$ on $[0,4]$. Find the total area between the curve and the $x$-axis (measuring all area as positive).

Answer: $17 / 3$

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

Answer: $\frac{\sin x}{2 \sqrt{x}}+\sqrt{x} \cos x$

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

Answer: $2 \sqrt{e^{t}+1}+C$

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

Answer: $\bar{x}=1 / 2, \bar{y}=2 / 5$

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Question: Find the derivative of the function: $\tan x /(1+\sin x) $

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

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Question: How much work is done in lifting a 100 kilogram weight from an orbit 1000 kilometers above the surface of the earth to an orbit 35,786 kilometers above the surface of the earth?

Answer: $\approx 4,457,854,041 \mathrm{~N}-\mathrm{m}$

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Question: Find a series representation for the function: $\sum_{n=0}^{\infty} \frac{2^{n}}{n !} x^{n} $

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

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

Answer: converges absolutely

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Question: For all $x, 2 x \leq g(x) \leq x^{4}-x^{2}+2$. Find $\lim _{x \rightarrow 1} g(x)$.

Answer: 2

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Question: Find all local maximum and minimum points of the function: $y=4 x+\sqrt{1-x} $

Answer: $\max$ at $x=63 / 64$

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Question: Compute the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x+8}-3}{x-1} $. If a limit does not exist, explain why.

Answer: $1 / 6$

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

Answer: $1+2^{x} \ln (2)$

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

Answer: converges

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Question: Find the volume of the solid obtained by revolving the region bounded by $y=x-x^{2}$ and the $x$-axis around the $x$-axis.

Answer: $\pi / 30$

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Question: Find a series representation for $\ln 2$.

Answer: the alternating harmonic series

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Question: A rotating beacon is located 2 miles out in the water. Let $A$ be the point on the shore that is closest to the beacon. As the beacon rotates at $10 \mathrm{rev} / \mathrm{min}$, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. Find the speed at which the point where the beam hits the shore is moving when the beam is lighting up a point 2 miles along the shore from the point $A$ .

Answer: $80 \pi \mathrm{mi} / \mathrm{min}$

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Question: Evaluate the integral: $\int t \sec ^{2} t d t $

Answer: $t \tan t+\ln |\cos t|+C$

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Question: Compute the limit: $\lim _{x \rightarrow 0^{+}} \frac{\sqrt{2-x^{2}}}{x} $. If a limit does not exist, explain why.

Answer: does not exist

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Question: Find all critical points of the function $f(x)=\sin ^{2} x $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\min$ at $n \pi, \max$ at $\pi / 2+n \pi$

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Question: Describe the concavity of the function: $y=3 x^{2}-\left(1 / x^{2}\right) $

Answer: concave up when $x<-1$ or $x>1$, concave down when $-1<x<0$ or $0<x<1$

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Question: Find the derivative of the function: $\left(x^{2}+5\right)^{3} $

Answer: $6 x\left(x^{2}+5\right)^{2}$

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

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

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Question: Determine whether the series converges: $\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+\frac{1}{5 \cdot 6}+\frac{1}{7 \cdot 8}+\cdots $

Answer: converges

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Question: Describe the concavity of the function: $y=\tan ^{2} x $

Answer: concave up everywhere

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Question: Does the improper integral $\int_{0}^{1} 1 / \sqrt{x} d x$ converge or diverge? If it converges, find the value.

Answer: 2

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

Answer: $\frac{-1}{2(1+\tan t)^{2}}+C$

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Question: A thin plate fills the upper half of the unit circle $x^{2}+y^{2}=1$. Find the centroid.

Answer: $\bar{x}=0, \bar{y}=4 /(3 \pi)$

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Question: Evaluate the definite integral: $\int_{0}^{\pi / 6}\left(\cos ^{2} x-\sin ^{2} x\right) d x $

Answer: $\sqrt{3} / 4$

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

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

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

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

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

Answer: $\{x \mid x \geq 1\}$

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

Answer: $24 x^{2}\left(1-4 x^{3}\right)^{-3}$

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Question: Find the antiderivative of the function: $\int\left(x^{2}+1\right)^{2} d x $

Answer: $x^{5} / 5+2 x^{3} / 3+x+C$

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Question: Find an algebraic expression for the difference quotient $(f(x+\Delta x)-f(x)) / \Delta x$ when $f(x)=$ $m x+b$. Simplify the expression as much as possible. Then determine what happens as $\Delta x$ approaches 0 . That value is $f^{\prime}(x)$.

Answer: $m$

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Question: Find the antiderivative: $\int \cos ^{2} x \sin ^{3} x d x $

Answer: $\left(\cos ^{5} x\right) / 5-\left(\cos ^{3} x\right) / 3+C$

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Question: Compute the value of the series $\sum_{n=0}^{\infty} \frac{4}{(-3)^{n}}-\frac{3}{3^{n}}$.

Answer: $-3 / 2$

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

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

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Question: Find the maxima and minima of the function $f(x)=\sec x $.

Answer: $\min$ at $2 n \pi, \max$ at $(2 n+1) \pi$

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Question: Compute $\frac{d}{d t} t^{5} \cos (6 t)$.

Answer: $5 t^{4} \cos (6 t)-6 t^{5} \sin (6 t)$

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Question: Describe all functions with derivative $\frac{1}{1+x^{2}}$.

Answer: $\arctan x+k$

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Question: Evaluate the definite integral: $\int_{-1}^{1}\left(2 x^{3}-1\right)\left(x^{4}-2 x\right)^{6} d x $

Answer: $-\left(3^{7}+1\right) / 14$

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Question: Find the derivative of the function: $\sqrt{r^{2}-x^{2}}, r$ is a constant

Answer: $-x / \sqrt{r^{2}-x^{2}}$

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Question: Determine whether the series $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{n^{n}} $ converges.

Answer: diverges

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

Answer: $\frac{-1}{\sin \arctan t}+C=-\sqrt{1+t^{2}} / t+C$

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

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

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Question: Find the antiderivative of the function: $\int x\left(x^{2}+1\right)^{100} d x $

Answer: $\left(x^{2}+1\right)^{101} / 202+C$

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Question: Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let $H$ and $R$ be the height and base radius of the larger cone, and let $h$ and $r$ be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating $h$ and $r$.)

