exam stringclasses 1
value | question_index int64 1 45 | subject stringclasses 1
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|---|---|---|---|---|---|---|---|---|
AP | 1 | calculus_bc | 2023_calculus_bc | item If $g(x)=x^{2} \sqrt{x+1}$, then $g^{\prime}(3)=$ | [
"A: $3 / 2$",
"B: $39 / 4$",
"C: $57 / 4$",
"D: 36"
] | C | ||
AP | 2 | calculus_bc | 2023_calculus_bc | item $\int_{0}^{4} 100 e^{\frac{x}{4}} d x=$ | [
"A: $25(e-1)$",
"B: $100(e-1)$",
"C: $400(e-1)$",
"D: $400\\left(e^{4}-1\\right)$ \\item"
] | C | ||
AP | 3 | calculus_bc | 2023_calculus_bc | An airplane's height above sea level is modeled by the differentiable function $y=h(t)$, where $t$ is the number of minutes after takeoff and $h(t)$ is measured in feet. The graph of $h$ is shown in the figure for $0 \leq t \leq 15$. Of the following, which is the best estimate of the rate at which the plane is rising ... | [
"A: $\\frac{37,275-150}{15-0}$",
"B: $\\frac{37,275-9,775}{15-5}$",
"C: $\\frac{37,275-16,859}{15-7}$",
"D: $\\frac{30,763-16,859}{11-7}$"
] | D | ||
AP | 4 | calculus_bc | 2023_calculus_bc | Let $f$ be a vector-valued function with $f(0)=\langle-2,3\rangle$. If the instantaneous rate of change of $f$ with respect to $t$ is given by $\left\langle 3 t^{2}+4 t+1,2 t^{3}+t-2\right\rangle$, what is $f(1)$ ? | [
"A: $\\langle 2,2\\rangle$",
"B: $\\langle 4,-1\\rangle$",
"C: $\\langle 8,10\\rangle$",
"D: $\\langle 10,7\\rangle$"
] | A | ||
AP | 5 | calculus_bc | 2023_calculus_bc | $\lim _{x \rightarrow 4} \frac{2-\sqrt{x}}{x-4}$ | [
"A: $-\\frac{1}{4}$",
"B: 0",
"C: $\\frac{1}{4}$",
"D: Nonexistent"
] | A | ||
AP | 6 | calculus_bc | 2023_calculus_bc | Let $y=f(x)$ be the solution to the differential equation $\frac{d y}{d x}=x+y^{2}$ with initial condition $f(1)=2$. What is the approximation for $f(1.4)$, if Euler's method is used, starting at $x=1$ with two steps of equal size? | [
"A: 2.4",
"B: 3.04",
"C: 4",
"D: 5.04"
] | D | ||
AP | 7 | calculus_bc | 2023_calculus_bc | \includegraphics[max width=\textwidth, center]{2025_09_09_3b57e8501b5d35d96302g-03} A portion of the slope field for the differential equation $\frac{d y}{d x}=x-y$ is shown in the figure. If the linear function $y=f(x)$ is the solution to the differential equation such that $f(-1)=-2$, then $f(3)=$ | [
"A: 0",
"B: 1",
"C: 2",
"D: 3"
] | C | ||
AP | 8 | calculus_bc | 2023_calculus_bc | If $f(x)=\sin \left(e^{3 x}\right)$, then $f^{\prime}(x)=$ | [
"A: $\\cos \\left(e^{3 x}\\right)$",
"B: $\\cos \\left(3 e^{3 x}\\right)$",
"C: $e^{3 x} \\cos \\left(e^{3 x}\\right)$",
"D: $3 e^{3 x} \\cos \\left(e^{3 x}\\right)$"
] | D | ||
AP | 9 | calculus_bc | 2023_calculus_bc | If the series converges, where $A \neq 0$ and $B \neq 0$, what is the sum of the series? $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{B^{n+1}}{A^{n}} $$ | [
"A: $\\frac{A B}{A-B}$",
"B: $\\frac{A B}{A+B}$",
"C: $\\frac{A}{A-B}$",
"D: $\\frac{A}{A+B}$"
] | B | ||
AP | 10 | calculus_bc | 2023_calculus_bc | Of the following series, which can be used with the comparison test to show that diverges? $$ \sum_{n=3}^{\infty} \frac{\ln n}{n} $$ | [
"A: $\\sum_{n=3}^{\\infty} \\frac{1}{n}$",
"B: $\\sum_{n=3}^{\\infty} \\frac{1}{n \\sqrt{n}}$",
"C: $\\sum_{n=3}^{\\infty} \\frac{1}{n^{2}}$",
"D: $\\sum_{n=3}^{\\infty} \\ln n$"
] | A | ||
AP | 11 | calculus_bc | 2023_calculus_bc | A particle moves in the $x y$-plane so that its position at any time $t$ is given by the parametric equations $x(t)=t^{2}-t^{3}$ and $y(t)=9-6 t$. What is the speed of the particle at time $t=2$ ? | [
"A: $\\frac{3}{4}$",
"B: 5",
"C: 10",
"D: 14"
] | C | ||
AP | 12 | calculus_bc | 2023_calculus_bc | Which of the following is the solution to the differential equation $\frac{d y}{d x}=-3 y$ with the initial condition $y(0)=7$ ? | [
"A: $y=7 e^{-3 x}$",
"B: $y=e^{-3 x}+6$",
"C: $y=-\\frac{3}{2} x^{2}+7$",
"D: $y=\\sqrt{49-6 x}$"
] | A | ||
AP | 13 | calculus_bc | 2023_calculus_bc | Consider the series. Which of the following statements is true? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\cos \left(\frac{1}{n}\right)} $$ | [
"A: The series converges absolutely.",
"B: The series converges conditionally.",
"C: The series diverges.",
"D: It cannot be determined whether the series converges or diverges from the information given."
