Question 1

Find the slowest growing and the fastest growing functions as x→∞ .
-$\begin{array} { l } 
y = x + 7 \
y = e ^ { x } \
y = x ^ { 3 } + \cos ^ { 2 } x \
y = 4 ^ { x }
\end{array}$

Accepted Answer

A)  Slowest: $y = x + 7$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { X } }$ 
B)  Slowest: $y = e ^ { x }$
Fastest: $y = x ^ { 3 } + \cos ^ { 2 } x$ 
C)  Slowest: $y = x + 7$
Fastest: $y = x ^ { 3 } + \cos ^ { 2 } x$ 
D)  Slowest: $y = x + 7$
Fastest: $y = 4 ^ { x }$ 
A)  Slowest: $y = x + 7$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { X } }$ 
B)  Slowest: $y = e ^ { x }$
Fastest: $y = x ^ { 3 } + \cos ^ { 2 } x$ 
C)  Slowest: $y = x + 7$
Fastest: $y = x ^ { 3 } + \cos ^ { 2 } x$ 
D)  Slowest: $y = x + 7$
Fastest: $y = 4 ^ { x }$

Question 2

Evaluate the integral.
-$$\int _ { - \ln 3 } ^ { 0 } 5 \sinh ^ { 2 } \left( \frac { x } { 2 } \right) d x$$

Accepted Answer

A)  $\frac { 5 } { 3 } \left( e ^ { 6 } - e ^ { - 6 } \right)$ 
B)  $\frac { 5 } { 2 } \left( \frac { 4 } { 3 } - \ln 3 \right)$ 
C)  $\frac { 5 } { 2 } \left( \frac { 4 } { 3 } + \ln 3 \right)$ 
D)  $- \frac { 320 } { 81 }$ 
A)  $\frac { 5 } { 3 } \left( e ^ { 6 } - e ^ { - 6 } \right)$ 
B)  $\frac { 5 } { 2 } \left( \frac { 4 } { 3 } - \ln 3 \right)$ 
C)  $\frac { 5 } { 2 } \left( \frac { 4 } { 3 } + \ln 3 \right)$ 
D)  $- \frac { 320 } { 81 }$

Question 3

Solve the problem.
-The barometric pressure $\mathrm { p }$ at an altitude of $\mathrm { h }$ miles above sea level satisfies the differential equation $\frac { d p } { d h } = - 0.2 p$. If the pressure at sea level is $29.92$ inches of mercury, find the barometric pressure at $29,000 \mathrm { ft }$.

Accepted Answer

A)  4.99 in. 
B)  9.97 in. 
C)  89.75 in. 
D)  0.09 in. 
A)  4.99 in. 
B)  9.97 in. 
C)  89.75 in. 
D)  0.09 in.

Question 4

Find the derivative of y with respect to the appropriate variable.
-$$y = ( 1 - 8 t ) \operatorname { coth } ^ { - 1 } \sqrt { 8 t }$$

Accepted Answer

A)  $( 1 - 8 \mathrm { t } ) \operatorname { coth } ^ { - 1 } \sqrt { 8 \mathrm { t } }$ 
B)  $\frac { \sqrt { 8 } } { 2 \sqrt { t } } - 8 \tanh ^ { - 1 } \sqrt { 8 t }$ 
C)  $- 4 \mathrm { t }$ 
D)  $\frac { \sqrt { 8 } } { 2 \sqrt { t } } - 8 \operatorname { coth } ^ { - 1 } \sqrt { 8 t }$ 
A)  $( 1 - 8 \mathrm { t } ) \operatorname { coth } ^ { - 1 } \sqrt { 8 \mathrm { t } }$ 
B)  $\frac { \sqrt { 8 } } { 2 \sqrt { t } } - 8 \tanh ^ { - 1 } \sqrt { 8 t }$ 
C)  $- 4 \mathrm { t }$ 
D)  $\frac { \sqrt { 8 } } { 2 \sqrt { t } } - 8 \operatorname { coth } ^ { - 1 } \sqrt { 8 t }$

Question 5

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi / 20 } \frac { \sec ^ { 2 } 5 x } { 5 + \tan 5 x } d x$$

