A rain barrel collects water off the roof of a house during three hours of heavy rainfall. The height of the water in the barrel increases at the rate of $r ( t ) = 4 t ^ { 3 } e ^ { - 1.5 t }$ feet per hour, where $t$ is the time in hours since the rain began. At time $t = 1$ hour, the height of the water is 0.75 foot. What is the height of the water in the barrel at time $t = 2$ hours?
(A) 1.361 ft
(B) 1.500 ft
(C) 1.672 ft
(D) 2.111 ft

Let $f$ be a function that is differentiable for all real numbers. The table below gives the values of $f$ and its derivative $f ^ { \prime }$ for selected points $x$ in the closed interval $- 1.5 \leq x \leq 1.5$. The second derivative of $f$ has the property that $f ^ { \prime \prime } ( x ) > 0$ for $- 1.5 \leq x \leq 1.5$.

$x$	- 1.5	- 1.0	- 0.5	0	0.5	1.0	1.5
$f ( x )$	- 1	- 4	- 6	- 7	- 6	- 4	- 1
$f ^ { \prime } ( x )$	- 7	- 5	- 3	0	3	5	7

(a) Evaluate $\int _ { 0 } ^ { 1.5 } \left( 3 f ^ { \prime } ( x ) + 4 \right) d x$. Show the work that leads to your answer.
(b) Write an equation of the line tangent to the graph of $f$ at the point where $x = 1$. Use this line to approximate the value of $f ( 1.2 )$. Is this approximation greater than or less than the actual value of $f ( 1.2 )$? Give a reason for your answer.
(c) Find a positive real number $r$ having the property that there must exist a value $c$ with $0 < c < 0.5$ and $f ^ { \prime \prime } ( c ) = r$. Give a reason for your answer.
(d) Let $g$ be the function given by $g ( x ) = \begin{cases} 2 x ^ { 2 } - x - 7 & \text { for } x < 0 \\ 2 x ^ { 2 } + x - 7 & \text { for } x \geq 0 . \end{cases}$ The graph of $g$ passes through each of the points $( x , f ( x ) )$ given in the table above. Is it possible that $f$ and $g$ are the same function? Give a reason for your answer.

Let $f$ be a function that is twice differentiable for all real numbers. The table above gives values of $f$ for selected points in the closed interval $2 \leq x \leq 13$.

$x$	2	3	5	8	13
$f(x)$	1	4	$-2$	3	6

(a) Estimate $f'(4)$. Show the work that leads to your answer.
(b) Evaluate $\int_{2}^{13} \left(3 - 5f'(x)\right) dx$. Show the work that leads to your answer.
(c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate $\int_{2}^{13} f(x) \, dx$. Show the work that leads to your answer.
(d) Suppose $f'(5) = 3$ and $f''(x) < 0$ for all $x$ in the closed interval $5 \leq x \leq 8$. Use the line tangent to the graph of $f$ at $x = 5$ to show that $f(7) \leq 4$. Use the secant line for the graph of $f$ on $5 \leq x \leq 8$ to show that $f(7) \geq \frac{4}{3}$.

The figure above shows the graph of $f'$, the derivative of a twice-differentiable function $f$, on the interval $[-3, 4]$. The graph of $f'$ has horizontal tangents at $x = -1$, $x = 1$, and $x = 3$. The areas of the regions bounded by the $x$-axis and the graph of $f'$ on the intervals $[-2, 1]$ and $[1, 4]$ are 9 and 12, respectively.
(a) Find all $x$-coordinates at which $f$ has a relative maximum. Give a reason for your answer.
(b) On what open intervals contained in $-3 < x < 4$ is the graph of $f$ both concave down and decreasing? Give a reason for your answer.
(c) Find the $x$-coordinates of all points of inflection for the graph of $f$. Give a reason for your answer.
(d) Given that $f(1) = 3$, write an expression for $f(x)$ that involves an integral. Find $f(4)$ and $f(-2)$.

The function $f$ is differentiable on the closed interval $[-6, 5]$ and satisfies $f(-2) = 7$. The graph of $f'$, the derivative of $f$, consists of a semicircle and three line segments, as shown in the figure above.
(a) Find the values of $f(-6)$ and $f(5)$.
(b) On what intervals is $f$ increasing? Justify your answer.
(c) Find the absolute minimum value of $f$ on the closed interval $[-6, 5]$. Justify your answer.
(d) For each of $f''(-5)$ and $f''(3)$, find the value or explain why it does not exist.