] | C | ||
AP | 14 | calculus_bc | 2023_calculus_bc | If $f$ is a function such that $\lim _{x \rightarrow \infty} f(x)=0$, which of the following could be an expression for $f(x)$ ? | [
"A: $\\frac{x^{2}}{x^{2}+4}$",
"B: $\\frac{\\sin x}{x}$",
"C: $\\cos x$",
"D: $e^{x}$"
] | D | ||
AP | 15 | calculus_bc | 2023_calculus_bc | $\int \frac{x+1}{x(x+3)} d x=$ | [
"A: $\\frac{1}{3} \\ln |x|+\\frac{2}{3} \\ln |x+3|+C$",
"B: $\\frac{1}{3} \\ln |x|+\\frac{2}{9} \\ln |x+3|+C$",
"C: $\\frac{2}{3} \\ln |x|+\\frac{1}{3} \\ln |x+3|+C$",
"D: $\\frac{2}{3} \\ln |x|+\\frac{1}{9} \\ln |x+3|+C$"
] | A | ||
AP | 16 | calculus_bc | 2023_calculus_bc | $f(x)=\left\{\begin{array}{r}x^{2}-2 x, x \leq 4 g(x), x>4\end{array}\right.$ Let $f$ be the function defined as shown. If $f$ is differentiable, which of the following could define $g(x)$ ? | [
"A: $6 x$",
"B: $x^{3 / 2}$",
"C: $x^{2}-8$",
"D: $\\frac{1}{8} x^{3}$"
] | A | ||
AP | 17 | calculus_bc | 2023_calculus_bc | The function $y=f(x)$ satisfies the differential equation $\frac{d y}{d x}=2 x y$ with the initial condition $f(0)=5$. What is the value of $f(2)$ ? | [
"A: $\\frac{5}{21}$",
"B: $\\sqrt{33}$",
"C: $4+e^{4}$",
"D: $5 e^{4}$"
] | D | ||
AP | 18 | calculus_bc | 2023_calculus_bc | Which of the following gives the length of the curve $y=\tan x$ from $x=-\frac{\pi}{4}$ to $x=\frac{\pi}{4} \quad$ ? | [
"A: $\\int_{-\\frac{\\pi}{4}}^{\\frac{\\pi}{4}} \\sqrt{1-\\tan x} d x$",
"B: $\\int_{-\\frac{\\pi}{4}}^{\\frac{\\pi}{4}} \\sqrt{1+\\tan ^{2} x} d x$",
"C: $\\int_{-\\frac{\\pi}{4}}^{\\frac{\\pi}{4}} \\sqrt{1+\\sec ^{2} x} d x$",
"D: $\\int_{-\\frac{\\pi}{4}}^{\\frac{\\pi}{4}} \\sqrt{1+\\sec ^{4} x} d x$"
] | D | ||
AP | 19 | calculus_bc | 2023_calculus_bc | If $y=f(x)$ is a function such that $\frac{d y}{d x}=\frac{2 x-y}{x}$, then $\frac{d^{2} y}{d x^{2}}=$ | [
"A: $\\frac{2 y-2 x}{x^{2}}$",
"B: $\\frac{y-x y}{x^{2}}$",
"C: $\\frac{y-x}{x^{2}}$",
"D: $\\frac{y}{x^{2}}$"
] | A | ||
AP | 20 | calculus_bc | 2023_calculus_bc | Which of the following series converges to $\frac{1}{e}$ ? | [
"A: $\\sum_{n=0}^{\\infty} \\frac{1}{n!}$",
"B: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{n!}$",
"C: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n)!}$",
"D: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n+1}}{(2 n+1)!}$"
] | B | ||
AP | 21 | calculus_bc | 2023_calculus_bc | The function $g$ has a continuous second derivative. If $g(0)=2, g^{\prime}(0)=4$, $g(3)=5$ and $g^{\prime}(3)=6$, what is the value of $\int_{0}^{3} x \cdot g^{\prime \prime}(x) d x ?$ | [
"A: 9",
"B: 15",
"C: 21",
"D: 27"
] | B | ||
AP | 22 | calculus_bc | 2023_calculus_bc | The Maclaurin series for the function $f$ is given below. What is the value of $f^{(4)}(0)$ ? $$ \sum_{n=1}^{\infty} \frac{(-2)^{n} x^{2 n}}{n} $$ | [
"A: 2",
"B: 4",
"C: 8",
"D: 48"
] | D | ||
AP | 23 | calculus_bc | 2023_calculus_bc | Let $f$ be a differentiable function such that $f(\pi)=2$ and $f^{\prime}(\pi)=-3$. What is the slope of the line tangent to the polar curve $r=f(\theta)$ at $\theta=\pi$ ? | [
"A: -3",
"B: -2",
"C: $-\\frac{3}{2}$",
"D: $-\\frac{2}{3}$"
] | D | ||
AP | 24 | calculus_bc | 2023_calculus_bc | $\int_{-\infty}^{1} x^{2} 2^{x^{3}} d x$ is | [
"A: $\\frac{6}{\\ln 2}$",
"B: $\\frac{2}{3 \\ln 2}$",
"C: $\\frac{2 \\ln 2}{3}$",
"D: divergent"
] | B | ||
AP | 25 | calculus_bc | 2023_calculus_bc | $\int \frac{d x}{x^{2}+4 x+5}=$ | [
"A: $\\tan ^{-1}(x+2)+C$",
"B: $\\ln \\left|x^{2}+4 x+5\\right|+C$",
"C: $\\frac{1}{2 x+4} \\ln \\left|x^{2}+4 x+5\\right|+C$",
"D: $\\frac{1}{\\frac{1}{3} x^{3}+2 x^{2}+5 x}+C$"
] | A | ||
AP | 26 | calculus_bc | 2023_calculus_bc | Let $f$ be a function defined by $f(x)=x^{3}+x$. If $g$ is the reverse function of $f$, what is the slope of the line tangent to the graph of $g$ at the point $(2,1)$ ? | [
"A: $-\\frac{1}{4}$",
"B: $-\\frac{1}{13}$",
"C: $\\frac{1}{13}$",
