2023 AP Calculus BC Free-Response Questions
Advanced PlacementAP Calculus BCFree ResponseMathematics
6 questions54 pts
6 gradable54 rubric criteria0 dependencies
q1
Question 1
t (seconds) 0 60 90 120 135 150 f(t) (gallons per second) 0 0.1 0.15 0.1 0.05 0 1. A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function f, where f(t) is measured in gallons per second and t is measured in seconds since pumping began. Selected values of f(t) are given in the table. (a) Using correct units, interpret the meaning of $\int_{60}^{135} f(t)\,dt$ in the context of the problem. Use a right Riemann sum with the three subintervals [60, 90], [90, 120], and [120, 135] to approximate the value of $\int_{60}^{135} f(t)\,dt$. (b) Must there exist a value of c, for $60<c<120$, such that $f'(c)=0$? Justify your answer. (c) The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t)=\left(\frac{t}{500}\right)\cos\left(\left(\frac{t}{120}\right)^2\right)$ for $0\le t\le 150$. Using this model, find the average rate of flow of gasoline over the time interval $0\le t\le 150$. Show the setup for your calculations. (d) Using the model $g$ defined in part (c), find the value of $g'(140)$. Interpret the meaning of your answer in the context of the problem. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.
Rubric Criteria (9)
C1The response interprets $\int_{60}^{135} f(t)\,dt$ as the total number of gallons of gasoline pumped into the gas tank from time $t=60$ seconds to time $t=135$ seconds, referencing both the quantity (gallons) and the time interval.
1.0000pts
C2The response presents a right Riemann sum using the subintervals $[60,90]$, $[90,120]$, and $[120,135]$ that contains at least five of the six correct factors, or presents the three correct products $(0.15)(30)+(0.1)(30)+(0.05)(15)$; a completely correct left Riemann sum with accompanying work also earns this point.
1.0000pts
C3The response gives the correct value $8.25$ (or equivalent) for the right Riemann sum approximation, supported by work that shows either all six factors, the three correct products, or the equation $f(90)(30)+f(120)(30)+f(135)(15)=8.25$; an answer of only $8.25$ with no supporting work does not earn this point.
1.0000pts
C4The response presents $f(120)-f(60)=0$, $0.1-0.1=0$ (as a numerator or difference), or $f(60)=f(120)$.
1.0000pts
C5The response answers that there is such a $c$ ("yes" or equivalent), states that $f$ is continuous on $[60,120]$ because $f$ is differentiable (or equivalent), and references the Mean Value Theorem or Rolle's Theorem.
1.0000pts
C6The response presents the average value formula $\frac{1}{150}\int_0^{150} g(t)\,dt$ (or equivalent correct setup for the average rate of flow over $0\leq t\leq 150$).
1.0000pts
C7The response gives the average rate of flow as approximately $0.096$ (or $0.095$ or $0.0959967$) or as the exact equivalent $\frac{12}{125}\sin\left(\frac{25}{16}\right)$.
1.0000pts
C8The response presents a numerical value for $g'(140)$, such as approximately $-0.004908$, $-0.005$, or $-0.004$ (or exact equivalent).
1.0000pts
C9The response interprets the numerical value of $g'(140)$ as the rate at which the rate of flow of gasoline is changing at $t=140$ seconds, using the declared value and correct units, and correctly states that the flow is decreasing when the value is negative.
1.0000pts
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q2
Question 2
2. For $0 \leq t \leq \pi$, a particle is moving along the curve shown so that its position at time $t$ is $(x(t), y(t))$, where $x(t)$ is not explicitly given and $y(t) = 2\sin t$. It is known that $\frac{dx}{dt} = e^{\cos t}$. At time $t = 0$, the particle is at position $(1, 0)$. (a) Find the acceleration vector of the particle at time $t = 1$. Show the setup for your calculations. (b) For $0 \leq t \leq \pi$, find the first time $t$ at which the speed of the particle is 1.5. Show the work that leads to your answer. (c) Find the slope of the line tangent to the path of the particle at time $t = 1$. Find the $x$-coordinate of the position of the particle at time $t = 1$. Show the work that leads to your answers. (d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq \pi$. Show the setup for your calculations. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.
