Unit 6 Overview (College Board AP® Calculus AB): Exam Questions

The continuous function $f$ is defined on the closed interval $-6\le x\le 5$. The figure below shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $(5,3)$. It is known that the point $\left(3,3-\sqrt{5}\right)$ is on the graph of $f$.

If ${\int }_{-6}^{5}f\left(x\right)dx=7$, find the value of ${\int }_{-6}^{-2}f\left(x\right)dx$. Show the work that leads to your answer.

Evaluate ${\int }_3^5\left(2f\text{'}\left(x\right)+4\right)dx$.

The function $g$ is given by $g\left(x\right)={\int }_{-2}^{x}f\left(t\right)dt$. Find the absolute maximum value of $g$ on the interval $-2\le x\le 5$. Justify your answer.

Find $\underset{x\to 1}{lim}\frac{{10}^x-3f\text{'}\left(x\right)}{f\left(x\right)-arctanx}$.

The continuous function $f$ is defined on the closed interval $-6\le x\le 12$. The graph of $f$, consisting of two semicircles and one line segment, is shown in the figure.

Let $g$ be the function defined by $g\left(x\right)={\int }_6^{x}f\left(t\right) \text{d}t$.

Find $g\text{'}\left(8\right)$. Give a reason for your answer.

Find all values of $x$ in the open interval $-6<x<12$ at which the graph of $g$ has a point of inflection. Give a reason for your answer.

Find $g\left(12\right)$ and $g\left(0\right)$. Label your answers.

Find the value of $x$ at which $g$ attains an absolute minimum on the closed interval $-6\le x\le 12$. Justify your answer.

The graph of the continuous function $g$, the derivative of the function $f$, is shown below. The function $g$ is piecewise linear for $-5\le x<3$, and $g\left(x\right)=2(x-4{)}^2$ for $3\le x\le 6$.

If $f\left(1\right)=3$, what is the value of $f(-5)$?

Evaluate ${\int }_1^{6}g\left(x\right) dx$.

For $-5<x<6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer.

Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.

An ice sculpture melts in such a way that it can be modelled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$, where $r\left(t\right)$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r\text{'}\left(t\right)$, the rate of change of the radius, over the time interval $0\le t\le 12$.

Approximate $r\text{'}\text{'}(8.5)$ using the average rate of change of $r\text{'}$ over the interval $7\le t\le 10$. Show the computations that lead to your answer, and indicate units of measure.

Is there a time $t$, $0\le t\le 3$, for which $r\text{'}\left(t\right)=-6$? Justify your answer.

Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of ${\int }_0^{12}r\text{'}\left(t\right) \text{d}t$.

The height of the cone decreases at a rate of 2 centimetres per day. At time $t=3$ days, the radius is 100 centimetres and the height is 50 centimetres. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t=3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\pi r^{2}h$.)

The function $f$ is differentiable on the closed interval $[-6,5]$ and satisfies $f(-2)=7$. The graph of $f\text{'}$, the derivative of $f$, consists of a semicircle and three line segments, as shown below.

Find the values of $f(-6)$ and $f\left(5\right)$.

On what intervals is $f$ increasing? Justify your answer.

Find the absolute minimum value of $f$ on the closed interval $[-6,5]$. Justify your answer.

For each of $f\text{'}\text{'}(-5)$ and $f\text{'}\text{'}\left(3\right)$, find the value or explain why it does not exist.

Let $f$ be a continuous function defined on the closed interval $-4\le x\le 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G\left(x\right)={\int }_0^{x}f\left(t\right)\text{d}t$.

On what open intervals is the graph of $G$ concave up? Give a reason for your answer.

Let $P$ be the function defined by $P\left(x\right)=G\left(x\right)·f\left(x\right)$. Find $P\text{'}\left(3\right)$.

Find $\underset{x\to 2}{\lim }\frac{G\left(x\right)}{x^2-2x}$.

Find the average rate of change of $G$ on the interval $[-4,2]$. Does the Mean Value Theorem guarantee a value $c$, $-4<c<2$, for which $G\text{'}\left(c\right)$ is equal to this average rate of change? Justify your answer.

The figure above shows the graph of the piecewise-linear function $f$. For $-4\le x\le 12$, the function $g$ is defined by

$g\left(x\right)={\int }_2^{x}f\left(t\right) dt$

Does $g$ have a relative minimum, a relative maximum, or neither at $x=10$? Justify your answer.

Does the graph of $g$ have a point of inflection at $x=4$? Justify your answer.

Find the absolute minimum value and the absolute maximum value of $g$ on the interval $-4\le x\le 12$. Justify your answers.

For $-4\le x\le 12$, find all intervals for which $g\left(x\right)\le 0$.

The graph of the differentiable function $f$, shown below for $-6\le x\le 7$, has a horizontal tangent at $x=-2$ and is linear for $0\le x\le 7$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x=-6$, and the $x$- and $y$-axes. Region $R$ has area 12.

The function $g$ is defined by $g\left(x\right)={\int }_0^{x}f\left(t\right) \text{d}t$. Find the values of $g(-6)$, $g\left(4\right)$, and $g\left(6\right)$.

For the function $g$ defined in part (a), find all values of $x$ in the interval $0\le x\le 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.

The function $h$ is defined by $h\left(x\right)={\int }_{-6}^{x}f\text{'}\left(t\right) \text{d}t$. Find the values of $h\left(6\right)$, $h\text{'}\left(6\right)$, and $h\text{'}\text{'}\left(6\right)$. Show the work that leads to your answers.

The functions $f$ and $g$ are twice differentiable. The table gives values of the functions and their first derivatives at selected values of $x$.

Let $h$ be the function defined by $h\left(x\right)=f\left(g\right(x\left)\right)$. Find $h\text{'}\left(7\right)$. Show the work that leads to your answer.

Let $k$ be a differentiable function such that $k\text{'}\left(x\right)=\left(f\right(x){)}^2·g(x)$. Is the graph of $k$ concave up or concave down at the point where $x=4$? Give a reason for your answer.

Let $m$ be the function defined by $m\left(x\right)=5x^3+{\int }_0^{x}f\text{'}\left(t\right)\text{d}t$. Find $m\left(2\right)$. Show the work that leads to your answer.

Is the function $m$ defined in part (c) increasing, decreasing, or neither at $x=2$? Justify your answer.

The functions $f$ and $g$ have continuous second derivatives. The table below gives values of the functions and their derivatives at selected values of $x$.

Let $k\left(x\right)=f\left(g\left(x\right)\right)$. Write an equation for the line tangent to the graph of $k$ at $x=3$.

Let $h\left(x\right)=\frac{g\left(x\right)}{f\left(x\right)}$. Find $h\text{'}\left(1\right)$.

Evaluate ${\int }_1^{3}f\text{'}\text{'}\left(2x\right)dx$.