"D: $\\frac{1}{4}$ \\begin{center} \\begin{tabular}{|c|c|c|c|c|} \\hline $x$ & 0 & 2 & 3 & 5 \\hline $f(x)$ & 2 & 3 & 6 & 12 \\hline \\end{tabular} \\end{center} \\begin{enumerate}"
] | D | ||
AP | 27 | calculus_bc | 2023_calculus_bc | item The function $f$ is continuous on the closed interval $[0,5]$ and differentiable on the open interval $(0,5)$. Selected values of $f$ are given in the table shown. The value $c=$ 4 satisfies the conclusion of the Mean Value Theorem for $f$ on the closed interval $[0,5]$. What is the slope of the line tangent to gr... | [
"A: 2",
"B: 2.8",
"C: 4.5",
"D: 10"
] | A | ||
AP | 28 | calculus_bc | 2023_calculus_bc | item Let $f$ be a function with $f(0)=0$ and derivative given by $f^{\prime}(x)=\cos \left(x^{2}\right)$. Which of the following is the Maclaurin series for $f$ ? | [
"A: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n} x^{2 n}}{(2 n)!}$",
"B: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n} x^{4 n}}{(2 n)!}$",
"C: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n} x^{4 n+1}}{(4 n+1)(2 n)!}$",
"D: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}(4 n) x^{4 n-1}}{(2 n)!}$ \\includegraphics[max width=\\textwidth, cen... | C | ||
AP | 29 | calculus_bc | 2023_calculus_bc | item Let $R$ be the region bounded by the graph of $y=\ln x$, the vertical line $x=e^{2}$, and the $x$-axis, as shown in the figure. Which of the following integrals are equal to the area of $R$ ? I. $\int_{1}^{e^{2}} \ln x d x$ II. $\int_{0}^{2} e^{y} d y$ III. $\int_{0}^{2}\left(e^{2}-e^{y}\right) d y$ | [
"A: I only",
"B: III only",
"C: I and II only",
"D: I and III only"
] | D | ||
AP | 30 | calculus_bc | 2023_calculus_bc | \item $\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\frac{3}{n} k\right)^{2}\left(\frac{3}{n}\right)$ is | [
"A: 1",
"B: 3",
"C: 9",
"D: nonexistent"
] | C | ||
AP | 31 | calculus_bc | 2023_calculus_bc | item If $f^{\prime}(x)=1+2 x-x^{3}-\sqrt{x^{4}+1}$, then $f$ has a local minimum at $x=$ | [
"A: -1.836",
"B: -1.117",
"C: 0",
"D: 1.181"
] | C | ||
AP | 32 | calculus_bc | 2023_calculus_bc | item The function $f$ has derivatives of all orders for all real numbers. The table shown gives values of $f$ and its first three derivatives at selected values of $x$. What is the value of $\int_{-1}^{1} f^{\prime \prime}(x) d x$ ? | [
"A: -36",
"B: -10",
"C: -8",
"D: 48"
] | A | ||
AP | 33 | calculus_bc | 2023_calculus_bc | item On a certain day, an industrial plant manufactures steel at a rate modeled by $R(t)= 10+30 e^{-0.6 t}$ tons per hour, where $t$ is measured in hours since the plant opened that day. What is the average rate at which the plant manufactures steel, in tons per hour, over the time interval $0 \leq t \leq 5$ ? | [
"A: 11.494",
"B: 19.502",
"C: 25.747",
"D: 97.511"
] | C | ||
AP | 34 | calculus_bc | 2023_calculus_bc | item The power series $\sum_{n=0}^{\infty} a_{n}(x+1)^{n}$ converges at $x=3$. Which of the following must be true? | [
"A: $\\sum_{n=0}^{\\infty} a_{n}(x+1)^{n}$ converges at $x=-5$",
"B: $\\sum_{n=0}^{\\infty} a_{n}(x+1)^{n}$ diverges at $x=-5$",
"C: $\\sum_{n=0}^{\\infty} a_{n}(x+1)^{n}$ converges at $x=-4$",
"D: $\\sum_{n=0}^{\\infty} a_{n}(x+1)^{n}$ diverges at $x=4$"
] | B | ||
AP | 35 | calculus_bc | 2023_calculus_bc | item A thin rectangular sheet of metal expands in a hot oven. At the instant when the width of the rectangular sheet of metal is 200 millimeters and the length is 300 millimeters, the width is increasing at a rate of 2 millimeters per minute, and the length is increasing at a rate of 3 millimeters per minute. What is t... | [
"A: 5",
"B: 6",
"C: 600",
"D: 1200"
] | C | ||
AP | 36 | calculus_bc | 2023_calculus_bc | item A particle travels along $x$-axis so that its velocity at time $t \geq 0$ is given by $\frac{d y}{d x}=\sin \left(t^{2}\right)$. What is the total distance the particle travels from $t=0$ to $t=2.7$ ? | [
"A: 0.519",
"B: 0.845",
"C: 1.448",
"D: 2.887"
] | D | ||
