The graph of a differentiable function $f$ is shown above for $- 3 \leq x \leq 3$. The graph of $f$ has horizontal tangent lines at $x = - 1 , x = 1$, and $x = 2$. The areas of regions $A , B , C$, and $D$ are 5, 4, 5, and 3, respectively. Let $g$ be the antiderivative of $f$ such that $g ( 3 ) = 7$.
(a) Find all values of $x$ on the open interval $- 3 < x < 3$ for which the function $g$ has a relative maximum. Justify your answer.
(b) On what open intervals contained in $- 3 < x < 3$ is the graph of $g$ concave up? Give a reason for your answer.
(c) Find the value of $\lim _ { x \rightarrow 0 } \frac { g ( x ) + 1 } { 2 x }$, or state that it does not exist. Show the work that leads to your answer.
(d) Let $h$ be the function defined by $h ( x ) = 3 f ( 2 x + 1 ) + 4$. Find the value of $\int _ { - 2 } ^ { 1 } h ( x ) d x$.

The figure above shows the graph of the continuous function $g$ on the interval $[ 0,8 ]$. Let $h$ be the function defined by $h ( x ) = \int _ { 3 } ^ { x } g ( t ) \, d t$. On what intervals is $h$ increasing?
(A) $[ 2,5 ]$ only
(B) $[ 1,7 ]$
(C) $[ 0,1 ]$ and $[ 3,7 ]$
(D) $[ 1,3 ]$ and $[ 7,8 ]$

35. If $F$ and $f$ are differentiable functions such that $F ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, and if $F ( a ) = - 2$ and $F ( b ) = - 2$ where $a < b$, which of the following must be true?
(A) $\quad f ( x ) = 0$ for some $x$ such that $a < x < b$.
(B) $\quad f ( x ) > 0$ for all $x$ such that $a < x < b$.
(C) $f ( x ) < 0$ for all $x$ such that $a < x < b$.
(D) $\quad F ( x ) \leq 0$ for all $x$ such that $a < x < b$.
(E) $\quad F ( x ) = 0$ for some $x$ such that $a < x < b$.

15. If $F ( x ) = \int _ { 0 } ^ { x } \sqrt { t ^ { 3 } + 1 } d t$, then $F ^ { \prime } ( 2 ) =$
(A) - 3
(B) - 2
(C) 2
(D) 3
(E) 18

The graph of the function $f$, consisting of three line segments, is given above. Let $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$.
(a) Compute $g ( 4 )$ and $g ( - 2 )$.
(b) Find the instantaneous rate of change of $g$, with respect to $x$, at $x = 1$.
(c) Find the absolute minimum value of $g$ on the closed interval $[ - 2, 4 ]$. Justify your answer.
(d) The second derivative of $g$ is not defined at $x = 1$ and $x = 2$. How many of these values are $x$-coordinates of points of inflection of the graph of $g$ ? Justify your answer.

The graph of the function $f$ above consists of three line segments.
(a) Let $g$ be the function given by $g ( x ) = \int _ { - 4 } ^ { x } f ( t ) d t$.
For each of $g ( - 1 ) , g ^ { \prime } ( - 1 )$, and $g ^ { \prime \prime } ( - 1 )$, find the value or state that it does not exist.
(b) For the function $g$ defined in part (a), find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $- 4 < x < 3$. Explain your reasoning.
(c) Let $h$ be the function given by $h ( x ) = \int _ { x } ^ { 3 } f ( t ) d t$. Find all values of $x$ in the closed interval $- 4 \leq x \leq 3$ for which $h ( x ) = 0$.
(d) For the function $h$ defined in part (c), find all intervals on which $h$ is decreasing. Explain your reasoning.

$$h ( x ) = \int _ { 1 } ^ { x } t ( t ) d t$$
a) $h ( 1 ) = \int ^ { \prime } f ( t ) d t = 0$
b) $$\begin{aligned} & h h ^ { \prime } ( x ) = \frac { d } { d x } \int _ { 0 } ^ { x } f ( t ) d t = f ( x ) \\ & h ^ { \prime } ( 4 ) = f ( 4 ) = 2 \end{aligned}$$
oncave up on $( 1,3 )$ and $( 6,7 )$ since $h ^ { \prime } ( x )$ is increa. those intervals
d) h(x) has no relotive minimum on [1,7] since $h ^ { \prime } ( x )$ does
change sign from - ve to tre
$\therefore$ Minimum must accur at either enopoint
$$\begin{aligned} & h ( 1 ) = 0 \\ & h ( 7 ) = \int _ { 1 } ^ { 7 } f ( t ) d t \end{aligned}$$
$h ( 7 ) > h ( 1 ) \quad$ since $h ( x )$ has a relative maximum $a t \quad x = 5$, and connot decrease to zero, $\int ^ { 5 } h ^ { \prime } ( x ) > \int _ { 5 } ^ { 7 } h ^ { \prime } ( x ) d x$

The graph of the function $f$ shown above consists of two line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( - 1 ) , g ^ { \prime } ( - 1 )$, and $g ^ { \prime \prime } ( - 1 )$.
(b) For what values of $x$ in the open interval $( - 2, 2 )$ is $g$ increasing? Explain your reasoning.
(c) For what values of $x$ in the open interval $( - 2, 2 )$ is the graph of $g$ concave down? Explain your reasoning.
(d) On the axes provided, sketch the graph of $g$ on the closed interval $[ - 2, 2 ]$.