Answer: $4 / 27$

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Question: Consider the function $f(x)=(x+2)(x+1)(x-1)(x-2)$ on $[-2,2]$. Find the total area between the curve and the $x$-axis (measuring all area as positive).

Answer: 8

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Question: Find the antiderivative: $\int \cos ^{3} x d x $

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

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

Answer: converges

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Question: Find the arc length of $f(x)=x^{2} / 8-\ln x$ on the interval $[1,2]$.

Answer: $\ln (2)+3 / 8$

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

Answer: $-2 x \ln (3)(1 / 3)^{x^{2}}$

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

Answer: diverges

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

Answer: converges

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Question: Find all local maximum and minimum points of the function: $y=3 x^{4}-4 x^{3} $

Answer: $\min$ at $x=1$

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Question: Find the area bounded by the curves: $x=1-y^{2}$ and $y=-x-1 $

Answer: $9 / 2$

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Question: Compute the limit of $\lim _{x \rightarrow \infty} \sqrt{x^{2}+x}-\sqrt{x^{2}-x} $.

Answer: 1

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Question: Determine whether the series converges: $\frac{1}{2 \cdot 3 \cdot 4}+\frac{2}{3 \cdot 4 \cdot 5}+\frac{3}{4 \cdot 5 \cdot 6}+\frac{4}{5 \cdot 6 \cdot 7}+\cdots $

Answer: converges

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

Answer: $\frac{3 t}{8}+\frac{\sin 2 t}{4}+\frac{\sin 4 t}{32}+C$

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Question: Find the antiderivative: $\int \frac{1}{x^{2}\left(1+x^{2}\right)} d x $

Answer: $-\arctan x-1 / x+C$

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Question: Compute the limit: $\lim _{x \rightarrow 0^{+}} \frac{\sqrt{2-x^{2}}}{x+1} $. If a limit does not exist, explain why.

Answer: $\sqrt{2}$

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Question: Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for $x \cos \left(x^{2}\right)$.

Answer: $\sum_{n=0}^{\infty}(-1)^{n} x^{4 n+1} /(2 n)$ !

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

Answer: undefined

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Question: Find the derivative of the function: $x \sqrt{169-x^{2}} $

Answer: $\sqrt{169-x^{2}}-x^{2} / \sqrt{169-x^{2}}$

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Question: A thin plate lies in the region between $y=x^{2}$ and the $x$-axis between $x=1$ and $x=2$. Find the centroid.

Answer: $\bar{x}=45 / 28, \bar{y}=93 / 70$

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Question: Compute the value of the integral: $\int_{0}^{5} e^{x} d x $

Answer: $e^{5}-1$

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Question: Find the domain of the function: $y=f(x)=\sqrt{r^{2}-(x-h)^{2}}$, where $r$ and $h$ are positive constants.

Answer: $\{x \mid h-r \leq x \leq h+r\}$

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Question: Find the arc length of $f(x)=x^{3 / 2}$ on the interval $[0,2]$.

Answer: $(22 \sqrt{22}-8) / 27$

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Question: Find the derivative of the function: $\frac{1}{\sqrt{5-\sqrt{x}}} \cdot $

Answer: $\frac{1}{4 \sqrt{x}(5-\sqrt{x})^{3 / 2}}$

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Question: For a cylinder with surface area 50, 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 lines $3 x+6 y=7$ and $2 x+4 y=5$ are parallel.

Answer: yes

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Question: Find all local maximum and minimum points of the function: $y=x+1 / x $

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

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

Answer: $\max$ at $n \pi$, min at $\pi / 2+n \pi$

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

Answer: diverges

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Question: Find an equation for the tangent line to $\left(x^{2}-4 x+5\right) \sqrt{25-x^{2}}$ at $(3,8)$.

Answer: $y=13 x / 2-23 / 2$

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

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

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

Answer: converges

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Question: Evaluate the integral: $\int \frac{1}{t(\ln t)^{2}} d t $

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

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

Answer: $x^{2}-3 x$

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Question: Evaluate the integral: $\int \frac{1}{t\left(9+4 t^{2}\right)} d t $

Answer: $\frac{\ln |\sin (\arctan (2 t / 3))|}{9}+C=$ $\left(\ln \left(4 t^{2}\right)-\ln \left(9+4 t^{2}\right)\right) / 18+C$

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Question: Find an equation for the tangent line to $f(x)=(x-2) /\left(x^{3}+4 x-1\right)$ at $x=1$.

Answer: $y=11 x / 16-15 / 16$

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Question: A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at a rate of $0.6 \mathrm{~m} / \mathrm{sec}$. Find the rate at which the top of the ladder is sliding down the wall when the foot of the ladder is $5 \mathrm{~m}$ from the wall.

Answer: $1 / 4 \mathrm{~m} / \mathrm{s}$

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Question: Find a series representation for the function: $2^{x} $

Answer: $\sum_{n=0}^{\infty} \frac{(\ln (2))^{n}}{n !} x^{n}$

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Question: Find all critical points of the function $f(x)=\left|x^{2}-121\right| $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\max$ at $x=0$, min at $x= \pm 11$

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

Answer: converges absolutely

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Question: Find all critical points of the function $y=\cos (2 x)-x $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\min$ at $x=7 \pi / 12+k \pi, \max$ at $x=-\pi / 12+k \pi$, for integer $k$.

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Question: Find the antiderivative: $\int \sin ^{3} x d x $

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

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Question: A car rental firm has the following charges for a certain type of car: $\$ 25$ per day with 100 free miles included, $\$ 0.15$ per mile for more than 100 miles. Suppose you want to rent a car for one day, and you know you'll use it for more than 100 miles. What is the equation relating the cost $y$ to the number of miles $x$ that you drive the car?

Answer: $y=0.15 x+10$

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Question: Evaluate the integral: $\int \frac{\cos 3 t}{\sqrt{\sin 3 t}} d t $

Answer: $\frac{2}{3} \sqrt{\sin 3 t}+C$

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Question: Approximate the value of the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n^{4}}$ to two decimal places.

Answer: 0.95

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

Answer: $-\csc ^{2} x$

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Question: For the velocity function $v=\cos (\pi t)$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 2.5$.