AP | 37 | calculus_bc | 2023_calculus_bc | item The function $f$ is continuous on the closed interval $[a, b]$, where $a$ and $b$ are constants and $a<b$. If $f(a)<0<f(b)$, which of the following statements must be true? | [
"A: There is at least one value of $x$ in the interval $(a, b)$ such that $f(x)=0$",
"B: There is at least one value of $x$ in the interval $(a, b)$ such that $f^{\\prime}(x)=\\frac{f(b)-f(a)}{b-a}$",
"C: $f^{\\prime}(x)>0$ for all $x$ in the interval $(a, b)$",
"D: $f(a)<f(x)<f(b)$ for all $x$ in the interva... | C | ||
AP | 38 | calculus_bc | 2023_calculus_bc | item The alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2 n+2}$ converges to $S$. If the sum of the first six terms of the series is used to approximate $S$. What is the alternating series error bound? | [
"A: $1 / 6$",
"B: 1/7",
"C: $1 / 14$",
"D: $1 / 16$ \\includegraphics[max width=\\textwidth, center]{2025_09_09_3b57e8501b5d35d96302g-14}"
] | A | ||
AP | 39 | calculus_bc | 2023_calculus_bc | item The graph of a twice-differentiable function $f$ is shown for $0<x<4$. The graph of $f$ has a horizontal tangent at $x=2$ and points of inflection at $x=1$ and $x=3$. On what intervals do $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ have the same sign? | [
"A: $0<x<1$ only",
"B: $2<x<4$",
"C: $0<x<1$ and $2<x<3$",
"D: $1<x<2$ and $3<x<4$"
] | D | ||
AP | 40 | calculus_bc | 2023_calculus_bc | item The table shown gives values of the twice- differentiable function $f$ and $g$ and their derivatives $f^{\prime}$ and $g^{\prime}$ at selected values of $x$. What is $\lim _{x \rightarrow 2} \frac{g\left(x^{2}\right)}{3 x+f(x)}$ ? | [
"A: $\\frac{1}{2}$",
"B: $\\frac{20}{13}$",
"C: 2",
"D: The limit does not exist."
] | C | ||
AP | 41 | calculus_bc | 2023_calculus_bc | item The velocity of a particle moving along the $x$-axis is given by $v(t)=\frac{1-t^{3}}{1+t}$ for time $t \geq 0$. Which of the following statements is true about the particle at time $t=$ 1.5? | [
"A: The acceleration is negative, and the speed is decreasing",
"B: The acceleration is negative, and the speed is increasing",
"C: The acceleration is positive, and the speed is decreasing",
"D: The acceleration is positive, and the speed is increasing \\includegraphics[max width=\\textwidth, center]{2025_09... | C | ||
AP | 42 | calculus_bc | 2023_calculus_bc | item The figure shows the graphs of the polar curve $r=4(\sin (2 \theta))^{2}$ and $r=1$ for $0 \leq \theta \leq \frac{\pi}{2}$. What is the area of the shaded region? | [
"A: 8.267",
"B: 4.134",
"C: 3.927",
"D: 1.913"
] | B | ||
AP | 43 | calculus_bc | 2023_calculus_bc | item The graph of the function $f$ shown consists of a semicircle centered at the origin and three line segments. If $g(x)=\int_{-2}^{x}(f(t)+1) d t$, then $g(8)=$ | [
"A: $2 \\pi+6$",
"B: $2 \\pi+10$",
"C: $2 \\pi+15$",
"D: $2 \\pi+19$"
] | B | ||
AP | 44 | calculus_bc | 2023_calculus_bc | item The position of a particle moving in the $x y$-plane is given by the parametric equation $x(t)=5 \ln (t+5)$ and $y(t)=5 \sin (\pi t)$. At any point on the curve, the slope of the line tangent to the path of the particle is given by $\frac{d y}{d x}=\pi \cos (\pi t)(t+5)$. What is the value of $\frac{d^{2} y}{d x^{... | [
"A: -18.850",
"B: -3.770",
"C: -3.142",
"D: 0.200"
] | C | ||
AP | 1 | calculus_bc | 2024_calculus_bc | item $\frac{d}{d x}\left(\left(x^{2}-11 x+24\right)^{7}\right)=$ | [
"A: $7(2 x-11)^{6}$",
"B: $7(2 x-11)^{6}\\left(x^{2}-11 x+24\\right)$",
"C: $7\\left(x^{2}-11 x+24\\right)^{6}$",
"D: $7\\left(x^{2}-11 x+24\\right)^{6}(2 x-11)$"
] | D | ||
AP | 2 | calculus_bc | 2024_calculus_bc | item If the function $f$ is defined as shown, then $f$ is differentiable for | [
"A: all values of $x$",
"B: all values of $x$ except $x=0$",
"C: all values of $x$ except $x=2$",
"D: all values of $x$ except $x=0$ and $x=2$ \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-03}"
] | C | ||
AP | 3 | calculus_bc | 2024_calculus_bc | The graph of $f^{\prime}$ ，the derivative of the function $f$ ，is shown．Which of the following could be the graph of $f$ ？ | [
