Question 1

Evaluate the integral.   $\int _ { 0 } ^ { 1 } x ^ { 2 } ( x + 2 ) ^ { 2 } \mathrm {~d} x$

Accepted Answer

A)  $- \frac { 253 } { 60 }$ 
B)  $\frac { 38 } { 15 }$ 
C)  $\frac { 38 } { 5 }$ 
D)  $\frac { 38 } { 3 }$ 
E)  $\frac { 481 } { 30 }$ 
A)  $- \frac { 253 } { 60 }$ 
B)  $\frac { 38 } { 15 }$ 
C)  $\frac { 38 } { 5 }$ 
D)  $\frac { 38 } { 3 }$ 
E)  $\frac { 481 } { 30 }$

Question 2

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate.   $R ( t ) = 50,000$ , $0 \leq t \leq 15$ , at 9%

Accepted Answer

A)  $T V = \$ 50,000$ , $P V = \$ 1,587,458.63$ 
B)  $T V = \$ 750,000$ , $P V = \$ 411,533.19$ 
C)  $T V = \$ 0$ , $P V = \$ 37,037.99$ 
D)  $T V = \$ 750,000$ , $P V = \$ 2,698,569.74$ 
E)  $T V = \$ 0$ , $P V = \$ 411,533.19$ 
A)  $T V = \$ 50,000$ , $P V = \$ 1,587,458.63$ 
B)  $T V = \$ 750,000$ , $P V = \$ 411,533.19$ 
C)  $T V = \$ 0$ , $P V = \$ 37,037.99$ 
D)  $T V = \$ 750,000$ , $P V = \$ 2,698,569.74$ 
E)  $T V = \$ 0$ , $P V = \$ 411,533.19$

Question 3

Evaluate the integral. $\int \left( 5 z ^ { 2 } e ^ { z } - 5 z e ^ { z ^ { 2 } } \right) \mathrm { d } z$

Accepted Answer

A)  $5 \left( z ^ { 2 } + 2 z - 2 \right) e ^ { z } - \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
B)  $5 \left( z ^ { 2 } + 2 z - 2 \right) e ^ { z } + \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
C)  $5 \left( z ^ { 2 } - 2 z + 2 \right) e ^ { z } - \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
D)  $5 \left( z ^ { 2 } - 2 z + 2 \right) e ^ { z } + \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
E)  $5 \left( z ^ { 2 } - 2 z - 2 \right) e ^ { z } - \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
A)  $5 \left( z ^ { 2 } + 2 z - 2 \right) e ^ { z } - \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
B)  $5 \left( z ^ { 2 } + 2 z - 2 \right) e ^ { z } + \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
C)  $5 \left( z ^ { 2 } - 2 z + 2 \right) e ^ { z } - \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
D)  $5 \left( z ^ { 2 } - 2 z + 2 \right) e ^ { z } + \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$ 
E)  $5 \left( z ^ { 2 } - 2 z - 2 \right) e ^ { z } - \frac { 5 } { 2 } e ^ { z ^ { 2 } } + C$

Question 4

Nancy's Chocolates estimates that the elasticity of demand for its dark chocolate truffles is   $E = 0.06 p - 2$  
Where p is the price per pound. Nancy's sells 19 pounds of truffles per week when the price is $23 per pound. Find the formula expressing the demand q as a function of p. Recall that the elasticity of demand is given by $E = - \frac { \mathrm { d } q } { \mathrm {~d} p } \times \frac { p } { q }$ .
 
NOTE: Round the value of the constant in your answer to four decimal places.

Accepted Answer

A)  $q = 0.1428 e ^ { - 0.06 p }$ 
B)    $q = 0.1428 e ^ { - 0.06 p } p ^ { 2 }$ 
C)    $q = e ^ { - 0.06 p } p ^ { - 2 }$ 
D)    $q = 0.1428 e ^ { 0.06 p } p ^ { 2 }$ 
E)    $q = 0.1428 e ^ { - 2.06 p }$ 
A)  $q = 0.1428 e ^ { - 0.06 p }$ 
B)    $q = 0.1428 e ^ { - 0.06 p } p ^ { 2 }$ 
C)    $q = e ^ { - 0.06 p } p ^ { - 2 }$ 
D)    $q = 0.1428 e ^ { 0.06 p } p ^ { 2 }$ 
E)    $q = 0.1428 e ^ { - 2.06 p }$