Answer: $1 / \pi, 5 / \pi$

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Question: Find a series representation for the function: $\frac{1}{1+x^{2}} $

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

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Question: A force of 2 Newtons will compress a spring from 1 meter (its natural length) to 0.8 meters. How much work is required to stretch the spring from 1.1 meters to 1.5 meters?

Answer: $6 / 5 \mathrm{~N}-\mathrm{m}$

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Question: A woman $5 \mathrm{~ft}$ tall walks at a speed of $3.5 \mathrm{~ft} / \mathrm{sec}$ away from a streetlight that is $12 \mathrm{~ft}$ above the ground. Find the rate at which the tip of her shadow is moving. Find the rate at which her shadow is lengthening.

Answer: tip: $6 \mathrm{ft} / \mathrm{s}$, length: $5 / 2 \mathrm{ft} / \mathrm{s}$

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Question: Evaluate the integral: $\int e^{t} \sin t d t $

Answer: $\frac{e^{t} \sin t-e^{t} \cos t}{2}+C$

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Question: An object is shot upwards from ground level with an initial velocity of 100 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground.

Answer: 25000/49 meters, 1000/49 seconds

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

Answer: $\frac{\left(t^{2}+1\right)^{5 / 2}}{5}-\frac{\left(t^{2}+1\right)^{3 / 2}}{3}+C$

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Question: Find a series representation for the function: $\arctan (x) $

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

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Question: Find the antiderivative of the function: $\int_{0}^{\sqrt{\pi} / 2} x \sec ^{2}\left(x^{2}\right) \tan \left(x^{2}\right) d x $

Answer: $1 / 4$

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

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

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Question: Describe the concavity of the function: $y=x^{3}-9 x^{2}+24 x $

Answer: concave down when $x<3$, concave up when $x>3$

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Question: Compute the limit: $\lim _{x \rightarrow 2}\left(x^{2}+4\right)^{3} $.

Answer: 512

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Question: Find all local maximum and minimum points of the function: $y=x^{4}-2 x^{2}+3 $

Answer: $\min$ at $x= \pm 1, \max$ at $x=0$.

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

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

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Question: Find the antiderivative of the function: $\int \sec ^{2} x \tan x d x $

Answer: $\tan ^{2}(x) / 2+C$

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Question: Find all values of $\theta$ such that $\cos (2 \theta)=1 / 2$; give your answer in radians.

Answer: $n \pi \pm \pi / 6$, any integer $n$

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Question: Use an angle sum identity to compute $\tan (5 \pi / 12)$.

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

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Question: Does the improper integral $\int_{0}^{\pi / 2} \sec ^{2} x d x$ converge or diverge? If it converges, find the value.

Answer: diverges

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Question: Find the area of the largest rectangle that fits inside a semicircle of radius $r$ (one side of the rectangle is along the diameter of the semicircle).

Answer: $r^{2}$

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Question: A 20 meter long steel cable has density 2 kilograms per meter, and is hanging straight down. How much work is required to lift the entire cable to the height of its top end?

Answer: $3920 \mathrm{~N}-\mathrm{m}$

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Question: Find the antiderivative of the function: $(5 x+1)^{2} $

Answer: $(5 x+1)^{3} / 15+C$

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

Answer: 5

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Question: An object is shot upwards from ground level with an initial velocity of 2 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground.

Answer: $10 / 49$ meters, $20 / 49$ seconds

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

Answer: converges

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Question: Find a series representation for the function: $\ln \left(\frac{1+x}{1-x}\right) $

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

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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, a=5 $

Answer: $\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-5)^{n}}{5^{n+1}}, R=5$

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Question: Determine whether the series converges: $1+\frac{5^{2}}{2^{2}}+\frac{5^{4}}{(2 \cdot 4)^{2}}+\frac{5^{6}}{(2 \cdot 4 \cdot 6)^{2}}+\frac{5^{8}}{(2 \cdot 4 \cdot 6 \cdot 8)^{2}}+\cdots $

Answer: converges

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Question: A hyperbola passing through $(8,6)$ consists of all points whose distance from the origin is a constant more than its distance from the point $(5,2)$. Find the slope of the tangent line to the hyperbola at $(8,6) . $

Answer: 1

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Question: Find all critical points of the function $y=x^{4}-2 x^{2}+3 $. Identify them as local maximum points, local minimum points, or neither.

Answer: $\min$ at $x= \pm 1, \max$ at $x=0$.

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

Answer: converges

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Question: Find the area bounded by the curves: $y=x^{2}-2 x$ and $y=x-2 $

Answer: $1 / 6$

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Question: Find the area bounded by the curves: $y=\sqrt{x}$ and $y=x^{2} $

Answer: $1 / 3$

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Question: Compute the limit: $\lim _{x \rightarrow 0} \frac{\cot (4 x)}{\csc (3 x)} $

Answer: $3 / 4$

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Question: Find the dimensions of the rectangle of largest area having fixed perimeter $P$.

Answer: $P / 4 \times P / 4$

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

Answer: 0

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Question: A box with square base and no top is to hold a volume 100. Find 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.

Answer: $w=l=2 \cdot 5^{2 / 3}, h=5^{2 / 3}, h / w=$ $1 / 2$

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Question: Find the antiderivative of the function: $\int \frac{x^{2}}{\sqrt{1-x^{3}}} d x $

Answer: $-2 \sqrt{1-x^{3}} / 3+C$

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Question: Find the derivative of the function: $y=f(x)=a x^{2}+b x+c$ (where $a, b$, and $c$ are constants).

Answer: $2 a x+b$

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Question: Find all local maximum and minimum points of the function: $y=\cos (2 x)-x $

Answer: $\min$ at $x=7 \pi / 12+n \pi$, $\max$ at $x=-\pi / 12+n \pi$, for integer $n$.

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Question: Describe the concavity of the function: $y=3 x^{4}-4 x^{3} $

Answer: concave up when $x<0$ or $x>2 / 3$, concave down when $0<x<2 / 3$

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Question: Find the antiderivative of the function: $\int x \sqrt{100-x^{2}} d x $

Answer: $-\left(100-x^{2}\right)^{3 / 2} / 3+C$

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Question: Compute the limit: $\lim _{x \rightarrow 0} \sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{x}} $.