"A: \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-03(3)}",
"B: \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-03(2)}",
"C: \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-03(4)}",
"D: \\includegraphics[max... | B | ||
AP | 4 | calculus_bc | 2024_calculus_bc | If $a$ is a nonzero constant，then $\lim _{x \rightarrow a} \frac{x-a}{a^{2}-x^{2}}$ is | [
"A: 0",
"B: $-\\frac{1}{2 a}$",
"C: $-\\frac{1}{a}$",
"D: nonexistent \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-05}"
] | B | ||
AP | 5 | calculus_bc | 2024_calculus_bc | The figure shown is a slope field for which of the following differential equations？ | [
"A: $\\frac{d y}{d x}=\\frac{y^{2}}{(x-2)^{2}}$",
"B: $\\frac{d y}{d x}=\\frac{1}{(x-2)^{3}}$",
"C: $\\frac{d y}{d x}=\\frac{1}{(x+2)^{2}}$",
"D: $\\frac{d y}{d x}=\\frac{1}{(x-2)^{2}}$"
] | D | ||
AP | 6 | calculus_bc | 2024_calculus_bc | For $x \geq 0$ ，the function $g$ is continuous and $g(x)>0$ ．If the area bounded by the graph of $y=g(x)$ and the $x$－axis from $x=0$ to $x=2$ is 3 ，what is the average value of $g$ on the interval $[0,2]$ ？ | [
"A: $\\frac{2}{3}$",
"B: $\\frac{3}{2}$",
"C: 2",
"D: 3 \\begin{center} \\begin{tabular}{|c||c|c|c|c|c|c|} \\hline $x$ & 0 & 1 & 2 & 3 & 4 & 5 \\hline $f^{\\prime}(x)$ & 1 & -2 & 3 & 0 & 5 & 0 \\hline \\end{tabular} \\end{center}"
] | B | ||
AP | 7 | calculus_bc | 2024_calculus_bc | Let $f$ be a twice－differentiable function．Selected values of $f^{\prime}$ ，the first derivative of $f$ ，are given in the table． On which of the following intervals is $f$ guaranteed to attain a local maximum？ | [
"A: $(0,1)$",
"B: $(1,2)$",
"C: $(2,4)$",
"D: $(3,5)$"
] | D | ||
AP | 8 | calculus_bc | 2024_calculus_bc | $\lim _{h \rightarrow 0} \frac{e^{-(1+h)}-e^{-(1)}}{h}$ is | [
"A: $-\\frac{1}{e}$",
"B: 0",
"C: $\\frac{1}{e}$",
"D: nonexistent"
] | A | ||
AP | 9 | calculus_bc | 2024_calculus_bc | Let $f$ be the function defined by $f(x)=\sin \left(x^{2}\right)$ ．What are the first three nonzero terms of the Maclaurin series for $f^{\prime}$ ，the derivative of $f$ ？ | [
"A: $-2 x^{3}+\\frac{x^{7}}{3}-\\frac{x^{11}}{60}$",
"B: $1-\\frac{x^{4}}{2}+\\frac{x^{8}}{24}$",
"C: $2 x-x^{5}+\\frac{x^{9}}{12}$",
"D: $2 x+2 x^{3}+x^{5}$"
] | B | ||
AP | 10 | calculus_bc | 2024_calculus_bc | For time $t>0$ ，the position of a particle moving along a curve is given by $(x(t), y(t))$ ，where $x(t)$ and $y(t)$ are differentiable functions of $t$ ．An equation of the line tangent to the curve at the point where $t=1$ is $y=4+3 x$ ． If $x^{\prime}(t)=12 \sin \left(\frac{\pi t}{2}\right)$ ，what is the value of $y^{... | [
"A: 3",
"B: 4",
"C: 36",
"D: 40"
] | C | ||
AP | 11 | calculus_bc | 2024_calculus_bc | A cell phone was plugged into a charger．The battery life of the cell phone，in percent of battery life remaining， can be modeled by a differentiable function $P(t)$ ，where $t$ is the number of minutes after the phone was plugged into the charger and $0 \leq t \leq 20$ ．Which of the following is the best interpretation o... | [
"A: Ten minutes after the phone was plugged into the charger，the battery life was changing at a rate of 1.5 percent per minute．",
"B: Ten minutes after the phone was plugged into the charger，the rate of change in the battery life was changing by 1.5 percent per minute per minute．",
"C: During the first ten minu... | B | ||
AP | 12 | calculus_bc | 2024_calculus_bc | What are all values of $x$ for which the series $\sum_{n=0}^{\infty}(-1)^{n}(3 x+1)^{n}$ converges？ | [
"A: $-\\frac{1}{3} \\leq x<\\frac{1}{3}$",
"B: $-\\frac{1}{3}<x<\\frac{1}{3}$",
"C: $-\\frac{2}{3} \\leq x<0$",
"D: $-\\frac{2}{3}<x<0$"
] | D | ||
AP | 13 | calculus_bc | 2024_calculus_bc | For $-4 \leq x \leq 6$ ，the graph of the function $f$ is piecewise linear，as shown．If $g(x)=\int_{0}^{x} f(t) d t$ ，what values of $x$ in the closed interval $-4 \leq x \leq 6$ satisfy the equation $g(x)=0 ?$ | [