The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$.
(a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ?
(b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) \, dx$.
(c) 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.
(d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.

Let $f$ be a function that is twice differentiable for all real numbers. The table below gives values of $f$ for selected points in the closed interval $2 \leq x \leq 13$.

$x$	2	3	5	8	13
$f(x)$	1	4	$-2$	3	6

(a) Estimate $f'(4)$. Show the work that leads to your answer.
(b) Evaluate $\int_{2}^{13} \left(3 - 5f'(x)\right) dx$. Show the work that leads to your answer.
(c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate $\int_{2}^{13} f(x) \, dx$. Show the work that leads to your answer.
(d) Suppose $f'(5) = 3$ and $f''(x) < 0$ for all $x$ in the closed interval $5 \leq x \leq 8$. Use the line tangent to the graph of $f$ at $x = 5$ to show that $f(7) \leq 4$. Use the secant line for the graph of $f$ on $5 \leq x \leq 8$ to show that $f(7) \geq \frac{4}{3}$.

The graph of $f ^ { \prime }$, the derivative of a function $f$, consists of two line segments and a semicircle, as shown in the figure above. If $f ( 2 ) = 1$, then $f ( - 5 ) =$
(A) $2 \pi - 2$
(B) $2 \pi - 3$
(C) $2 \pi - 5$
(D) $6 - 2 \pi$
(E) $4 - 2 \pi$

The derivative $f ^ { \prime } ( x )$ of a polynomial function $f ( x )$ is $f ^ { \prime } ( x ) = 6 x ^ { 2 } + 4$. If the graph of $y = f ( x )$ passes through the point $( 0,6 )$, find the value of $f ( 1 )$. [4 points]

For a function $f ( x )$, if $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 5$ and $f ( 0 ) = 4$, find the value of $f ( 1 )$. [3 points]

For a function $f ( x )$ with $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x$ and $f ( 0 ) = 2$, find the value of $f ( 1 )$. [3 points]

A polynomial function $f(x)$ satisfies $$f'(x) = 3x(x-2), \quad f(1) = 6$$ Find the value of $f(2)$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a polynomial function $f(x)$, $f'(x) = 9x^{2} + 4x$ and $f(1) = 6$. What is the value of $f(2)$? [3 points]

Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$?
(A) $h(t)$ depends on $t$
(B) $h(t)$ is not defined for all $t$
(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$
(D) None of the above

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
$\int _ { - 1 } ^ { 1 } g ^ { \prime } ( x ) d x =$
(A) $2 g ( - 1 )$
(B) 0
(C) $- 2 g ( 1 )$
(D) $2 g ( 1 )$

$$f''(x) = 6x - 2, \quad f'(0) = 4, \quad f(0) = 1$$
For the function $f$ that satisfies these conditions, what is the value of $f(1)$?
A) 4
B) 5
C) 6
D) 7
E) 8

$$\begin{aligned} & f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 3 \\ & f ( 0 ) = 2 \end{aligned}$$
Given this, what is the value of $\mathbf { f } ( - \mathbf { 1 } )$?
A) - 2
B) - 1
C) 0
D) 1
E) 2

Below, the graph of the derivative of a function f is given. Given that $f ( 0 ) = 1$, what is the value of $f ( 2 )$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { - 1 } { 2 }$
E) $\frac { - 1 } { 3 }$

The derivative of a function f that is defined and differentiable on the set of real numbers is given as
$$f ^ { \prime } ( x ) = \begin{cases} 1 , & \text{if } x \leq 1 \\ x , & \text{if } x > 1 \end{cases}$$
Given that $f ( 1 ) = 1$, what is the value of $f ( 0 ) + f ( 3 )$?
A) 2
B) 3
C) 4
D) 5
E) 6

In the rectangular coordinate plane, the graph of $f ^ { \prime }$, the derivative of function $f$, is given on the closed interval $[ 0,10 ]$. The areas of the regions between this graph and the x-axis are shown as follows.
$$f ( 0 ) = \frac { - 1 } { 2 }$$
Given that, how many different roots does the function $f$ have on the interval $[ 0 , 10 ]$?
A) 1
B) 2
C) 3
D) 4
E) 5