Learn Extracted exam questions AP Calculus BC 2023 Free Response
2023 Free Response
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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.
| $t$ (seconds) | 0 | 60 | 90 | 120 | 135 | 150 |
|---|---|---|---|---|---|---|
| $f(t)$ (gallons per second) | 0 | 0.1 | 0.15 | 0.1 | 0.05 | 0 |
Using correct units, interpret the meaning of $\displaystyle\int_{60}^{135} f(t)\,dt$ 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 $\displaystyle\int_{60}^{135} f(t)\,dt$.
Must there exist a value of $c$, for $60 < c < 120$, such that $f'(c) = 0$? Justify your answer.
The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t) = \left(\dfrac{t}{500}\right)\cos\!\left[\left(\dfrac{t}{120}\right)^2\right]$ 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.
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.
[Graph of a curve in the $xy$-plane; $x$-axis from $O$ to 5, $y$-axis with gridlines at 1 and 2. The curve starts at the origin $(0,0)$, rises as a straight line through about $(1,1)$ up to a peak near $(4,2)$, then curves down steeply to end at $(5,0)$.]
For $0 \le t \le \pi$, 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 $\dfrac{dx}{dt} = e^{\cos t}$. At time $t = 0$, the particle is at position $(1, 0)$.
Find the acceleration vector of the particle at time $t = 1$. Show the setup for your calculations.
For $0 \le t \le \pi$, find the first time $t$ at which the speed of the particle is $1.5$. Show the work that leads to your answer.
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.
Find the total distance traveled by the particle over the time interval $0 \le t \le \pi$. Show the setup for your calculations.
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 ($^\circ\text{C}$) and $t$ is the number of minutes since the bottle was placed in the pan. $M$ satisfies the differential equation $\dfrac{dM}{dt} = \dfrac{1}{4}(40 - M)$. At time $t = 0$, the temperature of the milk is $5^\circ\text{C}$. It can be shown that $M(t) < 40$ for all values of $t$.
A slope field for the differential equation $\dfrac{dM}{dt} = \dfrac{1}{4}(40 - M)$ is shown. Sketch the solution curve through the point $(0, 5)$.
[Slope field diagram: axes labeled $M(t)$ (vertical) and $t$ (horizontal); vertical axis marked at 5 and 70, horizontal axis marked at 15. Short line segments fill the plane: below $M=40$ the segments have positive slope (steeper near $M=5$, shallower approaching $M=40$); at $M=40$ the segments are horizontal (slope 0); above $M=40$ the segments have negative slope. The point $(0,5)$ is marked on the vertical axis.]
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.
Write an expression for $\dfrac{d^2M}{dt^2}$ in terms of $M$. Use $\dfrac{d^2M}{dt^2}$ 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.
Use separation of variables to find an expression for $M(t)$, the particular solution to the differential equation $\dfrac{dM}{dt} = \dfrac{1}{4}(40 - M)$ with initial condition $M(0) = 5$.
[Graph of $f'$ on the $xy$-plane; $x$-axis from $-2$ to 8 with gridlines at each integer, $y$-axis with gridlines at $-2, -1, 1, 2$. The graph of $f'$ consists of two line segments and a semicircle: a line segment from $(-2, 2)$ down to $(0, -2)$; a line segment from $(0, -2)$ up to $(4, 2)$; then a semicircle (below the segment, dipping down to touch the $x$-axis near $(6,0)$) from $(4, 2)$ to $(8, 2)$.] Graph of $f'$
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.
Does $f$ have a relative minimum, a relative maximum, or neither at $x = 6$? Give a reason for your answer.
On what open intervals, if any, is the graph of $f$ concave down? Give a reason for your answer.
Find the value of $\displaystyle\lim_{x \to 2} \dfrac{6f(x) - 3x}{x^2 - 5x + 6}$, or show that it does not exist. Justify your answer.
Find the absolute minimum value of $f$ on the closed interval $[-2, 8]$. Justify your answer.
[Graph on the $xy$-plane; $x$-axis from $O$ to 3 with gridlines at 1, 2, 3, $y$-axis with gridlines at 1, 2, 3, 4. Two curves $y = f(x)$ and $y = g(x)$ are shown for $0 \le x \le 3$, both starting at $(0, 4)$ and ending near $(3, 2)$, with $y = f(x)$ lying above $y = g(x)$ in between (concave, bulging shape); the region between the two curves is shaded gray.]
The graphs of the functions $f$ and $g$ are shown in the figure for $0 \le x \le 3$. It is known that $g(x) = \dfrac{12}{3+x}$ for $x \ge 0$. The twice-differentiable function $f$, which is not explicitly given, satisfies $f(3) = 2$ and $\displaystyle\int_0^3 f(x)\,dx = 10$.
Find the area of the shaded region enclosed by the graphs of $f$ and $g$.
Evaluate the improper integral $\displaystyle\int_0^{\infty} (g(x))^2\,dx$, or show that the integral diverges.
Let $h$ be the function defined by $h(x) = x \cdot f'(x)$. Find the value of $\displaystyle\int_0^3 h(x)\,dx$.
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 $f'''(x) = -2x \cdot f'(x^2)$.
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.
The fourth-degree Taylor polynomial for $f$ about $x = 0$ is used to approximate $f(0.1)$. Given that $\left|f^{(5)}(x)\right| \le 15$ for $0 \le x \le 0.5$, use the Lagrange error bound to show that this approximation is within $\dfrac{1}{10^5}$ of the exact value of $f(0.1)$.
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 $x = 0$.