Question 1

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

Accepted Answer

A)  x  1 
D)  x > -1 
A)  x  1 
D)  x > -1

Question 2

Find the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 12$ on the interval -1 $\le$  x $\le$ 2.

Accepted Answer

Maximum is

Question 3

A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can expect to receive for the machines will depend on the number of machines made available. (For example, if only a few of the machines are placed on the market, competitive bidding among prospective purchasers will tend to drive the price up.) It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market, the machines will sell for $60 - \frac { x } { 5 } + \frac { y } { 20 }$ thousand dollars apiece at home and for $50 - \frac { y } { 10 } + \frac { x } { 20 }$ thousand dollars apiece abroad. If the manufacturer can produce the machines at a cost of $45,000 apiece, how many should be supplied to each market to generate the largest possible profit?

Accepted Answer

The answer of A manufacturer with exclusive rights to a...

Question 4

Find the points of intersection (if any) of the given pair of curves.y = x + 9 and y = 2x + 4

Accepted Answer

A)  (1, -4) 
B)  (0, 6) 
C)  (14, 23) 
D)  (5, 14) 
A)  (1, -4) 
B)  (0, 6) 
C)  (14, 23) 
D)  (5, 14)

Question 5

Find two non-negative numbers whose sum is 12 for which the product of their squares is as large as possible.

Accepted Answer

The answer of Find two non-negative numbers whose sum is...

Question 6

An equation for the tangent line to the curve $y = \left( x ^ { 7 } + x - 1 \right) ^ { 8 }$ at the point where x = 1 is

Accepted Answer

The answer of An equation for the tangent line to...

Question 7

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 6 }$ at the point where x = 1 is

Accepted Answer

A)  y = 18x 
B)  y = 18x - 1 
C)  y = 18x - 17 
D)  y = 2x + 1 
A)  y = 18x 
B)  y = 18x - 1 
C)  y = 18x - 17 
D)  y = 2x + 1

Question 8

The equation of the tangent line to the curve $f ( x ) = \left( 7 x ^ { 5 } - 5 x ^ { 2 } + 2 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -2) is y = 2x - 2.

Accepted Answer

A) True 
 B)False

Question 9

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1, 1).

Accepted Answer

A)  1 
B)  3 
C)  5 
D)  -5 
A)  1 
B)  3 
C)  5 
D)  -5

Question 10

Compute $f _ { y }$ for $f ( x , y ) = 5 x y ^ { 3 }$

Accepted Answer

A)  $15 x y ^ { 2 }$ 
B)  $y ^ { 3 }$ 
C)  5x 
D)  $15 x y ^ { 2 } + 5 y ^ { 3 }$ 
A)  $15 x y ^ { 2 }$ 
B)  $y ^ { 3 }$ 
C)  5x 
D)  $15 x y ^ { 2 } + 5 y ^ { 3 }$

Question 11

Decide if the given function is continuous at the specified value of x. $$f ( x ) = \left\{ \begin{array} { r l } 
x + 5 & \text { if } x < 1 \
3 x + 3 & \text { if } x \geq 1
\end{array} ; \quad x = 1 \right.$$

Accepted Answer

A)  Yes, the function is continuous at x = 1. 
B)  No, the function is not continuous at x = 1. 
A)  Yes, the function is continuous at x = 1. 
B)  No, the function is not continuous at x = 1.

Question 12

To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 90 days ago, the club has collected 45,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, because bottles are accumulating faster than they can be recycled, the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass. Assume that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible. What is the most advantageous time for the club to conclude its project and deliver the bottles?

Accepted Answer

The answer of To raise money, a service club has...

Question 13

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Accepted Answer

Concave up

Question 14

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 5 }$ the ratio in the atmosphere. Approximately how old is the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years. Round to the nearest whole year.

