Question 1

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

Accepted Answer

A)  Slope: $\frac { 3 } { 2 }$ ; y-intercept: $\frac { 1 } { 2 }$ 
B)  Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3 
C)  Slope: $\frac { 3 } { 2 }$ ; y-intercept: 1 
D)  Slope: $\frac { 1 } { 2 }$ ; y-intercept: 1 
A)  Slope: $\frac { 3 } { 2 }$ ; y-intercept: $\frac { 1 } { 2 }$ 
B)  Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3 
C)  Slope: $\frac { 3 } { 2 }$ ; y-intercept: 1 
D)  Slope: $\frac { 1 } { 2 }$ ; y-intercept: 1

Question 2

Compute $f _ { x }$ for $f ( x , y ) = 5 x y ^ { 7 }$

Accepted Answer

A)  $35 x y ^ { 6 } + 5 y ^ { 7 }$ 
B)  $5 y ^ { 7 }$ 
C)  $35 x y ^ { 6 }$ 
D)  5x 
A)  $35 x y ^ { 6 } + 5 y ^ { 7 }$ 
B)  $5 y ^ { 7 }$ 
C)  $35 x y ^ { 6 }$ 
D)  5x

Question 3

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x$ .

Accepted Answer

The answer of Evaluate \(\int x ^ { 7 }...

Question 4

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

Accepted Answer

The answer of Find the maximum value of f (x,...

Question 5

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 1$ that passes through (9, 4) is y = 2x + 1.

Accepted Answer

A) True 
 B)False

Question 6

Compute f (1, 5) if $f ( x , y ) = 7 x y ^ { 3 }$ .

Accepted Answer

The answer of Compute f (1, 5) if \(f (...

Question 7

Simplify: $\frac { 2 ^ { 3 } } { 2 ^ { 2 } }$

Accepted Answer

The answer of Simplify: \(\frac { 2 ^ { 3...

Question 8

Sketch the region R and then use calculus to find the area of R. R is the region between the curve $y = x ^ { 3 }$ and the line y = 20x for x $\ge$  0.

Accepted Answer

The answer of Sketch the region R and then use...

Question 9

Evaluate the following integral: $\int 7 e ^ { 3 x } d x$

Accepted Answer

A)  $\frac { 7 } { 3 } e ^ { 3 x } + C$ 
B)  $21 e ^ { 3 x } + C$ 
C)  $\frac { 7 } { 4 } e ^ { 4 x } + C$ 
D)  $7 e ^ { 3 x } + C$ 
A)  $\frac { 7 } { 3 } e ^ { 3 x } + C$ 
B)  $21 e ^ { 3 x } + C$ 
C)  $\frac { 7 } { 4 } e ^ { 4 x } + C$ 
D)  $7 e ^ { 3 x } + C$

Question 10

The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.3 t }$ What fraction can be expected to fail before 5 years of use? Round your answer to two decimal places, if necessary.

Accepted Answer

A)  4.48 
B)  0.78 
C)  1.00 
D)  0.22 
A)  4.48 
B)  0.78 
C)  1.00 
D)  0.22

Question 11

An investment will generate income continuously at the constant rate of $1,500 per year for 5 years. If the prevailing annual interest rate remains fixed at 13% compounded continuously, what is the present value of the investment?

Accepted Answer

A)  $551.49 
B)  $55.15 
C)  $6,688.26 
D)  $5,514.86 
A)  $551.49 
B)  $55.15 
C)  $6,688.26 
D)  $5,514.86

Question 12

Evaluate $\int x \sqrt { x ^ { 2 } + 6 } d x$ .

Accepted Answer

The answer of Evaluate \(\int x \sqrt { x ^...

Question 13

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 4 }$ with respect to t when t = 17?