Rubric Criteria (9)
C1The response presents $x''(1)$ with correct setup, such as $\frac{d}{dt}(e^{\cos t})\big|_{t=1}$, evaluating to approximately $-1.444$ (or exact $-e^{\cos 1}\sin 1$).
1.0000pts
C2The response presents $y''(1)$ with correct setup, such as $\frac{d}{dt}(2\cos t)\big|_{t=1}$, evaluating to approximately $-1.683$ (or $-1.682$) (or exact $-2\sin 1$).
1.0000pts
C3The response presents a correct speed equation equating $\sqrt{(e^{\cos t})^2+(2\cos t)^2}$ to $1.5$ (or equivalent correct setup using $\frac{dx}{dt}$ and $\frac{dy}{dt}$) for $0\leq t\leq \pi$; a response with only a parenthesis error in the squared terms is not eligible for this point.
1.0000pts
C4The response gives the first time $t\approx 1.254$ (or $1.254472$) at which the speed is $1.5$.
1.0000pts
C5The response presents $\frac{dy}{dx}\big|_{t=1}$ with supporting work, such as $\frac{2\cos 1}{e^{\cos 1}}\approx 0.630$ (or $0.629$), using $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$.
1.0000pts
C6The response presents the definite integral $\int_0^1 e^{\cos t}\,dt$ or $\int_0^1 \frac{dx}{dt}\,dt$; the integral written without the differential also earns this point.
1.0000pts
C7The response gives the $x$-coordinate at $t=1$ as approximately $3.342$ (or $3.341$), computed with the initial condition $x(0)=1$ and the definite integral.
1.0000pts
C8The response presents the definite integral $\int_0^{\pi}\sqrt{(e^{\cos t})^2+(2\cos t)^2}\,dt$ (or equivalent correct integrand for total distance).
1.0000pts
C9The response gives the total distance as approximately $6.035$ (or $6.034$).
1.0000pts
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q3
Question 3
3. A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function M models the temperature of the milk at time t, where M(t) is measured in degrees Celsius (°C) and t is the number of minutes since the bottle was placed in the pan. M satisfies the differential equation dM/dt = 1/4(40-M). At time t = 0, the temperature of the milk is 5°C. It can be shown that M(t) < 40 for all values of t. (a) A slope field for the differential equation dM/dt = 1/4(40-M) is shown. Sketch the solution curve through the point (0, 5). (b) Use the line tangent to the graph of M at t = 0 to approximate M(2), the temperature of the milk at time t = 2 minutes. (c) Write an expression for d²M/dt² in terms of M. Use d²M/dt² to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of M(2). Give a reason for your answer. (d) Use separation of variables to find an expression for M(t), the particular solution to the differential equation dM/dt = 1/4(40-M) with initial condition M(0) = 5. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part. © 2023 College Board. Visit College Board on the web: collegeboard.org.
Rubric Criteria (9)
C1The response sketches a solution curve through $(0,5)$ that extends reasonably close to the left and right edges of the slope field, has no obvious conflicts with the given slope lines, and lies entirely below $M=40$.
1.0000pts
C2The response presents $\frac{dM}{dt}\big|_{t=0}=\frac{35}{4}$ (or equivalent) with supporting work.
1.0000pts
C3The response presents a tangent-line approximation for $M(2)$ using a line through $(0,5)$ with slope $\frac{35}{4}$ (or a declared nonzero slope equal to $\frac{dM}{dt}\big|_{t=0}$), yielding $22.5$ (or unsimplified $5+\frac{35}{4}\cdot 2$).
1.0000pts
C4The response presents an expression for $\frac{d^2M}{dt^2}$ in terms of $M$, such as $-\frac{1}{16}(40-M)$ (or equivalent unsimplified form).