"A: 0 only",
"B: 0，2 and 6 only",
"C: －2，1，and 4",
"D: $-4,-2,0,2$ ，and 6"
] | B | ||
AP | 14 | calculus_bc | 2024_calculus_bc | A particle moves in the $x y$－plane so that its position at any time $t,-10 \leq t \leq 10$ ，is given by the parametric equations $x(t)=t^{4}-4 t^{3}+3$ and $y(t)=t^{3}-27 t+26$ ．For what values of $t$ is the particle at rest？ | [
"A: 3 only",
"B: －3 and 3 only",
"C: 0 and 3 only",
"D: $-3,0$ ，and 3"
] | A | ||
AP | 15 | calculus_bc | 2024_calculus_bc | Let $f$ and $g$ be twice－differentiable functions such that $\lim _{x \rightarrow \infty} f(x)=\infty, \lim _{x \rightarrow \infty} g(x)=\infty, \lim _{x \rightarrow \infty} f^{\prime}(x)=8$ ，and $\lim _{x \rightarrow \infty} g^{\prime}(x)=2$ ．What is $\lim _{x \rightarrow \infty}\left(\frac{f(x)}{g(x)}+6\right)$ ？ | [
"A: 4",
"B: 7",
"C: 10",
"D: nonexistent"
] | C | ||
AP | 16 | calculus_bc | 2024_calculus_bc | Let $y=f(x)$ be the solution to the differential equation $\frac{d y}{d x}=2 x(y-x)$ with the initial condition $f(1)=0$ ． What is the approximation for $f(0)$ if Euler＇s method is used，starting at $x=1$ with two steps of equal size？ | [
"A: $-\\frac{9}{2}$",
"B: $-\\frac{7}{4}$",
"C: $\\frac{3}{4}$",
"D: 2"
] | B | ||
AP | 17 | calculus_bc | 2024_calculus_bc | item $\int_{3}^{5} \frac{x}{x^{2}-9} d x$ is | [
"A: $\\frac{1}{2} \\ln \\left(\\frac{5}{3}\\right)$",
"B: $\\ln \\left(\\frac{5}{3}\\right)$",
"C: $\\frac{1}{2} \\ln (16)$",
"D: divergent \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-13(3)}"
] | D | ||
AP | 18 | calculus_bc | 2024_calculus_bc | The figure shows the graph of the function $f$ ．If $g(x)=\int_{1}^{x} f(t) d t$ ，which of the following could be the graph of $g$ ？ | [
"A: \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-13(1)}",
"B: \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-13(4)}",
"C: \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-13(2)}",
"D: \\includegraphics[max... | D | ||
AP | 19 | calculus_bc | 2024_calculus_bc | Which of the following is the solution to the differential equation $\frac{d y}{d x}=x(y-2)$ with the initial condition $y(2)=3$ ？ | [
"A: $y=2-e^{\\left(x^{2} / 2\\right)-2}$",
"B: $y=2+e^{\\left(x^{2} / 2\\right)-2}$",
"C: $y=2+e^{2-\\left(x^{2} / 2\\right)}$",
"D: $y=3-e^{2}+e^{x^{2} / 2}$"
] | A | ||
AP | 20 | calculus_bc | 2024_calculus_bc | $\int_{0}^{\sqrt{\pi} / 2} x \cos \left(x^{2}\right) d x=$ | [
"A: $-\\frac{\\sqrt{2}}{2}$",
"B: $-\\frac{\\sqrt{2}}{4}$",
"C: $\\frac{\\sqrt{2}}{4}$",
"D: $\\frac{\\sqrt{2}}{2}$"
] | C | ||
AP | 21 | calculus_bc | 2024_calculus_bc | The first derivative of the function $f$ is given by $f^{\prime}(x)=4 x-8$ ．At what value of $x$ does $f$ attain an absolute maximum on the closed interval $[0,3]$ ？ | [
"A: 0",
"B: $\\frac{3}{2}$",
"C: 2",
"D: 3"
] | D | ||
AP | 22 | calculus_bc | 2024_calculus_bc | $\int x^{2} e^{2 x} d x=$ | [
"A: $\\frac{1}{2} x^{2} e^{2 x}-\\frac{1}{2} x e^{2 x}+\\frac{1}{4} e^{2 x}+C$",
"B: $\\frac{1}{2} x^{2} e^{2 x}+\\frac{1}{2} x e^{2 x}+\\frac{1}{4} e^{2 x}+C$",
"C: $x^{2} e^{2 x}-2 x e^{2 x}+2 e^{2 x}+C$",
"D: $2 x^{2} e^{2 x}-8 x e^{2 x}+16 e^{2 x}+C$"
] | D | ||
AP | 23 | calculus_bc | 2024_calculus_bc | Let $f$ be the function defined as shown．For what value of $b$ is $f$ continuous at $x=0$ ？ | [
"A: 0",
"B: $\\frac{1}{5}$",
"C: 1",
"D: 5 $$ f(0)=0 \\quad \\lim _{x \\rightarrow-1} f(x)=\\frac{2}{5} \\quad \\lim _{x \\rightarrow-1} \\frac{f(x)-f(-1)}{x+1}=-\\frac{3}{5} $$"
] | C | ||
AP | 24 | calculus_bc | 2024_calculus_bc | The function $f$ satisfies the properties shown．What is the slope of the line tangent to the graph of $f$ at $x=-1$ ？ | [
"A: $-\\frac{3}{5}$",
"B: $-\\frac{2}{5}$",
"C: $\\frac{2}{5}$",
"D: 1"
] | A | ||
AP | 25 | calculus_bc | 2024_calculus_bc | Which of the following series diverges by the $n$th term test？ | [
"A: $\\sum_{n=1}^{\\infty} \\frac{n^{2}+\\sin n}{3 n^{2}+1}$",
"B: $\\sum_{n=1}^{\\infty} \\frac{\\cos (n \\pi)}{n}$",