5. The graph of a function $f$ consists of a semicircle and two line segments as shown above. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( 3 )$.
(b) Find all values of $x$ on the open interval $( - 2,5 )$ at which $g$ has a relative maximum. Justify your answer.
(c) Write an equation for the line tangent to the graph of $g$ at $x = 3$.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $( - 2,5 )$. Justify your answer.

The graph of the function $f$ consists of three line segments.
(a) Let $g$ be the function given by $g(x) = \int_{-4}^{x} f(t)\, dt$. For each of $g(-1)$, $g'(-1)$, and $g''(-1)$, find the value or state that it does not exist.
(b) For the function $g$ defined in part (a), find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $-4 < x < 3$. Explain your reasoning.
(c) Let $h$ be the function given by $h(x) = \int_{x}^{3} f(t)\, dt$. Find all values of $x$ in the closed interval $-4 \leq x \leq 3$ for which $h(x) = 0$.
(d) For the function $h$ defined in part (c), find all intervals on which $h$ is decreasing. Explain your reasoning.

The graph of the function $f$ shown above consists of six line segments. Let $g$ be the function given by $g(x) = \int_{0}^{x} f(t)\, dt$.
(a) Find $g(4)$, $g'(4)$, and $g''(4)$.
(b) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 1$? Justify your answer.
(c) Suppose that $f$ is defined for all real numbers $x$ and is periodic with a period of length 5. The graph above shows two periods of $f$. Given that $g(5) = 2$, find $g(10)$ and write an equation for the line tangent to the graph of $g$ at $x = 108$.

The continuous function $f$ is defined on the interval $-4 \leq x \leq 3$. The graph of $f$ consists of two quarter circles and one line segment, as shown in the figure. Let $g(x) = 2x + \int_{0}^{x} f(t)\,dt$.
(a) Find $g(-3)$. Find $g'(x)$ and evaluate $g'(-3)$.
(b) Determine the $x$-coordinate of the point at which $g$ has an absolute maximum on the interval $-4 \leq x \leq 3$. Justify your answer.
(c) Find all values of $x$ on the interval $-4 < x < 3$ for which the graph of $g$ has a point of inflection. Give a reason for your answer.
(d) Find the average rate of change of $f$ on the interval $-4 \leq x \leq 3$. There is no point $c$, $-4 < c < 3$, for which $f'(c)$ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.

Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$.
(a) Find the values of $g ( 2 )$ and $g ( - 2 )$.
(b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist.
(c) 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 nor a maximum at the point. Justify your answers.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.

Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$.
(a) Find the values of $g ( 2 )$ and $g ( - 2 )$.
(b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist.
(c) 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 nor a maximum at the point. Justify your answers.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.

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 $g ( x ) = \int _ { - 3 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( 3 )$.
(b) 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.
(c) The function $h$ is defined by $h ( x ) = \dfrac { g ( x ) } { 5 x }$. Find $h ^ { \prime } ( 3 )$.
(d) The function $p$ is defined by $p ( x ) = f \left( x ^ { 2 } - x \right)$. Find the slope of the line tangent to the graph of $p$ at the point where $x = - 1$.

The figure above shows the graph of the piecewise-linear function $f$. For $- 4 \leq x \leq 12$, the function $g$ is defined by $g ( x ) = \int _ { 2 } ^ { x } f ( t ) \, dt$.
(a) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 10$? Justify your answer.
(b) Does the graph of $g$ have a point of inflection at $x = 4$? Justify your answer.
(c) Find the absolute minimum value and the absolute maximum value of $g$ on the interval $- 4 \leq x \leq 12$. Justify your answers.
(d) For $- 4 \leq x \leq 12$, find all intervals for which $g ( x ) \leq 0$.

The continuous function $f$ is defined on the closed interval $-6 \leq x \leq 5$. The figure above 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 $(3, 3 - \sqrt{5})$ is on the graph of $f$.
(a) If $\int_{-6}^{5} f(x)\, dx = 7$, find the value of $\int_{-6}^{-2} f(x)\, dx$. Show the work that leads to your answer.
(b) Evaluate $\int_{3}^{5} \left(2f'(x) + 4\right) dx$.
(c) The function $g$ is given by $g(x) = \int_{-2}^{x} f(t)\, dt$. Find the absolute maximum value of $g$ on the interval $-2 \leq x \leq 5$. Justify your answer.
(d) Find $\lim_{x \to 1} \dfrac{10^x - 3f'(x)}{f(x) - \arctan x}$.