1.0000pts
C5The response concludes that the approximation is an overestimate and gives a valid reason, such as $\frac{d^2M}{dt^2}<0$ (or $\frac{dM}{dt}$ is decreasing, or the graph of $M$ is concave down) on the relevant domain.
1.0000pts
C6The response correctly separates variables to obtain $\frac{dM}{40-M}=\frac{1}{4}dt$ (or equivalent).
1.0000pts
C7The response finds correct antiderivatives yielding $-\ln|40-M|=\frac{1}{4}t+C$ (or equivalent).
1.0000pts
C8The response includes a constant of integration in an equation and correctly substitutes $t=0$ and $M=5$ to determine the constant (or equivalent work leading to the correct particular solution).
1.0000pts
C9The response solves for $M$ to obtain $M=40-35e^{-t/4}$ (or equivalent).
1.0000pts
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q4
Question 4
Graph of f′ 4. The function f is defined on the closed interval [−2, 8] and satisfies f(2) = 1. The graph of f′, the derivative of f, consists of two line segments and a semicircle, as shown in the figure. (a) Does f have a relative minimum, a relative maximum, or neither at x = 6 ? Give a reason for your answer. (b) On what open intervals, if any, is the graph of f concave down? Give a reason for your answer. (c) Find the value of $\lim_{x\to 2}\frac{6f(x)-3x}{x^{2}-5x+6}$ , or show that it does not exist. Justify your answer. (d) Find the absolute minimum value of f on the closed interval [−2, 8]. Justify your answer.
Rubric Criteria (9)
C1The response answers "neither" and gives a reason that $f'(x)$ does not change sign at $x=6$ (or equivalent, e.g., $f'(x)>0$ on both sides), with any presented intervals being correct.
1.0000pts
C2The response gives the intervals on which $f$ is concave down as $(-2,0)$ and $(4,6)$, possibly including endpoints.
1.0000pts
C3The response gives a correct reason based on the behavior of $f'$ (e.g., $f'$ is decreasing, or slopes of $f'$ are negative) on the identified intervals, and has earned the intervals point.
1.0000pts
C4The response presents the separate limits of the numerator and denominator as $x\to 2$, showing $\lim_{x\to 2}(6f(x)-3x)=0$ and $\lim_{x\to 2}(x^2-5x+6)=0$ (or equivalent).
1.0000pts
C5The response applies L'Hôpital's Rule by presenting at least one correct derivative in the limit of a ratio of derivatives, such as $\lim_{x\to 2}\frac{6f'(x)-3}{2x-5}$.
1.0000pts
C6The response gives the correct limit value $3$ with supporting work.
1.0000pts
C7The response states $f'(x)=0$ (or equivalent) to identify critical points; merely listing the zeros is not sufficient.
1.0000pts
C8The response justifies the absolute minimum by considering critical points and both endpoints $x=-2$ and $x=8$, correctly evaluating $f$ at each candidate or appropriately eliminating non-minimum candidates, and comparing values.
1.0000pts
C9The response states that the absolute minimum value of $f$ is $1$.
1.0000pts
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q5
Question 5
5. The graphs of the functions f and g are shown in the figure for $0 \leq x \leq 3$. It is known that $g(x) = \frac{12}{3+x}$ for $x \geq 0$. The twice-differentiable function $f$, which is not explicitly given, satisfies $f(3) = 2$ and $\int_0^3 f(x) \, dx = 10$. (a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$. (b) Evaluate the improper integral $\int_0^{\infty} \left( g(x) \right)^2 dx$, or show that the integral diverges. (c) Let $h$ be the function defined by $h(x) = x \cdot f'(x)$. Find the value of $\int_0^3 h(x) \, dx$.
Rubric Criteria (9)
C1The response presents a definite integral with a correct integrand for the enclosed area, such as $f(x)-g(x)$, $g(x)-f(x)$, $|f(x)-g(x)|$, or $|g(x)-f(x)|$ on $[0,3]$, or presents $10-\int_0^3 g(x)\,dx$ with implied integrand for $f$.
1.0000pts
C2The response finds an antiderivative of the form $a\ln(3+x)$ or $a\ln|3+x|$ for the $g(x)$ part.
