Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = \ln x$. (a) Find the area of the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$. (b) Find the volume of the solid generated when the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$ is revolved about the line $y = 4$. (c) Let $h$ be the function given by $h ( x ) = f ( x ) - g ( x )$. Find the absolute minimum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$, and find the absolute maximum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$. Show the analysis that leads to your answers.
The rate at which people enter an amusement park on a given day is modeled by the function $E$ defined by $$E ( t ) = \frac { 15600 } { \left( t ^ { 2 } - 24 t + 160 \right) }$$ The rate at which people leave the same amusement park on the same day is modeled by the function $L$ defined by $$L ( t ) = \frac { 9890 } { \left( t ^ { 2 } - 38 t + 370 \right) }$$ Both $E ( t )$ and $L ( t )$ are measured in people per hour and time $t$ is measured in hours after midnight. These functions are valid for $9 \leq t \leq 23$, the hours during which the park is open. At time $t = 9$, there are no people in the park. (a) How many people have entered the park by 5:00 P.M. ( $t = 17$ )? Round your answer to the nearest whole number. (b) The price of admission to the park is $\$15$ until 5:00 P.M. ( $t = 17$ ). After 5:00 P.M., the price of admission to the park is $\$11$. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. (c) Let $H ( t ) = \int _ { 9 } ^ { t } ( E ( x ) - L ( x ) ) d x$ for $9 \leq t \leq 23$. The value of $H ( 17 )$ to the nearest whole number is 3725. Find the value of $H ^ { \prime } ( 17 )$, and explain the meaning of $H ( 17 )$ and $H ^ { \prime } ( 17 )$ in the context of the amusement park. (d) At what time $t$, for $9 \leq t \leq 23$, does the model predict that the number of people in the park is a maximum?
The position of a roller coaster car at time $t$ seconds can be modeled parametrically by $$\begin{aligned}
& x ( t ) = 10 t + 4 \sin t \\
& y ( t ) = ( 20 - t ) ( 1 - \cos t ) ,
\end{aligned}$$ where $x$ and $y$ are measured in meters, over the time interval $0 \leq t \leq 18$ seconds. The derivatives of these functions are given by $$\begin{aligned}
& x ^ { \prime } ( t ) = 10 + 4 \cos t \\
& y ^ { \prime } ( t ) = ( 20 - t ) \sin t + \cos t - 1
\end{aligned}$$ (a) Find the slope of the path at time $t = 2$. Show the computations that lead to your answer. (b) Find the acceleration vector of the car at the time when the car's horizontal position is $x = 140$. (c) Find the time $t$ at which the car is at its maximum height, and find the speed, in $\mathrm { m } / \mathrm { sec }$, of the car at this time. (d) For $0 < t < 18$, there are two times at which the car is at ground level ( $y = 0$ ). Find these two times and write an expression that gives the average speed, in $\mathrm { m } / \mathrm { sec }$, of the car between these two times. Do not evaluate the expression.
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 $[ - 2, 2 ]$.
Consider the differential equation $\frac { d y } { d x } = 2 y - 4 x$. (a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through the point $( 0, 1 )$ and sketch the solution curve that passes through the point $( 0 , - 1 )$. (b) Let $f$ be the function that satisfies the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with a step size of 0.1, to approximate $f ( 0.2 )$. Show the work that leads to your answer. (c) Find the value of $b$ for which $y = 2 x + b$ is a solution to the given differential equation. Justify your answer. (d) Let $g$ be the function that satisfies the given differential equation with the initial condition $g ( 0 ) = 0$. Does the graph of $g$ have a local extremum at the point $( 0, 0 )$? If so, is the point a local maximum or a local minimum? Justify your answer.