Answer: 0

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Question: Find an $N$ such that $\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\sum_{n=1}^{N} \frac{1}{n^{4}} \pm 0.005$.

Answer: $N=5$

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Question: Use integration to find the volume of the solid obtained by revolving the region bounded by $x+y=2$ and the $x$ and $y$ axes around the $x$-axis.

Answer: $8 \pi / 3$

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

Answer: $N=1687$

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

Answer: $2 e^{2 x}$

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Question: A man 1.8 meters tall walks at a speed of 1 meter per second toward a streetlight that is 4 meters above the ground. Find the rate at which the tip of his shadow is moving. Find the rate at which his shadow is shortening.

Answer: tip: $20 / 11 \mathrm{~m} / \mathrm{s}$, length: $9 / 11 \mathrm{~m} / \mathrm{s}$

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Question: Compute the limit: $\lim _{x \rightarrow 0} \frac{\sin (5 x)}{x} $

Answer: 5

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

Answer: $\min$ at $x=1 / 2$

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Question: The function $f(x)=x^{3}-3 x^{2}-3 x+6$ has a root between 3 and 4 , because $f(3)=-3$ and $f(4)=10$. Approximate the root to two decimal places.

Answer: 3.36

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

Answer: $2 \arcsin (x / 2)-x \sqrt{4-x^{2}} / 2+C$

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

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

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Question: An object moves so that its velocity at time $t$ is $v(t)=\sin t$. Set up and evaluate a single definite integral to compute the net distance traveled between $t=0$ and $t=2 \pi$.

Answer: $\int_{0}^{2 \pi} \sin t d t=0$

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Question: Describe the concavity of the function: $y=x+1 / x $

Answer: concave up on $(0, \infty)$

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Question: Find the radius and interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{(x+5)^{n}}{n(n+1)} $

Answer: $R=1, I=(-6,-4)$

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

Answer: diverges

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

Answer: $\pi x^{\pi-1}$

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Question: Compute the limit: $\lim _{x \rightarrow 0} \frac{\tan x}{x} $

Answer: 1

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Question: Find the derivative of the function: $\sqrt{1+x^{4}} $

Answer: $2 x^{3} / \sqrt{1+x^{4}}$

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Question: A beam 4 meters long has density $\sigma(x)=x^{3}$ at distance $x$ from the left end of the beam. Find the center of mass $\bar{x}$.

Answer: $16 / 5$

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Question: A farmer wants to build a fence along a river. He has 500 feet of fencing and wants to enclose a rectangular pen on three sides (with the river providing the fourth side). If $x$ is the length of the side perpendicular to the river, determine the area of the pen as a function of $x$. What is the domain of this function?

Answer: $A=x(500-2 x),\{x \mid 0 \leq x \leq 250\}$

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Question: Compute the limit: $\lim _{x \rightarrow 0} \frac{4 x-5 x^{2}}{x-1} $. If a limit does not exist, explain why.

Answer: 0

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Question: Find the first four terms of the Maclaurin series for $\tan x$ (up to and including the $x^{3}$ term).

Answer: $x+x^{3} / 3$

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Question: Find the antiderivative: $\int \sin x(\cos x)^{3 / 2} d x $

Answer: $-2(\cos x)^{5 / 2} / 5+C$

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

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

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

Answer: $\left(4 x\left(x^{2}+1\right)+\frac{4 x^{3}+4 x}{2 \sqrt{1+\left(x^{2}+1\right)^{2}}}\right) /$ $2 \sqrt{\left(x^{2}+1\right)^{2}+\sqrt{1+\left(x^{2}+1\right)^{2}}}$

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Question: Find the antiderivative of the function: $(x-6)^{2} $

Answer: $(x-6)^{3} / 3+C$

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Question: Find a power series representation for $1 /(1-x)^{3}$. What is the radius of convergence?

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

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Question: Describe the concavity of the function: $y=x^{4}-2 x^{2}+3 $

Answer: concave up when $x<-1 / \sqrt{3}$ or $x>1 / \sqrt{3}$, concave down when $-1 / \sqrt{3}<x<1 / \sqrt{3}$

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

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

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Question: The sun is setting at a rate of $1 / 4 \mathrm{deg} / \mathrm{min}$ and appears to be climbing into the sky perpendicular to the horizon. Find the rate at which the shadow of a 25 meter wall is lengthening at the moment when the shadow is 50 meters.

Answer: $25 \pi / 144 \mathrm{~m} / \mathrm{min}$

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Question: Find the derivative of the function: $x^{-100} $

Answer: $-100 x^{-101}$

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Question: Find the area bounded by the curves: $x=0$ and $x=25-y^{2} $

Answer: $500 / 3$

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Question: Find the derivative of the function: $\sin \left(e^{x}\right) $

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

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

Answer: $\ln \left|t^{2}+t+3\right|+C$

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Question: Compute the limit: $\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x-a} $.

Answer: $3 a^{2}$

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Question: An object moves along a straight line with acceleration given by $a(t)=\sin (\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t), v(t)$, and the maximum speed of the object. Describe the motion of the object.

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

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Question: Find all local maximum and minimum points of the function: $y=x^{2}+1 / x $

Answer: $\min$ at $2^{-1 / 3}$

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

Answer: diverges

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Question: Evaluate the integral: $\int \frac{\arctan 2 t}{1+4 t^{2}} d t $

Answer: $\frac{(\arctan (2 t))^{2}}{4}+C$

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Question: A piece of carboard is 1 meter by $1 / 2$ meter. A square is to be cut from each corner and the sides folded up to make an open-top box. What are the dimensions of the box with maximum possible volume?

Answer: $\frac{\sqrt{3}}{6} \times \frac{\sqrt{3}}{6}+\frac{1}{2} \times \frac{1}{4}-\frac{\sqrt{3}}{12}$

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Question: Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n !}{8^{n}} $

Answer: diverges

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Question: A police helicopter is flying at a speed of 200 kilometers per hour at a constant altitude of $1 \mathrm{~km}$ above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 2 kilometers from the helicopter, and that this distance is decreasing at a rate of $250 \mathrm{~kph}$. Find the speed of the car.