Accepted Answer

A)  21 
B)  -21 
C)  $- \frac { 1 } { 21 }$ 
D)  $\frac { 1 } { 21 }$ 
A)  21 
B)  -21 
C)  $- \frac { 1 } { 21 }$ 
D)  $\frac { 1 } { 21 }$

Question 14

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 40 days ago, the club has collected 16,000 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

A)  40 days from now 
B)  Today 
C)  30 days from now 
D)  80 days from now 
A)  40 days from now 
B)  Today 
C)  30 days from now 
D)  80 days from now

Question 15

Use a double integral to find the area of R.R is the region bounded by y = 15x, y = ln x, y = 0, and y = 1.

Accepted Answer

A)  $\int _ { y = 0 } ^ { y = 1 } \int _ { x = \frac { y } { 15 } } ^ { x = e ^ { y } } 1 d x d y = e - \frac { 31 } { 30 }$ 
B)  $\int _ { y = 0 } ^ { y = 1 } \int _ { x = 0 } ^ { x =e ^ { y } } 1 d x d y = e - 1$ 
C)  $\int _ { x= 0 } ^ { x - 2 } \int _ { y = \ln x } ^ { y =1 } 1 d y d x = e$ 
D)  $\int _ { x = 0 } ^ { x= e } \int _ { y = \ln x } ^ { y = 15 x } 1 d y d x = \frac { 15 e ^ { 2 } } { 2 }$ 
A)  $\int _ { y = 0 } ^ { y = 1 } \int _ { x = \frac { y } { 15 } } ^ { x = e ^ { y } } 1 d x d y = e - \frac { 31 } { 30 }$ 
B)  $\int _ { y = 0 } ^ { y = 1 } \int _ { x = 0 } ^ { x =e ^ { y } } 1 d x d y = e - 1$ 
C)  $\int _ { x= 0 } ^ { x - 2 } \int _ { y = \ln x } ^ { y =1 } 1 d y d x = e$ 
D)  $\int _ { x = 0 } ^ { x= e } \int _ { y = \ln x } ^ { y = 15 x } 1 d y d x = \frac { 15 e ^ { 2 } } { 2 }$

Question 16

The function y = ln 2x is concave downward everywhere.

Accepted Answer

A) True 
 B)False

Question 17

Differentiate: $f ( x ) = x ^ { 6 } + 5$

Accepted Answer

The answer of Differentiate: \(f ( x ) = x...

Question 18

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 8 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 8 \ln | x | + C$

Accepted Answer

A) True 
 B)False

Question 19

Use Lagrange multipliers to find the maximum value of f (x, y, z) = 4xyz subject to 3x + 3y + 2z = 54.

Accepted Answer

A)  f (7, 4, 9) = 1,008 
B)  f (0, 0, 0) = 0 
C)  f (9, 7, 7) = 1,764 
D)  f (6, 6, 9) = 1,296 
A)  f (7, 4, 9) = 1,008 
B)  f (0, 0, 0) = 0 
C)  f (9, 7, 7) = 1,764 
D)  f (6, 6, 9) = 1,296

Question 20

The equation of the tangent line to $f ( x ) = 8 \ln x ^ { 3 }$ at x = e is

Accepted Answer

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

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

Compute $f _ { x }$ for $f ( x , y ) = 5 x y ^ { 7 }$

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x$ .

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 1$ that passes through (9, 4) is y = 2x + 1.

Compute f (1, 5) if $f ( x , y ) = 7 x y ^ { 3 }$ .

Simplify: $\frac { 2 ^ { 3 } } { 2 ^ { 2 } }$

Sketch the region R and then use calculus to find the area of R. R is the region between the curve $y = x ^ { 3 }$ and the line y = 20x for x $\ge$ 0.

Evaluate the following integral: $\int 7 e ^ { 3 x } d x$

The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.3 t }$ What fraction can be expected to fail before 5 years of use? Round your answer to two decimal places, if necessary.

An investment will generate income continuously at the constant rate of $1,500 per year for 5 years. If the prevailing annual interest rate remains fixed at 13% compounded continuously, what is the present value of the investment?

Evaluate $\int x \sqrt { x ^ { 2 } + 6 } d x$ .

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 4 }$ with respect to t when t = 17?