Accepted Answer

A)  $\frac { 1 } { 5 } \ln \left| \frac { 1 } { 5 } \right|$ 
B)  $\ln \left| \frac { 6 } { 5 } \right|$ 
C)  $\frac { 1 } { 5 } \ln \left| \frac { 6 } { 5 } \right|$ 
D)  $e ^ { 6 / 5 }$ 
A)  $\frac { 1 } { 5 } \ln \left| \frac { 1 } { 5 } \right|$ 
B)  $\ln \left| \frac { 6 } { 5 } \right|$ 
C)  $\frac { 1 } { 5 } \ln \left| \frac { 6 } { 5 } \right|$ 
D)  $e ^ { 6 / 5 }$

Question 6

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi / 8 } \left( 1 + e ^ { \tan 2 x } \right) \sec ^ { 2 } 2 x d x$$

Accepted Answer

A)  $2 \mathrm { e }$ 
B)  $- \frac { e } { 2 }$ 
C)  $\frac { e } { 2 }$ 
D)  e 
A)  $2 \mathrm { e }$ 
B)  $- \frac { e } { 2 }$ 
C)  $\frac { e } { 2 }$ 
D)  e

Question 7

Evaluate the integral.
-$$\int _ { \ln 2 } ^ { \ln 4 } 12 e ^ { t } \cosh t d t$$

Accepted Answer

A)  $36 + 6 \ln 2$ 
B)  $60 + 6 \ln 8$ 
C)  $36 + \ln 2$ 
D)  $12 + 6 \ln 2$ 
A)  $36 + 6 \ln 2$ 
B)  $60 + 6 \ln 8$ 
C)  $36 + \ln 2$ 
D)  $12 + 6 \ln 2$

Question 8

Rewrite the expression in terms of exponentials and simplify the results.
-$16 \cosh ( \ln x ) + 6 \sinh ( \ln x )$

Accepted Answer

A)  0 
B)  $11 x + \frac { 5 } { x }$ 
C)  $11 \left( x + \frac { 1 } { x } \right)$ 
D)  $11 \mathrm { x }$ 
A)  0 
B)  $11 x + \frac { 5 } { x }$ 
C)  $11 \left( x + \frac { 1 } { x } \right)$ 
D)  $11 \mathrm { x }$

Question 9

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$$\begin{array} { l } 
y = \frac { 1 } { 3 x } \int _ { 1 } ^ { 3 x } \frac { e ^ { t } } { t } d t \
x ^ { 2 } y ^ { \prime } + x y = \frac { 1 } { 3 } e ^ { 3 x }
\end{array}$$

Accepted Answer

A) True 
 B)False

Question 10

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function.
-$\cosh x = \frac { 17 } { 15 } , x < 0 , \operatorname { csch } x =$

Accepted Answer

A)  $\frac { 15 } { 8 }$ 
B)  $- \frac { 8 } { 15 }$ 
C)  $- \frac { 15 } { 8 }$ 
D)  $- \frac { 15 } { 17 }$ 
A)  $\frac { 15 } { 8 }$ 
B)  $- \frac { 8 } { 15 }$ 
C)  $- \frac { 15 } { 8 }$ 
D)  $- \frac { 15 } { 17 }$

Question 11

Solve the differential equation.
-$$\frac { d y } { d x } = 3 x ^ { 2 } \cos ^ { 2 } y$$

Accepted Answer

A)  $y = x ^ { 3 } + C$ 
B)  $y = \tan ^ { - 1 } \left( x ^ { 3 } + C \right)$ 
C)  $y = \tan ^ { - 1 } \left( x ^ { 2 } + C \right)$ 
D)  $y = \tan \left( x ^ { 3 } + C \right)$ 
A)  $y = x ^ { 3 } + C$ 
B)  $y = \tan ^ { - 1 } \left( x ^ { 3 } + C \right)$ 
C)  $y = \tan ^ { - 1 } \left( x ^ { 2 } + C \right)$ 
D)  $y = \tan \left( x ^ { 3 } + C \right)$

Question 12

Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l } 
y = 2 x ^ { 2 } + 9 x \
y = e ^ { x } \
y = e ^ { x } / 6 \
y = \log _ { 4 } x
\end{array}$$

Accepted Answer

A)  Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { X } }$ 
B)  Slowest: $2 x ^ { 2 } + 9 x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ 
C)  Slowest: $y = e ^ { x } / 6$
Fastest: $2 x ^ { 2 } + 9 x$ 
D)  Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ and $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } / 6$ grow at the same rate. 
A)  Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { X } }$ 
B)  Slowest: $2 x ^ { 2 } + 9 x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ 
C)  Slowest: $y = e ^ { x } / 6$
Fastest: $2 x ^ { 2 } + 9 x$ 
D)  Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ and $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } / 6$ grow at the same rate.