Answer: $81 \mathrm{~km} / \mathrm{hr}$

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Question: Find the antiderivative of the function: $3 t^{2}+1 $

Answer: $t^{3}+t+C$

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Question: Compute the value of the integral: $\int_{0}^{\pi} \sin t d t $

Answer: 2

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Question: Find the derivative of the function: $x^{100} $

Answer: $100 x^{99}$

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Question: Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=2 t+2$, and that at $t=1$ the object is at position 5 . Find the position of the object at $t=2 . $

Answer: 10

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

Answer: $R=e, I=(-e, e)$

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

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

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Question: Find an equation for the tangent line to $f(x)=(x-2)^{1 / 3} /\left(x^{3}+4 x-1\right)^{2}$ at $x=1$.

Answer: $y=23 x / 96-29 / 96$

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

Answer: converges

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

Answer: $(5-4 x) /\left((2 x+1)^{2}(x-3)^{2}\right)$

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Question: Find the antiderivative: $\int \frac{1}{4-x^{2}} d x $

Answer: $-\ln |x-2| / 4+\ln |x+2| / 4+C$

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Question: A balloon is at a height of 50 meters and is rising at a constant rate of $5 \mathrm{~m} / \mathrm{sec}$. A bicyclist passes beneath it, traveling in a straight line at a constant speed of $10 \mathrm{~m} / \mathrm{sec}$. Find the rate at which the distance between the bicyclist and the balloon is increasing 2 seconds later.

Answer: $5 \sqrt{10} / 2 \mathrm{~m} / \mathrm{s}$

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

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

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Question: A spring has constant $k=10 \mathrm{~kg} / \mathrm{s}^{2}$. How much work is done in compressing it $1 / 10$ meter from its natural length?

Answer: $0.05 \mathrm{~N}-\mathrm{m}$

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Question: Find the average height of $1 / x^{2}$ over the interval $[1, A]$.

Answer: $1 / A$

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Question: Find a series representation for the function: $x+\frac{1}{2} \frac{x^{3}}{3}+\frac{1 \cdot 3}{2 \cdot 4} \frac{x^{5}}{5}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^{7}}{7}+\cdots $

Answer: $(-1,1)$

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Question: Evaluate the integral: $\int \sin t \cos 2 t d t $

Answer: $\cos t-\frac{2}{3} \cos ^{3} t+C$

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Question: Compute the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1} $. If a limit does not exist, explain why.

Answer: 2

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Question: Evaluate the integral: $\int(t+4)^{3} d t $

Answer: $\frac{(t+4)^{4}}{4}+C$

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Question: For the velocity function $v=3(t-3)(t-1)$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 5$.

Answer: 20,28

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Question: Find an equation for the tangent line to $f(x)=x^{3} / 4-1 / x$ at $x=-2$.

Answer: $y=13 x / 4+5$

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Question: Find the antiderivative: $\int \cos ^{3} x \sin ^{2} x d x $

Answer: $\left(\sin ^{3} x\right) / 3-\left(\sin ^{5} x\right) / 5+C$

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

Answer: 7

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Question: Use differentials to estimate the amount of paint needed to apply a coat of paint $0.02 \mathrm{~cm}$ thick to a sphere with diameter 40 meters. (Recall that the volume of a sphere of radius $r$ is $V=(4 / 3) \pi r^{3}$. Notice that you are given that $d r=0.02$.)

Answer: $d V=32 \pi / 25$

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Question: Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $e^{x} $

Answer: $\sum_{n=0}^{\infty} x^{n} / n !, R=\infty$

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Question: Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n}{\left(n^{2}+4\right)^{2}} $

Answer: converges

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Question: Compute the value of the integral: $\int_{0}^{3} x^{3} d x $

Answer: $3^{4} / 4$

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

Answer: converges conditionally

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Question: Describe the concavity of the function: $y=\cos ^{2} x-\sin ^{2} x $

Answer: concave up on $(\pi / 4+n \pi, 3 \pi / 4+n \pi)$

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

Answer: $60 x^{4}+72 x^{3}+18 x^{2}+18 x-6$

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Question: Evaluate the integral: $\int\left(e^{t^{2}}+16\right) t e^{t^{2}} d t $

Answer: $\frac{\left(e^{t^{2}}+16\right)^{2}}{4}+C$

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

Answer: $\frac{3 \ln |t+3|}{4}+\frac{\ln |t-1|}{4}+C$

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Question: Find the dimensions of the lightest cylindrical can containing 0.25 liter $\left(=250 \mathrm{~cm}^{3}\right)$ if the top and bottom are made of a material that is twice as heavy (per unit area) as the material used for the side.

Answer: $r=5 /(2 \pi)^{1 / 3} \approx 2.7 \mathrm{~cm}$, $h=5 \cdot 2^{5 / 3} / \pi^{1 / 3}=4 r \approx 10.8 \mathrm{~cm}$

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Question: A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The top of the ladder is being pulled up the wall at a rate of 0.1 meters per second. Find the rate at which the foot of the ladder is approaching the wall when the foot of the ladder is $5 \mathrm{~m}$ from the wall.

Answer: $-6 / 25 \mathrm{~m} / \mathrm{s}$

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

Answer: $x \ln x-x+C$

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Question: Suppose the position of an object at time $t$ is given by $f(t)=-49 t^{2} / 10+5 t+10$. Find a function giving the speed of the object at time $t$. The acceleration of an object is the rate at which its speed is changing, which means it is given by the derivative of the speed function. Find the acceleration of the object at time $t . $

Answer: $-49 t / 5+5,-49 / 5$

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Question: Describe the concavity of the function: $y=4 x+\sqrt{1-x} $

Answer: concave down everywhere

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

Answer: converges

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Question: Find an equation for the tangent line to $\sqrt{\left(x^{2}+1\right)^{2}+\sqrt{1+\left(x^{2}+1\right)^{2}}}$ at $(1, \sqrt{4+\sqrt{5}})$.

