Learn Extracted exam questions AP Calculus AB 2014 Free Response
2014 Free Response
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Grass clippings are placed in a bin, where they decompose. For $0 \le t \le 30$, the amount of grass clippings remaining in the bin is modeled by $A(t) = 6.687(0.931)^t$, where $A(t)$ is measured in pounds and $t$ is measured in days.
Find the average rate of change of $A(t)$ over the interval $0 \le t \le 30$. Indicate units of measure.
Find the value of $A'(15)$. Using correct units, interpret the meaning of the value in the context of the problem.
Find the time $t$ for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval $0 \le t \le 30$.
For $t > 30$, $L(t)$, the linear approximation to $A$ at $t = 30$, is a better model for the amount of grass clippings remaining in the bin. Use $L(t)$ to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.
[Graph of $y = f(x)$ for $x \ge 0$: a curve starting near the origin, dipping down below the $x$-axis to a minimum, then rising steeply back up and crossing above $y = 0$; the region $R$ is labelled in the dip between the curve and the $x$-axis/line $y=4$, roughly between the two crossing points of $f(x)=4$.]
Let $R$ be the region enclosed by the graph of $f(x) = x^4 - 2.3x^3 + 4$ and the horizontal line $y = 4$, as shown in the figure above.
Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = -2$.
Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with a leg in $R$. Find the volume of the solid.
The vertical line $x = k$ divides $R$ into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value $k$.
[Graph of $f$ on the closed interval $[-5,4]$, consisting of three line segments: the graph starts at the open point $(-5,2)$ and descends to a point at approximately $(-2,0)$ on the $x$-axis; from there it rises steeply (steeper slope) through the origin area up to a peak near $x=0$ (peak value not labeled, appears above $y=3$); from the peak it descends steeply down to the point $(4,-4)$. Axis tick marks shown at $1$ on both the $x$-axis and $y$-axis. Labelled "Graph of $f$".]
The function $f$ is defined on the closed interval $[-5, 4]$. The graph of $f$ consists of three line segments and is shown in the figure above. Let $g$ be the function defined by $g(x) = \displaystyle\int_{-3}^{x} f(t)\,dt$.
Find $g(3)$.
On what open intervals contained in $-5 < x < 4$ is the graph of $g$ both increasing and concave down? Give a reason for your answer.
The function $h$ is defined by $h(x) = \dfrac{g(x)}{5x}$. Find $h'(3)$.
The function $p$ is defined by $p(x) = f(x^2 - x)$. Find the slope of the line tangent to the graph of $p$ at the point where $x = -1$.
Train $A$ runs back and forth on an east-west section of railroad track. Train $A$'s velocity, measured in meters per minute, is given by a differentiable function $v_A(t)$, where time $t$ is measured in minutes. Selected values for $v_A(t)$ are given in the table above.
| $t$ (minutes) | 0 | 2 | 5 | 8 | 12 |
|---|---|---|---|---|---|
| $v_A(t)$ (meters/minute) | 0 | 100 | 40 | -120 | -150 |
Find the average acceleration of train $A$ over the interval $2 \le t \le 8$.
Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v_B(t) = -5t^2 + 60t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.
The twice-differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f'$, $g$, and $g'$ for various values of $x$ are given in the table above.
| $x$ | $-2$ | $-2 < x < -1$ | $-1$ | $-1 < x < 1$ | $1$ | $1 < x < 3$ | $3$ |
|---|---|---|---|---|---|---|---|
| $f(x)$ | 12 | Positive | 8 | Positive | 2 | Positive | 7 |
| $f'(x)$ | $-5$ | Negative | 0 | Negative | 0 | Positive | $\tfrac{1}{2}$ |
| $g(x)$ | $-1$ | Negative | 0 | Positive | 3 | Positive | 1 |
| $g'(x)$ | 2 | Positive | $\tfrac{3}{2}$ | Positive | 0 | Negative | $-2$ |
Find the $x$-coordinate of each relative minimum of $f$ on the interval $[-2, 3]$. Justify your answers.
Explain why there must be a value $c$, for $-1 < c < 1$, such that $f''(c) = 0$.
The function $h$ is defined by $h(x) = \ln(f(x))$. Find $h'(3)$. Show the computations that lead to your answer.
Evaluate $\displaystyle\int_{-2}^{3} f'(g(x))g'(x)\,dx$.
Consider the differential equation $\dfrac{dy}{dx} = (3 - y)\cos x$. Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(0) = 1$. The function $f$ is defined for all real numbers.
A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point $(0, 1)$.
[Slope field diagram on the domain $-3 \le x \le 3$, $-3 \le y \le 3$, with tick marks at $-3$, $1$, $2$(implied), $3$ on both axes (visible ticks at $-3$ and $3$ on the $x$-axis, at $3$ and $-3$ on the $y$-axis, plus unlabelled tick marks at intermediate integer values). Short line segments show the slope at each grid point: segments are roughly horizontal (dashes) in the upper-left and upper-right regions (where $y$ is large, slope near 0 since $3-y$ is very negative but cos x oscillates - shown as near-flat dashes), segments tilt more steeply (steep near-vertical strokes) in the band around $y \approx 1$ to $2$ near $x=0$, and the pattern is symmetric in a banded way across the vertical line $x=0$, repeating roughly periodically along the $x$-axis consistent with the $\cos x$ factor.]
Write an equation for the line tangent to the solution curve in part (a) at the point $(0, 1)$. Use the equation to approximate $f(0.2)$.
Find $y = f(x)$, the particular solution to the differential equation with the initial condition $f(0) = 1$.