Question 13

Evaluate the integral.
-$$\int \frac { \sec x \tan x } { 5 + \sec x } d x$$

Accepted Answer

Question 14

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function.
-$\sinh x = - \frac { 12 } { 5 } , \operatorname { csch } x =$

Accepted Answer

A)  $\frac { 169 } { 25 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $\frac { 13 } { 5 }$ 
D)  $- \frac { 5 } { 12 }$ 
A)  $\frac { 169 } { 25 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $\frac { 13 } { 5 }$ 
D)  $- \frac { 5 } { 12 }$

Question 15

Evaluate the integral.
-$$\int 2 e ^ { 5 x } d x$$

Accepted Answer

A)  $2 \mathrm { e } ^ { 5 \mathrm { x } } + \mathrm { C }$ 
B)  $\frac { 1 } { 3 } \mathrm { e } ^ { 5 \mathrm { x } + 1 } + \mathrm { C }$ 
C)  $\frac { 1 } { 5 } \mathrm { e } ^ { 5 \mathrm { x } ^ { 2 } } + \mathrm { C }$ 
D)  $\frac { 2 } { 5 } \mathrm { e } ^ { 5 \mathrm { x } } + \mathrm { C }$ 
A)  $2 \mathrm { e } ^ { 5 \mathrm { x } } + \mathrm { C }$ 
B)  $\frac { 1 } { 3 } \mathrm { e } ^ { 5 \mathrm { x } + 1 } + \mathrm { C }$ 
C)  $\frac { 1 } { 5 } \mathrm { e } ^ { 5 \mathrm { x } ^ { 2 } } + \mathrm { C }$ 
D)  $\frac { 2 } { 5 } \mathrm { e } ^ { 5 \mathrm { x } } + \mathrm { C }$

Question 16

Evaluate the integral.
-$$\int _ { 1 } ^ { \sqrt { 2 } } x 4 ^ { x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 4 \sqrt { 2 } - 4 } { 2 \ln 4 }$ 
B)  $\frac { 4 } { \ln 4 }$ 
C)  6 
D)  $\frac { 6 } { \ln 4 }$ 
A)  $\frac { 4 \sqrt { 2 } - 4 } { 2 \ln 4 }$ 
B)  $\frac { 4 } { \ln 4 }$ 
C)  6 
D)  $\frac { 6 } { \ln 4 }$

Question 17

Evaluate the integral.
-$\int t \sqrt { 8 } - 1 d t$

Accepted Answer

A)  1 
B)  $\frac { \mathrm { t } \sqrt { 8 } } { \sqrt { 8 } } + C$ 
C)  $\frac { t \sqrt { 8 } - 1 } { \ln t } + C$ 
D)  $\frac { t ^ { \sqrt { 8 } } - 2 } { \sqrt { 8 } - 2 } + C$ 
A)  1 
B)  $\frac { \mathrm { t } \sqrt { 8 } } { \sqrt { 8 } } + C$ 
C)  $\frac { t \sqrt { 8 } - 1 } { \ln t } + C$ 
D)  $\frac { t ^ { \sqrt { 8 } } - 2 } { \sqrt { 8 } - 2 } + C$

Question 18

Evaluate the integral.
-$$\int _ { 1 } ^ { 2 } \frac { 4 ^ { \ln x } } { x } d x$$

Accepted Answer

A)  $\frac { 12 } { \ln 4 }$ 
B)  $\frac { 7 } { \ln 4 }$ 
C)  $\frac { 4 ^ { \ln 2 } - 1 } { \ln 4 }$ 
D)  $\frac { 4 ^ { \ln 2 } } { \ln 4 }$ 
A)  $\frac { 12 } { \ln 4 }$ 
B)  $\frac { 7 } { \ln 4 }$ 
C)  $\frac { 4 ^ { \ln 2 } - 1 } { \ln 4 }$ 
D)  $\frac { 4 ^ { \ln 2 } } { \ln 4 }$

Question 19

Solve the problem.
-The charcoal from a tree killed in a volcanic eruption contained 68.2% of the carbon-14 found in living matter. How old is the tree, to the nearest year? Use 5700 years for the half-life of carbon-14.