Answer: $y=\frac{20+2 \sqrt{5}}{5 \sqrt{4+\sqrt{5}}} x+\frac{3 \sqrt{5}}{5 \sqrt{4+\sqrt{5}}}$

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Question: Find the area bounded by the curves: $y=x^{3 / 2}$ and $y=x^{2 / 3} $

Answer: $1 / 5$

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Question: Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{1}{n^{\pi / 4}} $

Answer: diverges

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

Answer: $\min$ at $n \pi$

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Question: Find the derivative of the function: $\sqrt{\ln \left(x^{2}\right)} / x $

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

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Question: Find the antiderivative of the function: $\int_{0}^{\pi} \sin ^{5}(3 x) \cos (3 x) d x $

Answer: 0

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Question: Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $\cos x $

Answer: $\sum_{n=0}^{\infty}(-1)^{n} x^{2 n} /(2 n) !, R=\infty$

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Question: Find the antiderivative: $\int \frac{x^{3}}{\sqrt{4 x^{2}-1}} d x $

Answer: $\left(2 x^{2}+1\right) \sqrt{4 x^{2}-1} / 24+C$

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Question: Find the derivative of the function: $y=f(x)=x^{3}$.

Answer: $3 x^{2}$

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Question: You want to make cylindrical containers of a given volume $V$ using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $2 r$, so that $2(2 r)^{2}=8 r^{2}$ of material is needed (rather than $2 \pi r^{2}$, which is the total area of the top and bottom). Find the optimal ratio of height to radius.

Answer: $8 / \pi$

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

Answer: $-8(4 x-1)\left(2 x^{2}-x+3\right)^{-3}$

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Question: Find the antiderivative: $\int \frac{1}{x^{2}+10 x+29} d x $

Answer: $(1 / 2) \arctan (x / 2+5 / 2)+C$

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Question: Find the derivative of the function: $\frac{(x-1)(x-2)}{x-3} $

Answer: $\left(x^{2}-6 x+7\right) /(x-3)^{2}$

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Question: Find a series representation for the function: $\sum_{n=0}^{\infty} \frac{x^{n}}{1+3^{n}} $

Answer: $(-3,3)$

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Question: Find all values of $\theta$ such that $\sin (\theta)=-1$; give your answer in radians.

Answer: $2 n \pi-\pi / 2$, any integer $n$

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Question: Determine whether the series converges: $\sum_{n=1}^{\infty} \sin (1 / n) $. Find the interval and radius of convergence; you need not check the endpoints of the intervals.

Answer: diverges

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Question: Find a series representation for the function: $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2} 3^{n}} x^{2 n} $

Answer: $(-\sqrt{3}, \sqrt{3})$

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

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

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Question: Evaluate the definite integral: $\int_{-1}^{1} \sin ^{7} x d x $

Answer: 0

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Question: Find the average height of $x^{2}$ over the interval $[-2,2] . $

Answer: $4 / 3$

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

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

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Question: Approximate the value of the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n^{3}}$ to two decimal places.

Answer: 0.90

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Question: A thin plate lies in the region contained by $y=x^{1 / 3}$ and the $x$-axis between $x=0$ and $x=1$. Find the centroid.

Answer: $\bar{x}=4 / 7, \bar{y}=2 / 5$

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Question: Evaluate the given double integral: $\int_{0}^{1} \int_{\sqrt{x}}^{1} 24 x^{2} y d y d x$

Answer: $1 $


Question:
Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}$

Answer: 0


Question: For the points $P=(0,0,0), Q=(1,3,2), R=(1,0,1), S=(2,3,4)$, does $\overrightarrow{P Q}=\overrightarrow{R S}$ ?

Answer: No


Question: Find all local maxima and minima of the function $f(x, y) = 4x^{2} - 4xy + 2y^{2} + 10x - 6y$.

Answer: local min. $(-1,1 / 2) $


Question: Find the Laplacian of the function $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}$.

Answer: $12 \sqrt{x^{2}+y^{2}+z^{2}} $


Question:
Describe geometrically the set of points with position vector $\mathbf{x}$ satisfying the equation
$$(\mathbf{v} \times \mathbf{x}) \times \mathbf{x}=\mathbf{v}$$
for given vector $\mathbf{v} \neq \mathbf{0}$

Answer: A circle of radius $\frac{1}{\|\mathbf{v}\|}$ centered at the origin in the normal plane to $\mathbf{v}$.


Question: Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=x^{3}+y^{2}, R=[0,1] \times[0,1]$

Answer: $\frac{7}{12} $


Question: Evaluate the given triple integral: $\int_{0}^{\pi} \int_{0}^{x} \int_{0}^{x y} x^{2} \sin z d z d y d x$

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


Question:
Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 4\}, \delta(x, y)=2 y$

Answer: $(1,8 / 3)$


Question: State the domain and range of the given function: $f(x, y, z)=\sin (x y z)$

Answer: domain: $\mathbb{R}^{3}$, range: $[-1,1] $


Question: Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=(y-2 z) \mathbf{i}+x y \mathbf{j}+(2 x z+y) \mathbf{k} ; \quad C: x=t, y=2 t, z=t^{2}-1,0 \leq t \leq 1$

Answer: $67 / 15 $


Question: Evaluate the given triple integral: $\int_{1}^{2} \int_{2}^{4} \int_{0}^{3} 1 d x d y d z$

Answer: 6


Question: Evaluate the limit: $\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}-y^{2}}{x-y}$

Answer: $2 $


Question: Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x}{y}$

Answer: does not exist


Question: Evaluate the given double integral: $\int_{0}^{\pi / 2} \int_{0}^{y} \cos x \sin y d x d y$

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


Question: Evaluate the double integral: $\int_{0}^{2} \int_{0}^{1}(x+2) d x d y$

Answer: $5 $


Question: Evaluate the given double integral: $\int_{1}^{2} \int_{0}^{\ln x} 4 x d y d x$

Answer: $8 \ln 2-3$


Question:
Calculate the line integral $\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.
$f(x, y)=2 x+y ; \quad C$ : polygonal path from $(0,0)$ to $(3,0)$ to $(3,2)$

Answer: 23


Question: Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=4 x y, R=[0,1] \times[0,1]$

Answer: $1 $


Question: Find three positive numbers $x, y, z$ whose sum is 10 such that $x^{2}y^{2}z$ is a maximum.

Answer: $x=y=4, z=2$


Question: Find all local maxima and minima of the function $f(x, y) = \sqrt{x^{2} + y^{2}}$.

Answer: local min. $(0,0) $


Question: Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{3} - 3y$.