Let $f$ be a continuous function defined on the closed interval $-4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G(x) = \int_{0}^{x} f(t)\, dt$.
(a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer.
(b) Let $P$ be the function defined by $P(x) = G(x) \cdot f(x)$. Find $P^{\prime}(3)$.
(c) Find $\lim_{x \rightarrow 2} \frac{G(x)}{x^{2} - 2x}$.
(d) 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 $G^{\prime}(c)$ is equal to this average rate of change? Justify your answer.

The graph of the differentiable function $f$, shown for $-6 \leq x \leq 7$, has a horizontal tangent at $x = -2$ and is linear for $0 \leq x \leq 7$. 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.
(a) The function $g$ is defined by $g(x) = \int_{0}^{x} f(t)\, dt$. Find the values of $g(-6)$, $g(4)$, and $g(6)$.
(b) For the function $g$ defined in part (a), find all values of $x$ in the interval $0 \leq x \leq 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.
(c) The function $h$ is defined by $h(x) = \int_{-6}^{x} f'(t)\, dt$. Find the values of $h(6)$, $h'(6)$, and $h''(6)$. Show the work that leads to your answers.

The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 12$. The graph of $f$, consisting of two semicircles and one line segment, is shown in the figure.
Let $g$ be the function defined by $g ( x ) = \int _ { 6 } ^ { x } f ( t ) d t$.
A. Find $g ^ { \prime } ( 8 )$. Give a reason for your answer.
B. 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.
C. Find $g ( 12 )$ and $g ( 0 )$. Label your answers.
D. Find the value of $x$ at which $g$ attains an absolute minimum on the closed interval $- 6 \leq x \leq 12$. Justify your answer.

The graph of the function $f$, consisting of three line segments, is given above. Let $g(x) = \int_{1}^{x} f(t)\, dt$.
(a) Compute $g(4)$ and $g(-2)$.
(b) Find the instantaneous rate of change of $g$, with respect to $x$, at $x = 1$.
(c) Find the absolute minimum value of $g$ on the closed interval $[-2, 4]$. Justify your answer.
(d) The second derivative of $g$ is not defined at $x = 1$ and $x = 2$. How many of these values are $x$-coordinates of points of inflection of the graph of $g$? Justify your answer.

The graph of the function $f$ shown above consists of two line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( - 1 ) , g ^ { \prime } ( - 1 )$, and $g ^ { \prime \prime } ( - 1 )$.
(b) For what values of $x$ in the open interval $( - 2, 2 )$ is $g$ increasing? Explain your reasoning.
(c) For what values of $x$ in the open interval $( - 2, 2 )$ is the graph of $g$ concave down? Explain your reasoning.
(d) On the axes provided, sketch the graph of $g$ on the closed interval $[ - 2, 2 ]$.

The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.
(a) Find $g(6)$, $g'(6)$, and $g''(6)$.
(b) On what intervals is $g$ decreasing? Justify your answer.
(c) On what intervals is the graph of $g$ concave down? Justify your answer.
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.

Let $f$ be a function defined on the closed interval $[0,7]$. The graph of $f$, consisting of four line segments, is shown above. Let $g$ be the function given by $g(x) = \int_{2}^{x} f(t)\, dt$.
(a) Find $g(3)$, $g'(3)$, and $g''(3)$.
(b) Find the average rate of change of $g$ on the interval $0 \leq x \leq 3$.
(c) For how many values $c$, where $0 < c < 3$, is $g'(c)$ equal to the average rate found in part (b)? Explain your reasoning.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the interval $0 < x < 7$. Justify your answer.

3. The graph of the function $f$ shown above consists of six line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( 4 ) , g ^ { \prime } ( 4 )$, and $g ^ { \prime \prime } ( 4 )$.
(b) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 1$ ? Justify your answer.
(c) Suppose that $f$ is defined for all real numbers $x$ and is periodic with a period of length 5 . The graph above shows two periods of $f$. Given that $g ( 5 ) = 2$, find $g ( 10 )$ and write an equation for the line tangent to the graph of $g$ at $x = 108$.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.

END OF PART A OF SECTION II

No calculator is allowed for these problems.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \hline

$v ( t )$

(feet per second)

& 5 & 14 & 22 & 29 & 35 & 40 & 44 & 47 & 49 \hline \end{tabular}

1. Rocket $A$ has positive velocity $v ( t )$ after being launched upward from an initial height of 0 feet at time $t = 0$ seconds. The velocity of the rocket is recorded for selected values of $t$ over the interval $0 \leq t \leq 80$ seconds, as shown in the table above.
   (a) Find the average acceleration of rocket $A$ over the time interval $0 \leq t \leq 80$ seconds. Indicate units of measure.
   (b) Using correct units, explain the meaning of $\int _ { 10 } ^ { 70 } v ( t ) d t$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int _ { 10 } ^ { 70 } v ( t ) d t$.
   (c) Rocket $B$ is launched upward with an acceleration of $a ( t ) = \frac { 3 } { \sqrt { t + 1 } }$ feet per second per second. At time $t = 0$ seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time $t = 80$ seconds? Explain your answer.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.

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