"C: $\\sum_{n=1}^{\\infty} \\frac{n^{2}}{n^{3}+1}$",
"D: $\\sum_{n=1}^{\\infty} \\frac{1}{3 n+1}$"
] | A | ||
AP | 26 | calculus_bc | 2024_calculus_bc | $\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\frac{1}{2+\frac{k}{2 n}} \cdot \frac{1}{n}\right)=$ | [
"A: $\\int_{0}^{1 / 2} \\frac{1}{2+\\frac{x}{2}} d x$",
"B: $\\int_{0}^{1 / 2} \\frac{1}{2+x} d x$",
"C: $\\int_{0}^{1} \\frac{1}{2+\\frac{x}{2}} d x$",
"D: $\\int_{0}^{1} \\frac{1}{2+x} d x$"
] | C | ||
AP | 27 | calculus_bc | 2024_calculus_bc | Using the comparison test, to which of the following series should the series $\sum_{k=1}^{\infty} \frac{2+\cos k}{k^{3}}$ be compared to determine its convergence or divergence? | [
"A: $\\sum_{k=1}^{\\infty} \\frac{3}{k}$",
"B: $\\sum_{k=1}^{\\infty} \\frac{2}{k^{3}}$",
"C: $\\sum_{k=1}^{\\infty} \\frac{3}{k^{3}}$",
"D: $\\sum_{k=1}^{\\infty} \\frac{1}{k^{4}}$"
] | B | ||
AP | 28 | calculus_bc | 2024_calculus_bc | Let $g$ be the function defined by $g(x)=e^{x^{3}}$ ．Using the Maclaurin series for $e^{x}$ ，what is $g^{(21)}(0)$ ，the twenty－first derivative of $g$ at $x=0$ ？ | [
"A: $\\frac{1}{21!}$",
"B: $\\frac{1}{7!}$",
"C: 1",
"D: $\\frac{21!}{7!}$"
] | C | ||
AP | 29 | calculus_bc | 2024_calculus_bc | If $f$ is the function defined by $f(x)=\int_{5}^{x^{3}} \tan (t) d t$, then $f^{\prime}(x)=$ | [
"A: $\\sec ^{2}\\left(x^{3}\\right)$",
"B: $\\tan \\left(x^{3}\\right)$",
"C: $3 x^{2} \\sec ^{2}\\left(x^{3}\\right)$",
"D: $3 x^{2} \\tan \\left(x^{3}\\right)$"
] | D | ||
AP | 30 | calculus_bc | 2024_calculus_bc | $\int \frac{1}{x^{2}-4 x+3} d x=$ | [
"A: $\\ln \\left|x^{2}-4 x+3\\right|+C$",
"B: $\\ln |x-3|-\\ln |x-1|+C$",
"C: $\\frac{1}{2} \\ln |x-3|-\\frac{1}{2} \\ln |x-1|+C$",
"D: $2 \\ln |x-3|-2 \\ln |x-1|+C$"
] | C | ||
AP | 31 | calculus_bc | 2024_calculus_bc | In a community of 6000 people，the number of people informed about an upcoming event after $t$ weeks is modeled by a function $P(t)$ that satisfies a logistic differential equation．When 1000 people are informed of the event，the rate of growth of informed people is 100 people per week．If all people in the community event... | [
"A: $\\frac{d P}{d t}=0.12 t\\left(1-\\frac{t}{1000}\\right)$",
"B: $\\frac{d P}{d t}=0.12 t\\left(1-\\frac{t}{6000}\\right)$",
"C: $\\frac{d P}{d t}=0.12 P\\left(1-\\frac{P}{5000}\\right)$",
"D: $\\frac{d P}{d t}=0.12 P\\left(1-\\frac{P}{6000}\\right)$"
] | D | ||
AP | 32 | calculus_bc | 2024_calculus_bc | The graphs of the functions $f$ and $g$ are shown in the figure．The function $f$ has a removable discontinuity， and the function $g$ is continuous．What is $\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}$ ？ | [
"A: -4",
"B: -2",
"C: $-\\frac{1}{2}$",
"D: Nonexistent"
] | B | ||
AP | 33 | calculus_bc | 2024_calculus_bc | Let $f$ be the function with first derivative given by $f^{\prime}(x)=x^{2} \sin \left(e^{x}\right)$ ．How many relative extrema does $f$ have on the open interval $-4<x<2$ ？ | [
"A: One",
"B: Two",
"C: Three",
"D: Four \\begin{center} \\begin{tabular}{|c|c|c|} \\hline $x$ & $h(x)$ & $h^{\\prime}(x)$ \\hline 2 & -3 & 5 \\hline 4 & 7 & -1 \\hline \\end{tabular} \\end{center}"
] | C | ||
AP | 34 | calculus_bc | 2024_calculus_bc | The table gives values of a function $h$ and its derivative at selected values of $x$ ．If $f$ is the function defined by $f(x)=h(2 x)+3$ ，what is the value of $f^{\prime}(2)$ ？ | [
"A: -2",
"B: -1",
"C: 1",
"D: 10 \\includegraphics[max width=\\textwidth, center]{2025_09_10_41bb88c9cee3557b5381g-28}"
] | A | ||
AP | 35 | calculus_bc | 2024_calculus_bc | The rate of change of the water level in a lake at time $t$ is modeled by the function $r$ ，where $r(t)$ is measured in feet per month and $t$ is measured in months．The graph of $r$ is shown for $0 \leq t \leq 12$ ．At what times $t$ ， for $0<t \leq 12$ ，is the water level in the lake equal to the water level at time $t... | [
"A: $t=5$ only",
"B: $t=5$ and $t=12$",
"C: $t=2, t=5$ ，and $t=8$",