Question 5

Evaluate the integral.   $\int x ^ { 2 } \ln ( 2 x ) \mathrm { d } x$

Accepted Answer

A)  $\frac { 3 x ^ { 3 } \ln ( 2 x ) - x ^ { 2 } } { 9 } + C$ 
B)  $\frac { 3 x ^ { 2 } \ln ( 2 x ) - x ^ { 3 } } { 9 } + C$ 
C)  $\frac { 3 x ^ { 2 } \ln ( 2 x ) - x } { 9 } + C$ 
D)  $\frac { x ^ { 3 } ( 3 \ln ( 2 x ) + 1 ) } { 9 } + C$ 
E)  $\frac { x ^ { 3 } ( 3 \ln ( 2 x ) - 1 ) } { 9 } + C$ 
A)  $\frac { 3 x ^ { 3 } \ln ( 2 x ) - x ^ { 2 } } { 9 } + C$ 
B)  $\frac { 3 x ^ { 2 } \ln ( 2 x ) - x ^ { 3 } } { 9 } + C$ 
C)  $\frac { 3 x ^ { 2 } \ln ( 2 x ) - x } { 9 } + C$ 
D)  $\frac { x ^ { 3 } ( 3 \ln ( 2 x ) + 1 ) } { 9 } + C$ 
E)  $\frac { x ^ { 3 } ( 3 \ln ( 2 x ) - 1 ) } { 9 } + C$

Question 6

Calculate the 7-unit moving average of the function.   $f ( x ) = e ^ { 0.5 x }$

Accepted Answer

A)  $\bar { f } ( x ) = \frac { 1 } { 7 } \left( e ^ { 0.5 x } - e ^ { 0.5 ( x - 7 ) } \right)$ 
B)  $\bar { f } ( x ) = \frac { 2 } { 7 } \left( e ^ { 0.5 x } - e ^ { 0.5 ( x - 7 ) } \right)$ 
C)  $\bar { f } ( x ) = \frac { 2 } { 7 } e ^ { 0.5 ( x - 7 ) }$ 
D)  $\bar { f } ( x ) = \frac { 2 } { 7 } \left( e ^ { 0.5 x } + e ^ { 0.5 ( x - 7 ) } \right)$ 
E)  $\bar { f } ( x ) = \frac { 1 } { 7 } \left( e ^ { 0.5 x } - e ^ { 0.5 ( x + 7 ) } \right)$ 
A)  $\bar { f } ( x ) = \frac { 1 } { 7 } \left( e ^ { 0.5 x } - e ^ { 0.5 ( x - 7 ) } \right)$ 
B)  $\bar { f } ( x ) = \frac { 2 } { 7 } \left( e ^ { 0.5 x } - e ^ { 0.5 ( x - 7 ) } \right)$ 
C)  $\bar { f } ( x ) = \frac { 2 } { 7 } e ^ { 0.5 ( x - 7 ) }$ 
D)  $\bar { f } ( x ) = \frac { 2 } { 7 } \left( e ^ { 0.5 x } + e ^ { 0.5 ( x - 7 ) } \right)$ 
E)  $\bar { f } ( x ) = \frac { 1 } { 7 } \left( e ^ { 0.5 x } - e ^ { 0.5 ( x + 7 ) } \right)$

Question 7

Calculate the 12-unit moving average of the given function.
​ $f ( x ) = x ^ { 3 } - 9 x$

Accepted Answer

The answer of Calculate the 12-unit moving average of the...

Question 8

Find the area of the region enclosed by $y = 2 x ^ { 3 }$ and $y = 2 x ^ { 4 }$ .

Accepted Answer

A)  $\frac { 22 } { 10 }$ 
B)  $- \frac { 2 } { 20 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 20 }$ 
E)  1 
A)  $\frac { 22 } { 10 }$ 
B)  $- \frac { 2 } { 20 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 20 }$ 
E)  1

Question 9

For the given differential equation, find the particular solution $y = - 9 \ln 2$ when $x = - \ln 2$ .   $\frac { \mathrm { d } y } { \mathrm {~d} x } = 9 - e ^ { - x }$

Accepted Answer

A)  $y = 9 x + e ^ { x } - 2$ 
B)    $y = 9 x + e ^ { - x } - 2$ 
C)    $y = - 9 x - e ^ { - x } + 2$ 
D)    $y = 9 x + e ^ { - x }$ 
E)    $y = e ^ { - x } - 2$ 
A)  $y = 9 x + e ^ { x } - 2$ 
B)    $y = 9 x + e ^ { - x } - 2$ 
C)    $y = - 9 x - e ^ { - x } + 2$ 
D)    $y = 9 x + e ^ { - x }$ 
E)    $y = e ^ { - x } - 2$