Answer: local min. $(1,1)$; local max. $(-1,-1)$; saddle pts. $(1,-1),(-1,1) $


Question:
Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=y \mathbf{i}-x \mathbf{j} ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$

Answer: $-2 \pi$


Question:
For $\sigma>0$ and $\mu>0$, evaluate

$$
\int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}} d x
$$

Answer: 1


Question:
Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\left\{(x, y): y \geq 0, x^{2}+y^{2} \leq 1\right\}, \delta(x, y)=y$

Answer: $(0,3 \pi / 16) $


Question:
Find the center of mass of the solid $S$ with the given density function $\delta(x, y, z)$:
$S=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}, \delta(x, y, z)=x^{2}+y^{2}+z^{2}$

Answer: $(7 / 12,7 / 12,7 / 12)$


Question: Find the tangent line, the osculating plane, and the curvature at each point of the curve $\mathbf{f}(t)= (\cos t, \sin t, t)$.

Answer: $\frac{3 \pi \sqrt{5}}{2} $


Question:
Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=(2,1,4), \mathbf{w}=(1,-2,0)$

Answer: $(8,4,-5) $


Question: Find the velocity $\mathbf{v}(t)$ and acceleration $\mathbf{a}(t)$ of an object with the given position vector $\mathbf{r}(t)$: $\mathbf{r}(t)=(t, t-\sin t, 1-\cos t)$

Answer: $\mathbf{v}(t)=(1,1-\cos t, \sin t)$, $\mathbf{a}(t)=(0, \sin t, \cos t) $


Question: Find the equation of the tangent plane to the given surface at the point $P$: $x^{2}+y^{2}-z^{2}=0$, $P=(3,4,5)$.

Answer: $3 x+4 y-5 z=0$


Question: Evaluate the given triple integral: $\int_{0}^{3} \int_{0}^{2} \int_{0}^{1} x y z d x d y d z$

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


Question: Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2}+y^{3}$, $P=(1,1,2)$.

Answer: $2 x+3 y-z-3=0 $


Question:
Calculate the area of the triangle $\triangle P Q R$:
$P=(5,1,-2), Q=(4,-4,3), R=(2,4,0)$

Answer: $16.72 $


Question: Evaluate the double integral: $\iint_{R} f(x, y) d A$, where $f(x, y)=x^{2}+y$ and $R$ is the triangle with vertices $(0,0),(2,0)$ and $(0,1)$.

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


Question:
Find the center of mass of the solid $S$ with the given density function $\delta(x, y, z)$:
$S=\left\{(x, y, z): z \geq 0, x^{2}+y^{2}+z^{2} \leq a^{2}\right\}, \delta(x, y, z)=x^{2}+y^{2}+z^{2}$

Answer: $(0,0,5 a / 12) $


Question: State whether or not the vector field $\mathbf{f}(x, y, z)$ has a potential in $\mathbb{R}^{3}$ (you do not need to find the potential itself): $\mathbf{f}(x, y, z)=x y \mathbf{i}-\left(x-y z^{2}\right) \mathbf{j}+y^{2} z \mathbf{k}$

Answer: No


Question: Evaluate $\iint_{R} \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) d A$, where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. (Hint: Use the change of variables $u=(x+y) / 2, v=(x-y) / 2$.)

Answer: $1-\frac{\sin 2}{2}$


Question: Let $\mathbf{v}=(8,4,3)$ and $\mathbf{w}=(-2,1,4)$. Is $\mathbf{v} \perp \mathbf{w}$ ? Justify your answer.

Answer: Yes, since $\mathbf{v} \cdot \mathbf{w}=0


Question: Compute the gradient $\nabla f$ for the function $f(x, y, z)=\sin (x y z)$.

Answer: $\quad(y z \cos (x y z), x z \cos (x y z), x y \cos (x y z))$


Question: For $\mathbf{f}(\rho, \theta, \phi)=\mathbf{e}_{\rho}+\rho \cos \theta \mathbf{e}_{\theta}+\rho \mathbf{e}_{\phi}$ in spherical coordinates, find $\operatorname{div} \mathbf{f}$ and curlf.

Answer: $\operatorname{div} \mathbf{f}=\frac{2}{\rho}-\frac{\sin \theta}{\sin \phi}+\cot \phi, \operatorname{curl} \mathbf{f}=\cot \phi \cos \theta \mathbf{e}_{\rho}+$ $2 \mathbf{e}_{\theta}-2 \cos \theta \mathbf{e}_{\phi} \mathbf{6}$


Question: Evaluate the double integral: $\int_{0}^{2} \int_{1}^{4} x y d x d y$

Answer: $15 $


Question: Let $f(x, y, z)=\frac{z}{x^{2}+y^{2}}$ in Cartesian coordinates. Find $\nabla f$ in cylindrical coordinates.

Answer: $-\frac{2 z}{r^{3}} \mathbf{e}_{r}+\frac{1}{r^{2}} \mathbf{e}_{z}$


Question: Compute the gradient $\nabla f$ for the function $f(x, y, z)=x^{2}+y^{2}+z^{2}$.

Answer: $\quad(2 x, 2 y, 2 z) $


Question: Evaluate the double integral: $\int_{0}^{\pi / 2} \int_{0}^{1} x y \cos \left(x^{2} y\right) d x d y$

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


Question: Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=(8 x y+3) \mathbf{i}+4\left(x^{2}+y\right) \mathbf{j}$ ? If so, find one.

Answer: Yes. $F(x, y)=4 x^{2} y+2 y^{2}+3 x$


Question: Evaluate $\oint_{C}\left(x^{2}+y^{2}\right) d x+2 x y d y$ for $C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$.

Answer: $0 $


Question: Evaluate the given double integral: $\int_{0}^{2} \int_{0}^{y} 1 d x d y$

Answer: $2 $


Question: Find the equation of the tangent plane to the given surface at the point $P$: $\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1$, $P=\left(1,2, \frac{2 \sqrt{11}}{3}\right)$.

Answer: $\frac{1}{2}(x-1)+\frac{4}{9}(y-2)+\frac{\sqrt{11}}{12}(z-\frac{2 \sqrt{11}}{3})=0 $


Question: Find the tangent line, the osculating plane, and the curvature at each point of the curve $\mathbf{f}(t)= (t \sin t, t \cos t)$.

