The height of the water in a conical storage tank is modeled by a differentiable function $h$, where $h ( t )$ is measured in meters and $t$ is measured in hours. At time $t = 0$, the height of the water in the tank is 25 meters. The height is changing at the rate $h ^ { \prime } ( t ) = 2 - \frac { 24 e ^ { - 0.025 t } } { t + 4 }$ meters per hour for $0 \leq t \leq 24$. (a) When the height of the water in the tank is $h$ meters, the volume of water is $V = \frac { 1 } { 3 } \pi h ^ { 3 }$. At what rate is the volume of water changing at time $t = 0$ ? Indicate units of measure. (b) What is the minimum height of the water during the time period $0 \leq t \leq 24$ ? Justify your answer. (c) The line tangent to the graph of $h$ at $t = 16$ is used to approximate the height of the water in the tank. Using the tangent line approximation, at what time $t$ does the height of the water return to 25 meters?
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 graph of the piecewise-defined function $f$ is shown in the figure above. The graph has a vertical tangent line at $x = - 2$ and horizontal tangent lines at $x = - 3$ and $x = - 1$. What are all values of $x , - 4 < x < 3$, at which $f$ is continuous but not differentiable? (A) $x = 1$ (B) $x = - 2$ and $x = 0$ (C) $x = - 2$ and $x = 1$ (D) $x = 0$ and $x = 1$
An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of $2 \pi$ cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is 5 meters? (Note: For a sphere of radius $r$, the surface area is $4 \pi r ^ { 2 }$ and the volume is $\frac { 4 } { 3 } \pi r ^ { 3 }$.) (A) $\frac { 4 \pi } { 5 }$ (B) $40 \pi$ (C) $80 \pi ^ { 2 }$ (D) $100 \pi$
Shown above is a slope field for which of the following differential equations? (A) $\frac { d y } { d x } = x y + x$ (B) $\frac { d y } { d x } = x y + y$ (C) $\frac { d y } { d x } = y + 1$ (D) $\frac { d y } { d x } = ( x + 1 ) ^ { 2 }$
Let $f$ be the piecewise-linear function defined by $$f ( x ) = \begin{cases} 2 x - 2 & \text { for } x < 3 \\ 2 x - 4 & \text { for } x \geq 3 \end{cases}$$ Which of the following statements are true? I. $\lim _ { h \rightarrow 0 ^ { - } } \frac { f ( 3 + h ) - f ( 3 ) } { h } = 2$ II. $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( 3 + h ) - f ( 3 ) } { h } = 2$ III. $f ^ { \prime } ( 3 ) = 2$ (A) None (B) II only (C) I and II only (D) I, II, and III
The function $f$ is continuous for $- 4 \leq x \leq 4$. The graph of $f$ shown above consists of five line segments. What is the average value of $f$ on the interval $- 4 \leq x \leq 4$ ? (A) $\frac { 1 } { 8 }$ (B) $\frac { 3 } { 16 }$ (C) $\frac { 15 } { 16 }$ (D) $\frac { 3 } { 2 }$
Let $y = f ( t )$ be a solution to the differential equation $\frac { d y } { d t } = k y$, where $k$ is a constant. Values of $f$ for selected values of $t$ are given in the table below:
$t$
0
2
$f ( t )$
4
12
Which of the following is an expression for $f ( t )$ ? (A) $4 e ^ { \frac { t } { 2 } \ln 3 }$ (B) $e ^ { \frac { t } { 2 } \ln 9 } + 3$ (C) $2 t ^ { 2 } + 4$ (D) $4 t + 4$
The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following could be the graph of $f$? (A), (B), (C), (D) [graphs as shown in the exam]
The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = e ^ { \sin x } - \cos x - 1$ for $0 < x < 9$. On what intervals is $f$ decreasing? (A) $0 < x < 0.633$ and $4.115 < x < 6.916$ (B) $0 < x < 1.947$ and $5.744 < x < 8.230$ (C) $0.633 < x < 4.115$ and $6.916 < x < 9$ (D) $1.947 < x < 5.744$ and $8.230 < x < 9$
The temperature of a room, in degrees Fahrenheit, is modeled by $H$, a differentiable function of the number of minutes after the thermostat is adjusted. Of the following, which is the best interpretation of $H ^ { \prime } ( 5 ) = 2$ ? (A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is adjusted. (B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes after the thermostat is adjusted. (C) The temperature of the room is increasing at a constant rate of $\frac { 2 } { 5 }$ degree Fahrenheit per minute. (D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute, 5 minutes after the thermostat is adjusted.
A function $f$ is continuous on the closed interval $[ 2,5 ]$ with $f ( 2 ) = 17$ and $f ( 5 ) = 17$. Which of the following additional conditions guarantees that there is a number $c$ in the open interval $( 2,5 )$ such that $f ^ { \prime } ( c ) = 0$ ? (A) No additional conditions are necessary. (B) $f$ has a relative extremum on the open interval $( 2,5 )$. (C) $f$ is differentiable on the open interval $( 2,5 )$. (D) $\int _ { 2 } ^ { 5 } f ( x ) d x$ exists.
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
A race car is traveling on a straight track at a velocity of 80 meters per second when the brakes are applied at time $t = 0$ seconds. From time $t = 0$ to the moment the race car stops, the acceleration of the race car is given by $a ( t ) = - 6 t ^ { 2 } - t$ meters per second per second. During this time period, how far does the race car travel? (A) 188.229 m (B) 198.766 m (C) 260.042 m (D) 267.089 m