Use a double integral to find the area of R.R is the region bounded by y = 15x, y = ln x, y = 0, and y = 1.

The function y = ln 2x is concave downward everywhere.

Differentiate: $f ( x ) = x ^ { 6 } + 5$

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 8 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 8 \ln | x | + C$

Use Lagrange multipliers to find the maximum value of f (x, y, z) = 4xyz subject to 3x + 3y + 2z = 54.

The equation of the tangent line to $f ( x ) = 8 \ln x ^ { 3 }$ at x = e is

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

Find the slope and y-intercept of the line whose equation is given. x2+y3=1\frac { x } { 2 } + \frac { y } { 3 } = 12x​+3y​=1

Compute fxf _ { x }fx​ for f(x,y)=5xy7f ( x , y ) = 5 x y ^ { 7 }f(x,y)=5xy7

Evaluate ∫x7(x4−6)7dx\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x∫x7(x4−6)7dx .

Find the maximum value of f (x, y) = xy on the ellipse 4x2+9y2=364 x ^ { 2 } + 9 y ^ { 2 } = 364x2+9y2=36 .

The equation of the line tangent to the graph of f(x)=x+1f ( x ) = \sqrt { x } + 1f(x)=x​+1 that passes through (9, 4) is y = 2x + 1.

Compute f (1, 5) if f(x,y)=7xy3f ( x , y ) = 7 x y ^ { 3 }f(x,y)=7xy3 .

Simplify: 2322\frac { 2 ^ { 3 } } { 2 ^ { 2 } }2223​

Sketch the region R and then use calculus to find the area of R. R is the region between the curve y=x3y = x ^ { 3 }y=x3 and the line y = 20x for x ≥\ge≥ 0.

Evaluate the following integral: ∫7e3xdx\int 7 e ^ { 3 x } d x∫7e3xdx

An investment will generate income continuously at the constant rate of $1,500 per year for 5 years. If the prevailing annual interest rate remains fixed at 13% compounded continuously, what is the present value of the investment?

Evaluate ∫xx2+6dx\int x \sqrt { x ^ { 2 } + 6 } d x∫xx2+6​dx .

What is the rate of change of f(t)=4t−5t+4f ( t ) = \frac { 4 t - 5 } { t + 4 }f(t)=t+44t−5​ with respect to t when t = 17?

Use a double integral to find the area of R.R is the region bounded by y = 15x, y = ln x, y = 0, and y = 1.

The function y = ln 2x is concave downward everywhere.

Differentiate: f(x)=x6+5f ( x ) = x ^ { 6 } + 5f(x)=x6+5

∫x3−3x2+8xdx=x33−3x22+8ln⁡∣x∣+C\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 8 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 8 \ln | x | + C∫xx3−3x2+8​dx=3x3​−23x2​+8ln∣x∣+C

Use Lagrange multipliers to find the maximum value of f (x, y, z) = 4xyz subject to 3x + 3y + 2z = 54.

The equation of the tangent line to f(x)=8ln⁡x3f ( x ) = 8 \ln x ^ { 3 }f(x)=8lnx3 at x = e is

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

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

Compute $f _ { x }$ for $f ( x , y ) = 5 x y ^ { 7 }$

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x$ .

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 1$ that passes through (9, 4) is y = 2x + 1.

Compute f (1, 5) if $f ( x , y ) = 7 x y ^ { 3 }$ .

Simplify: $\frac { 2 ^ { 3 } } { 2 ^ { 2 } }$

Sketch the region R and then use calculus to find the area of R. R is the region between the curve $y = x ^ { 3 }$ and the line y = 20x for x $\ge$ 0.

Evaluate the following integral: $\int 7 e ^ { 3 x } d x$

Evaluate $\int x \sqrt { x ^ { 2 } + 6 } d x$ .

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 4 }$ with respect to t when t = 17?

Differentiate: $f ( x ) = x ^ { 6 } + 5$

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 8 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 8 \ln | x | + C$

The equation of the tangent line to $f ( x ) = 8 \ln x ^ { 3 }$ at x = e is