Accepted Answer

A)  14,325 years 
B)  13,305 years 
C)  28,650 years 
D)  13,335 years 
A)  14,325 years 
B)  13,305 years 
C)  28,650 years 
D)  13,335 years

Question 15

Evaluate $\int x \sqrt { x ^ { 2 } + 3 } d x$

Accepted Answer

A)  $\left( x ^ { 2 } + 3 \right) ^ { 3 / 2 } + C$ 
B)  $\frac { x ^ { 3 } } { 3 } + 3 x + C$ 
C)  $\frac { 3 \left( x ^ { 2 } + 3 \right) ^ { 3 / 2 } } { 4 } + C$ 
D)  $\frac { \left( x ^ { 2 } + 3 \right) ^ { 3 / 2 } } { 3 } + C$ 
A)  $\left( x ^ { 2 } + 3 \right) ^ { 3 / 2 } + C$ 
B)  $\frac { x ^ { 3 } } { 3 } + 3 x + C$ 
C)  $\frac { 3 \left( x ^ { 2 } + 3 \right) ^ { 3 / 2 } } { 4 } + C$ 
D)  $\frac { \left( x ^ { 2 } + 3 \right) ^ { 3 / 2 } } { 3 } + C$

Question 16

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$

Accepted Answer

A)  $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$ 
B)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
C)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
D)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$ 
A)  $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$ 
B)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
C)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
D)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$

Question 17

Money is transferred continuously into an account at the constant rate of $1,900 per year. Assume the account earns interest at the annual rate of 8% compounded continuously. Compute the future value of the income stream over a 9 year period.

Accepted Answer

A)  $100,171.16 
B)  $2,003.42 
C)  $25,042.79 
D)  $50,085.58 
A)  $100,171.16 
B)  $2,003.42 
C)  $25,042.79 
D)  $50,085.58

Question 18

Evaluate $\int e ^ { 8 x - 2 } d x$

Accepted Answer

A)  $e ^ { 8 x - 2 } + C$ 
B)  $( 8 x - 2 ) e ^ { 8 x - 2 } + C$ 
C)  $( 8 x - 2 ) C e ^ { 8 x - 2 }$ 
D)  $\frac { e ^ { 8 x - 2 } } { 8 } + C$ 
A)  $e ^ { 8 x - 2 } + C$ 
B)  $( 8 x - 2 ) e ^ { 8 x - 2 } + C$ 
C)  $( 8 x - 2 ) C e ^ { 8 x - 2 }$ 
D)  $\frac { e ^ { 8 x - 2 } } { 8 } + C$

Question 19

Evaluate the improper integral: $$\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 6 } / 5 } d x$$ Round to two decimal places, if necessary.

Accepted Answer

A)  1.67 
B)  0.83 
C)  Undefined 
D)  1.25 
A)  1.67 
B)  0.83 
C)  Undefined 
D)  1.25

Question 20

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$

Accepted Answer

A) Increasing for all x B) Decreasing for $x - \frac { 5 } { 2 }$ C) Decreasing for $x > - \frac { 5 } { 2 }$ ; increasing for $x < - \frac { 5 } { 2 }$ D) Decreasing for all x A) Increasing for all x B) Decreasing for $x - \frac { 5 } { 2 }$ C) Decreasing for $x > - \frac { 5 } { 2 }$ ; increasing for $x < - \frac { 5 } { 2 }$ D) Decreasing for all x

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

Find the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 12$ on the interval -1 $\le$ x $\le$ 2.

Find the points of intersection (if any) of the given pair of curves.y = x + 9 and y = 2x + 4

Find two non-negative numbers whose sum is 12 for which the product of their squares is as large as possible.

An equation for the tangent line to the curve $y = \left( x ^ { 7 } + x - 1 \right) ^ { 8 }$ at the point where x = 1 is

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 6 }$ at the point where x = 1 is

The equation of the tangent line to the curve $f ( x ) = \left( 7 x ^ { 5 } - 5 x ^ { 2 } + 2 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -2) is y = 2x - 2.

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1, 1).

Compute $f _ { y }$ for $f ( x , y ) = 5 x y ^ { 3 }$

Decide if the given function is continuous at the specified value of x. $f ( x ) = \left\{ \begin{array} { r l } x + 5 & \text { if } x < 1 \\3 x + 3 & \text { if } x \geq 1\end{array} ; \quad x = 1 \right.$

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Evaluate $\int x \sqrt { x ^ { 2 } + 3 } d x$

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$

Money is transferred continuously into an account at the constant rate of $1,900 per year. Assume the account earns interest at the annual rate of 8% compounded continuously. Compute the future value of the income stream over a 9 year period.