Accepted Answer

A)  5700 years 
B)  2182 years 
C)  1512 years 
D)  3147 years 
A)  5700 years 
B)  2182 years 
C)  1512 years 
D)  3147 years

Question 20

Evaluate the integral.
-$$\int _ { 0 } ^ { \ln 5 } 5 \cosh ^ { 2 } \left( \frac { x } { 2 } \right) d x$$

Accepted Answer

A)  $\frac { 5 } { 2 } \left( \frac { 13 } { 5 } - \ln 5 \right)$ 
B)  $\frac { 5 } { 2 } \left( \frac { 12 } { 5 } + \ln 5 \right)$ 
C)  $\frac { 5 } { 3 } \left( \mathrm { e } ^ { 15 } - \mathrm { e } ^ { - 15 } \right)$ 
D)  $\frac { 2197 } { 25 }$ 
A)  $\frac { 5 } { 2 } \left( \frac { 13 } { 5 } - \ln 5 \right)$ 
B)  $\frac { 5 } { 2 } \left( \frac { 12 } { 5 } + \ln 5 \right)$ 
C)  $\frac { 5 } { 3 } \left( \mathrm { e } ^ { 15 } - \mathrm { e } ^ { - 15 } \right)$ 
D)  $\frac { 2197 } { 25 }$

Find the slowest growing and the fastest growing functions as x→∞ . - y=x+7 y= y=+x y=

Evaluate the integral. - $\int _ { - \ln 3 } ^ { 0 } 5 \sinh ^ { 2 } \left( \frac { x } { 2 } \right) d x$

Find the derivative of y with respect to the appropriate variable. - $y = ( 1 - 8 t ) \operatorname { coth } ^ { - 1 } \sqrt { 8 t }$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 20 } \frac { \sec ^ { 2 } 5 x } { 5 + \tan 5 x } d x$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 8 } \left( 1 + e ^ { \tan 2 x } \right) \sec ^ { 2 } 2 x d x$

Evaluate the integral. - $\int _ { \ln 2 } ^ { \ln 4 } 12 e ^ { t } \cosh t d t$

Rewrite the expression in terms of exponentials and simplify the results. - $16 \cosh ( \ln x ) + 6 \sinh ( \ln x )$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - y=dt +xy=

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function. - $\cosh x = \frac { 17 } { 15 } , x < 0 , \operatorname { csch } x =$

Solve the differential equation. - $\frac { d y } { d x } = 3 x ^ { 2 } \cos ^ { 2 } y$

Find the slowest growing and the fastest growing functions as x→∞ . - y=2+9x y= y=/6 y=x

Evaluate the integral. - $\int \frac { \sec x \tan x } { 5 + \sec x } d x$

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function. - $\sinh x = - \frac { 12 } { 5 } , \operatorname { csch } x =$

Evaluate the integral. - $\int 2 e ^ { 5 x } d x$

Evaluate the integral. - $\int _ { 1 } ^ { \sqrt { 2 } } x 4 ^ { x ^ { 2 } } d x$

Evaluate the integral. - $\int t \sqrt { 8 } - 1 d t$

Evaluate the integral. - $\int _ { 1 } ^ { 2 } \frac { 4 ^ { \ln x } } { x } d x$

Solve the problem. -The charcoal from a tree killed in a volcanic eruption contained 68.2% of the carbon-14 found in living matter. How old is the tree, to the nearest year? Use 5700 years for the half-life of carbon-14.

Evaluate the integral. - $\int _ { 0 } ^ { \ln 5 } 5 \cosh ^ { 2 } \left( \frac { x } { 2 } \right) d x$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 8: Integrals and Transcendental Functions