Answer: $2\left(5^{3 / 2}-8\right) $


Question: Compute the gradient $\nabla f$ for the function $f(x, y)=\sqrt{x^{2}+y^{2}+4}$.

Answer: $\left(\frac{x}{\sqrt{x^{2}+y^{2}+4}}, \frac{y}{\sqrt{x^{2}+y^{2}+4}}\right) $


Question: Compute the gradient $\nabla f$ for the function $f(x, y)=\ln (x y)$.

Answer: $\left(\frac{1}{x}, \frac{1}{y}\right)$


Question: Find the trace of the hyperbolic paraboloid $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=\frac{z}{c}$ in the $x y$-plane

Answer: lines $\frac{x}{a}=\frac{y}{b}, z=0$ and $\frac{x}{a}=-\frac{y}{b}, z=0$


Question: Find the volume $V$ inside the paraboloid $z=x^{2}+y^{2}$ for $0 \leq z \leq 4$.

Answer: $8 \pi$


Question: Find the angle $\theta$ between the vectors $\mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}$ and $\mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}$.

Answer: $0^{\circ} $


Question: State the domain and range of the given function: $f(x, y)=x^{2}+y^{2}-1$

Answer: domain: $\mathbb{R}^{2}$, range: $[-1, \infty) $


Question:
Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=(5,1,-2), \mathbf{w}=(4,-4,3)$

Answer: $(-5,-23,-24) $


Question: Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2} y$, $P=(-1,1,1)$.

Answer: $-2 x+y-z-2=0$


Question: Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=y \mathbf{i}-x \mathbf{j}$ ? If so, find one.

Answer: No


Question: Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{2}$.

Answer: local min. $(1,0)$; saddle pt. $(-1,0)$


Question: Let $\mathbf{v}=(5,1,-2)$ and $\mathbf{w}=(4,-4,3)$. Calculate $\mathbf{v} \cdot \mathbf{w}$.

Answer: $10 $


Question: For a rectangular solid of volume 1000 cubic meters, find the dimensions that will minimize the surface area. (Hint: Use the volume condition to write the surface area as a function of just two variables.)

Answer: width $=$ height $=\operatorname{depth}=10$


Question: Determine if the given equation describes a sphere. If so, find its radius and center: $x^{2}+y^{2}+z^{2}-4 x-6 y-10 z+37=0$

Answer: radius: 1 , center: $(2,3,5)$


Question: Compute the gradient $\nabla f$ for the function $f(x, y)=x^{2}+y^{2}-1$.

Answer: $(2 x, 2 y)$


Question: Find the directional derivative of $f(x, y)=\sqrt{x^{2}+y^{2}+4}$ at the point $P=(1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$.

Answer: $\quad \frac{1}{\sqrt{3}}$


Question: Find the directional derivative of $f(x, y, z)=\sin (x y z)$ at the point $P=(1,1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$.

Answer: $\sqrt{3} \cos (1) ; 1$. increase: $(45,20)$, decrease: $(-45,-20)$


Question:
Find the volume of the parallelepiped with adjacent sides $\mathbf{u}, \mathbf{v}, \mathbf{w}$:
$\mathbf{u}=(1,1,3), \mathbf{v}=(2,1,4), \mathbf{w}=(5,1,-2)$

Answer: $9 $


Question: Find all local maxima and minima of the function $f(x, y) = 2x^{3} + 6xy + 3y^{2}$.

Answer: local min. $(1,-1)$, saddle pt. $(0,0) $


Question: Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=x y^{2} \mathbf{i}+x^{3} y \mathbf{j}$ ? If so, find one.

Answer: No


Question: Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=x y \mathbf{i}+(z-x) \mathbf{j}+2 y z \mathbf{k} ; \quad C$ : the polygonal path from $(0,0,0)$ to $(1,0,0)$ to $(1,2,0)$ to $(1,2,-2)$

Answer: 6


Question: Find the volume $V$ of the solid inside both $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}=1$.

Answer: $\frac{4 \pi}{3}\left(8-3^{3 / 2}\right)$


Question: Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=x \mathbf{i}-y \mathbf{j}$ ? If so, find one.

Answer: Yes. $F(x, y)=\frac{x^{2}}{2}-\frac{y^{2}}{2} $


Question: Calculate $\mathbf{f}^{\prime}(t)$ and find the tangent line at $\mathbf{f}(0)$ for the following function: $\mathbf{f}(t)=\left(t+1, t^{2}+1, t^{3}+1\right)$

Answer: $\mathbf{f}^{\prime}(t)=\left(1,2 t, 3 t^{2}\right), \quad x=1+t, \quad y=z=$ $1 $


Question:
Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=\left(x^{2}+y^{2}\right) \mathbf{i} ; \quad C: x=2+\cos t, y=\sin t, 0 \leq t \leq 2 \pi$

Answer: 0


Question: Find the directional derivative of $f(x, y)=x^{2}+y^{2}-1$ at the point $P=(1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$.

Answer: $\quad 2 \sqrt{2} $


Question: Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=y \mathbf{i}-x \mathbf{j}+z \mathbf{k} ; \quad C: x=\cos t, y=\sin t, z=t, 0 \leq t \leq 2 \pi$

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


Question: Calculate $\mathbf{f}^{\prime}(t)$ and find the tangent line at $\mathbf{f}(0)$ for the following function: $\mathbf{f}(t)=(\cos 2 t, \sin 2 t, t)$

Answer: $\mathbf{f}^{\prime}(t)=(-2 \sin 2 t, 2 \cos 2 t, 1) ; \quad x=1$, $y=2 t, \quad z=t $


Question: For the points $P=(1,-1,1), Q=(2,-2,2), R=(2,0,1), S=(3,-1,2)$, does $\overrightarrow{P Q}=\overrightarrow{R S}$ ?

Answer: Yes


Question: Find the Laplacian of the function $f(x, y, z)=x^{3}+y^{3}+z^{3}$.

Answer: $6(x+y+z) $


Question: Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=\frac{x}{y}+y+2 y z ; \quad C: x=t^{2}, y=t, z=1,1 \leq t \leq 2$

Answer: $(17 \sqrt{17}-5 \sqrt{5}) / 3 $