Evaluate $\int e ^ { 8 x - 2 } d x$

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 6 } / 5 } d x$ Round to two decimal places, if necessary.

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Filters

Exam 8: Appendix: Algebra Review

Determine where the graph of f(x)=x3−3x2−9x+1f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1f(x)=x3−3x2−9x+1 is concave down.

Find the absolute maximum and minimum of f(x)=x4−8x2+12f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 12f(x)=x4−8x2+12 on the interval -1 ≤\le≤ x ≤\le≤ 2.

Find the points of intersection (if any) of the given pair of curves.y = x + 9 and y = 2x + 4

Find two non-negative numbers whose sum is 12 for which the product of their squares is as large as possible.

An equation for the tangent line to the curve y=(x7+x−1)8y = \left( x ^ { 7 } + x - 1 \right) ^ { 8 }y=(x7+x−1)8 at the point where x = 1 is

An equation for the tangent line to the curve y=(x2+x−1)6y = \left( x ^ { 2 } + x - 1 \right) ^ { 6 }y=(x2+x−1)6 at the point where x = 1 is

The equation of the tangent line to the curve f(x)=(7x5−5x2+2)(x3+x−1)f ( x ) = \left( 7 x ^ { 5 } - 5 x ^ { 2 } + 2 \right) \left( x ^ { 3 } + x - 1 \right)f(x)=(7x5−5x2+2)(x3+x−1) at the point (0, -2) is y = 2x - 2.

Find the slope of the tangent line to the curve x2+3xy−y2=3x ^ { 2 } + 3 x y - y ^ { 2 } = 3x2+3xy−y2=3 at the point (1, 1).

Compute fyf _ { y }fy​ for f(x,y)=5xy3f ( x , y ) = 5 x y ^ { 3 }f(x,y)=5xy3

Decide if the given function is continuous at the specified value of x. f(x)={x+5 if x<13x+3 if x≥1;x=1f ( x ) = \left\{ \begin{array} { r l } x + 5 & \text { if } x < 1 \\3 x + 3 & \text { if } x \geq 1\end{array} ; \quad x = 1 \right.f(x)={x+53x+3​ if x<1 if x≥1​;x=1

Determine where the graph of f(x)=x23f ( x ) = \frac { x ^ { 2 } } { 3 }f(x)=3x2​ is concave up and concave down.

Evaluate ∫xx2+3dx\int x \sqrt { x ^ { 2 } + 3 } d x∫xx2+3​dx

Evaluate ∫x3(x2−3)7dx\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x∫x3(x2−3)7dx

Money is transferred continuously into an account at the constant rate of $1,900 per year. Assume the account earns interest at the annual rate of 8% compounded continuously. Compute the future value of the income stream over a 9 year period.

Evaluate ∫e8x−2dx\int e ^ { 8 x - 2 } d x∫e8x−2dx

Evaluate the improper integral: ∫0∞x5e−x6/5dx\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 6 } / 5 } d x∫0∞​x5e−x6/5dx Round to two decimal places, if necessary.

Find the intervals of increase and decrease for the function f(x)=x2+5x−3f ( x ) = x ^ { 2 } + 5 x - 3f(x)=x2+5x−3

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Filters

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

Find the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 12$ on the interval -1 $\le$ x $\le$ 2.

An equation for the tangent line to the curve $y = \left( x ^ { 7 } + x - 1 \right) ^ { 8 }$ at the point where x = 1 is

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 6 }$ at the point where x = 1 is

The equation of the tangent line to the curve $f ( x ) = \left( 7 x ^ { 5 } - 5 x ^ { 2 } + 2 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -2) is y = 2x - 2.

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1, 1).

Compute $f _ { y }$ for $f ( x , y ) = 5 x y ^ { 3 }$

Decide if the given function is continuous at the specified value of x. $f ( x ) = \left\{ \begin{array} { r l } x + 5 & \text { if } x < 1 \\3 x + 3 & \text { if } x \geq 1\end{array} ; \quad x = 1 \right.$

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Evaluate $\int x \sqrt { x ^ { 2 } + 3 } d x$

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$

Evaluate $\int e ^ { 8 x - 2 } d x$

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 6 } / 5 } d x$ Round to two decimal places, if necessary.

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$