"D: $t=1$ and $t=6$"
] | B | ||
AP | 36 | calculus_bc | 2024_calculus_bc | What is the length of the curve defined by the parametric equations $x(t)=t^{3}+t$ and $y(t)=t^{2}-6 t+1$ from $t=0$ to $t=4$ ？ | [
"A: 78.063",
"B: 72.217",
"C: 68.469",
"D: 49.041"
] | B | ||
AP | 37 | calculus_bc | 2024_calculus_bc | The velocity of a particle moving along the $x$－axis is given by $v(t)=t^{2}-9-4 \sin \left(t^{2}-9\right)$ for $t \geq 0$ ．At what time $t$ ，for $0 \leq t \leq 3$ ，is the particle farthest to the left？ | [
"A: 1.183",
"B: 2.009",
"C: 2.554",
"D: 2.772"
] | C | ||
AP | 38 | calculus_bc | 2024_calculus_bc | Bacteria are small，single－celled organisms．The weight of a population of bacteria，in grams，is modeled by the function $w(t)=0.1\left(t^{2}+3\right) e^{-0.2 t}$ ，where $t$ is measured in hours since the beginning of an experiment．What is the average weight of the population，in grams，during the first 3 hours of the exper... | [
"A: 0.120",
"B: 0.418",
"C: 0.479",
"D: 1.255"
] | B | ||
AP | 39 | calculus_bc | 2024_calculus_bc | What is the area of the region bounded by the polar curve $r=0.45 \theta+0.2 \cos (0.5 \theta)$ for $0 \leq \theta \leq \pi$ and the $x$－axis？ | [
"A: 1.283",
"B: 1.310",
"C: 2.567",
"D: 2.621"
] | B | ||
AP | 40 | calculus_bc | 2024_calculus_bc | Let $f$ be a function such that $f(1)=2$ and the $n$th derivative of $f$ at $x=1$ is given by $f^{(n)}(1)=\frac{-n}{(2 n-1)^{2}}$ ． What is the estimate for $f(2)$ using the third－degree Taylor polynomial for $f$ about $x=1$ ？ | [
"A: $-\\frac{8}{9}-\\frac{24}{25}$",
"B: $-\\frac{4}{9}-\\frac{4}{25}$",
"C: $1-\\frac{2}{9}-\\frac{3}{25}$",
"D: $1-\\frac{1}{9}-\\frac{1}{50}$"
] | D | ||
AP | 41 | calculus_bc | 2024_calculus_bc | The function $f$ has first derivative $f^{\prime}$ defined by $f^{\prime}(x)=x^{2} \tan (1-2 x)$ for $-0.25<x<0.75$ ．At which of the following values of $x$ does the graph of $f$ have a point of inflection？ | [
"A: $x=0.324$ only",
"B: $x=0.5$ only",
"C: $x=0$ and $x=0.324$",
"D: $x=0$ and $x=0.5$"
] | C | ||
AP | 42 | calculus_bc | 2024_calculus_bc | At time $t \geq 0$ ，a particle moving in the $x y$－plane has a velocity vector given by $\langle\sin (2 t), 2+\cos (3 t)\rangle$ ．What is the total distance the particle travels from time $t=0$ to time $t=1$ ？ | [
"A: 1.655",
"B: 2.226",
"C: 2.432",
"D: 5.259 \\begin{center} \\begin{tabular}{|c|c|c|c|c|} \\hline $n$ & 2 & 3 & 4 & 5 \\hline $a_{n}$ & 0.2 & 0.1835 & 0.1715 & 0.1622 \\hline \\end{tabular} \\end{center}"
] | B | ||
AP | 43 | calculus_bc | 2024_calculus_bc | The series $\sum_{n=1}^{\infty}(-1)^{n} a_{n}$ converges to $S$ ，and $a_{n}>a_{n+1}>0$ for all $n \geq 1$ ．Selected values of $a_{n}$ are shown in the table．What is the smallest value of $k$ for which the alternating series error bound guarantees that $\left|S-\sum_{n=1}^{k}(-1)^{n} a_{n}\right|<0.1667 ?$ | [
"A: 2",
"B: 3",
"C: 4",
"D: 5"
] | C | ||
AP | 44 | calculus_bc | 2024_calculus_bc | Which of the following power series has an interval of convergence of $0<x \leq 2$ ？ | [
"A: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n+1}(x-1)^{n+1}}{n+1}$",
"B: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n+1}(x-1)^{n+2}}{(n+1)(n+2)}$",
"C: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n+1}(x+1)^{n+1}}{n+1}$",
"D: $\\sum_{n=0}^{\\infty} \\frac{(-1)^{n+1}(x+1)^{n+2}}{(n+1)(n+2)}$"
] | A | ||
AP | 45 | calculus_bc | 2024_calculus_bc | Which of the following limits of a Riemann sum gives the value of the definite integral $\int_{-2}^{3}\left(x^{2}+4 x\right) d x$ ？ | [
"A: $\\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} \\frac{3}{n}\\left(-2+\\left(\\frac{3}{n} k\\right)^{2}+4\\left(\\frac{3}{n} k\\right)\\right)$",
"B: $\\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} \\frac{3}{n}\\left(\\left(-2+\\frac{3}{n} k\\right)^{2}+4\\left(-2+\\frac{3}{n} k\\right)\\right)$",
"C: $\\... | D |
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