Graphs of Functions & Their Derivatives (College Board AP® Calculus AB): Exam Questions

The function $f$ is defined on the closed interval [-5, 5]. The graph of $f^{\text{'}}$, 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 $x=-3$? 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.

It is known for $y=ax\sqrt{16-x^2}$, where $a$ is a positive non-zero constant, that the rate of change is $\frac{dy}{dx}=\frac{2\left(a{x}^2-8a\right)}{\sqrt{16-x^2}}$. For a particular value of $a$, the maximum value of $y$ is 40. Find the value of $a$.

Let $f$ and $g$ be the functions defined by $f\left(x\right)=\ln \left(x+5\right)$ and $g\left(x\right)=x^6+2x^5$. The graphs of $f$ and $g$, shown in the figure above, inersect at $x=A$ and $x=B$, where $A<-1$ and $B>0$.

For $A\le x\le B$ let $h\left(x\right)$ be the vertical distance between the graphs of $f$ and $g$. Is $h$ increasing or decreasing at $x=-1$? Give a reason for your answer.

Selected values of the differentiable function $f\left(x\right)$ are shown in the table above.

Justify why there must be at least one value of $x$ for $0.9\le x\le 8.4$, at which $f^{\text{'}}\left(x\right)$ is equal to zero.

Let $f$ be the continuous function defined on $\left[-3,6\right]$ whose graph, consisting of three straight line segments and a semicircle, is given above. Let $g$ be the function $g\left(x\right)={\int }_1^{x}f\left(t\right)dt$.

Find the $x$-coordinate of each point at which the graph of $g$ has a horizontal tangent line. For each of these points, determine whether $g$ has a relative minimum, relative maximum or neither a minimum or maximum at the point. Justify your answers.

For $-3<x<6$, find all the values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.

A procedure is used to administer medication to a patient. The amount, in milligrams, of the medication in the patient at time $t$ hours is modeled by a function $y=B\left(t\right)$ that satisfies the differential equation $\frac{\text{d}y}{\text{d}t}=3-\frac{y}{t+2}$. At time $t=1$ hour, there are 2.5 milligrams of the medication in the patient. Is the rate of change of the amount of medication in the patient increasing or decreasing at time $t=1$? Give a reason for your answer.

Let $f$ be a differentiable function. On the interval $0\le x\le 12$, the graph of $f^{\text{'}}$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.

Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0<x<12$. Justify your answer.

Let $g$ be the function defined by $g\left(x\right)=f\left(x\right)+2x$. On what intervals, if any, is $g$ decreasing for $0\le x\le 12$? Show the analysis that leads to your answer.

Given that $f\left(0\right)=0$, find the absolute maximum value of the function $g\left(x\right)$ defined in part (b) on the interval $0\le x\le 12$. Justify your answer.

Let $f$ be a continuous function defined on the closed interval $-6\le x\le 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G\left(x\right)={\int }_0^{x}f\left(t\right)\mathit{d}t.$

Find the average rate of change of $G$ on the interval [-6, 6]. Does the Mean Value Theorem guarantee a value $c$, $-6<c<6$ for which $G^{\text{'}}\left(c\right)$ is equal to this average rate of change? Justify your answer.

Birds enter an area of woodland at a rate modeled by the function $E\left(t\right)=40+12\sin \left(\frac{\pi t}{6}\right)$. Birds leave the area at a rate modeled by the function $L\left(t\right)=12+2^{0.01t^2}$. Both $E\left(t\right)$ and $L\left(t\right)$ are measured in birds per hour, and $t$ is measured in hours since midnight $\left(t=0\right)$.

At what value of $t$, for $0\le t\le 24$, is the greatest number of birds in the area of woodland? Justify your answer.

Is the rate of change in the number of birds in the area of woodland increasing or decreasing at noon $\left(t=12\right)$? Explain your reasoning.

The continuous function $f$ is defined on the closed interval $-2\le x\le 6$. The figure above shows the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $\left(6,-3\right)$.

The function $g$ is given by $g\left(x\right)={\int }_{-2}^{x}f\left(t\right)\mathit{d}t$. Find the absolute maximum value of $g$ on the interval $-2\le x\le 6$. Justify your answer.

The rate at which a hot object cools down is proportional to the difference between its current temperature and the ambient temperature of the surrounding environment. If $F\left(t\right)$ is the temperature of the object in degrees Farenheit at time $t$ hours, and the surrounding temperature is constant at $60$°F, the rate of change of the temperature is given by:

$\frac{dF}{dt}=-\frac{1}{4}\left(F-60\right)$

Initially, at $t=0$, the temperature of the object is $220$°F.

Find $\frac{d^{2}F}{d{t}^2}$ in terms of $F$. Use $\frac{d^{2}F}{d{t}^2}$ to explain why the graph of $F$ cannot resemble the following graph.

The velocity of a particle, P, moving along the $x$-axis is given by the differentiable function $v_P$, where $v_P\left(t\right)$ is measured in meters per hour and $t$ is measured in hours. Selected values of $v_P\left(t\right)$ are shown in the table below.

Particle P is at the origin at time $t=0$.

Justify why there must be at least one time $t$, for $0.3\le t\le 2.8$, at which $v_P\text{'}\left(t\right)$, the acceleration of particle P, equals 0 meters per hour per hour.

The function $f$ is defined on the closed interval $[-2,8]$ and satisfies $f\left(2\right)=1$. The graph of $f\text{'}$, 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.