1.0000pts
C3The response gives the correct area $10-12\ln 2$ (or equivalent correct exact or decimal answer), eligible only if the first two points were earned.
1.0000pts
C4The response correctly uses limit notation throughout the improper integral, such as $\lim_{b\to\infty}\int_0^b \frac{144}{(3+x)^2}\,dx$, without arithmetic with infinity.
1.0000pts
C5The response finds an antiderivative of the form $-\frac{a}{3+x}$ with $a>0$ for the improper integral, with or without correct limit notation.
1.0000pts
C6The response gives the value $48$ (or equivalent) for the improper integral.
1.0000pts
C7The response shows integration by parts with implied or explicit identification $u=x$ and $dv=f'(x)\,dx$ (or tabular method beginning with $x$ and $f'(x)$), or directly presents $x\cdot f(x) - \int f(x)\,dx$.
1.0000pts
C8The response presents $x\cdot f(x) - \int f(x)\,dx$ (or equivalent result of integration by parts), with or without limits of integration.
1.0000pts
C9The response gives the correct value $-4$ for $\int_0^3 h(x)\,dx$, eligible only if the first two points were earned.
1.0000pts
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q6
Question 6
6. The function f has derivatives of all orders for all real numbers. It is known that f(0) = 2, f'(0) = 3, f''(x) = -f(x^2), and f'''(x) = -2x \cdot f'(x^2). (a) Find f^{(4)}(x), the fourth derivative of f with respect to x. Write the fourth-degree Taylor polynomial for f about x = 0. Show the work that leads to your answer. (b) The fourth-degree Taylor polynomial for f about x = 0 is used to approximate f(0.1). Given that |f^{(5)}(x)| \leq 15 for 0 \leq x \leq 0.5, use the Lagrange error bound to show that this approximation is within \frac{1}{10^5} of the exact value of f(0.1). (c) Let g be the function such that g(0) = 4 and g'(x) = e^x f(x). Write the second-degree Taylor polynomial for g about x = 0. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.
Rubric Criteria (9)
C1The response presents a correct application of the product/chain rule to find the fourth derivative, such as $f^{(4)}(x) = -2\cdot f'(x^2) + (-2x)\cdot f''(x^2)\cdot 2x$ (or equivalent correct form).
1.0000pts
C2The response gives a completely correct expression for $f^{(4)}(x)$.
1.0000pts
C3The response correctly uses $f(0)=2$ and $f'(0)=3$ to produce the first two terms $2+3x$ of the fourth-degree Taylor polynomial.
1.0000pts
C4The response correctly includes the $x^2$, $x^3$, and $x^4$ terms (with $x^3$ coefficient $0$) to complete the fourth-degree Taylor polynomial $2+3x-x^2-\frac{1}{4}x^4$ (or equivalent unsimplified correct form), with no terms of degree greater than four.
1.0000pts
C5The response presents the Lagrange error bound term $\frac{\max_{0\leq x\leq 0.1}|f^{(5)}(x)|}{5!}\cdot(0.1)^5$ or $\frac{15}{5!}(0.1)^5$ (or equivalent).
1.0000pts
C6The response communicates the inequality $|Error|\leq \frac{15}{5!}(0.1)^5\leq \frac{1}{10^5}$ (or equivalent correct reasoning showing the error is at most $\frac{1}{10^5}$).
1.0000pts
C7The response presents $g''(x)=e^x f(x)+e^x f'(x)$, or evaluates $g''(0)=f(0)+f'(0)=5$ (or equivalent).
1.0000pts
C8The response presents the first two terms of the second-degree Taylor polynomial for $g$ about $x=0$ as $4+2x$ (or equivalent correct first two terms, with or without supporting work).
1.0000pts
C9The response writes the complete second-degree Taylor polynomial for $g$ about $x=0$ as $4+2x+\frac{5}{2}x^2$ (or equivalent correct form with supporting work for the $x^2$ coefficient and no terms of degree greater than two).
1.0000pts
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