Question 10

For the differential equation, find the particular solution if $y = \frac { 1 } { 2 }$ when $x = \frac { 1 } { 5 }$ .   $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { x ^ { 2 } }$

Accepted Answer

A)  $y = \frac { 1 } { 1 - 3 x }$ 
B)    $y = \frac { x } { 1 - 3 x }$ 
C)    $y = \frac { x } { 3 - x }$ 
D)    $y = - \frac { 3 } { 1 - x }$ 
E)    $y = \frac { x } { 1 + 3 x }$ 
A)  $y = \frac { 1 } { 1 - 3 x }$ 
B)    $y = \frac { x } { 1 - 3 x }$ 
C)    $y = \frac { x } { 3 - x }$ 
D)    $y = - \frac { 3 } { 1 - x }$ 
E)    $y = \frac { x } { 1 + 3 x }$

Question 11

The rate of U.S. per capita sales of bottled water for the period 1993-2003 could be approximated by $Q ( t ) = 0.05 t ^ { 2 } + 0.4 t + 9$ where t is the time in years since 1990.  
Compute the average annual sales of bottled water over the period 1996-2003, to the nearest gallon per year.

Accepted Answer

A) 16 gallons per year 
B)  15 gallons per year 
C)  17 gallons per year 
D)  18 gallons per year 
E)  19 gallons per year 
A) 16 gallons per year 
B)  15 gallons per year 
C)  17 gallons per year 
D)  18 gallons per year 
E)  19 gallons per year

Question 12

A fast-food outlet finds that the demand equation for its new side dish, "Sweetdough Tidbit", is given by $p = \frac { 6,750 } { ( q + 5 ) ^ { 2 } }$ where p is the price (in cents) per serving and q is the number of servings that can be sold per hour at this price. At the same time, the franchise is prepared to sell $q = 0.5 p - 5$ servings per hour at a price of p cents. Find the consumers' surplus at the equilibrium price level.

Accepted Answer

A) $6.06 
B)  $6.04 
C)  $6.00 
D)  $6.10 
E)  $6.02 
A) $6.06 
B)  $6.04 
C)  $6.00 
D)  $6.10 
E)  $6.02

Question 13

Decide whether the integral converges. If the integral converges, compute its value.   $\int _ { 1 } ^ { 5 } \frac { 9 } { x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x$

Accepted Answer

A)  $\frac { 27 } { 2 } \left( 5 ^ { \frac { 2 } { 3 } } \right)$ 
B)  $\frac { 27 } { 2 } \left( 5 ^ { \frac { 2 } { 3 } } - 1 \right)$ 
C)  $27 \left( 5 ^ { \frac { 2 } { 3 } } - 1 \right)$ 
D)  $\frac { 9 } { 2 } \left( 5 ^ { \frac { 2 } { 3 } } + 1 \right)$ 
E)  diverges 
A)  $\frac { 27 } { 2 } \left( 5 ^ { \frac { 2 } { 3 } } \right)$ 
B)  $\frac { 27 } { 2 } \left( 5 ^ { \frac { 2 } { 3 } } - 1 \right)$ 
C)  $27 \left( 5 ^ { \frac { 2 } { 3 } } - 1 \right)$ 
D)  $\frac { 9 } { 2 } \left( 5 ^ { \frac { 2 } { 3 } } + 1 \right)$ 
E)  diverges

Question 14

Find the area of the region enclosed by $y = 6 \ln x$ , $y = 7 - \ln x$ , and the line $x = 6$ .

Accepted Answer

A)  $42 \ln 6 + 84 - 7 e$ 
B)  $\ln 6 + 84 + 7 e$ 
C)  $\ln 6 - 84 - 7 e$ 
D)  $42 \ln 6 - 84 + 7 e$ 
E)  $- 42 \ln 6 - 84 + 7 e$ 
A)  $42 \ln 6 + 84 - 7 e$ 
B)  $\ln 6 + 84 + 7 e$ 
C)  $\ln 6 - 84 - 7 e$ 
D)  $42 \ln 6 - 84 + 7 e$ 
E)  $- 42 \ln 6 - 84 + 7 e$

Question 15

Find the area of the region between $y = \frac { 1 } { 7 } x$ and $y = \frac { 1 } { 7 } x ^ { 2 }$ for x in $[ - 1,1 ]$ .

Accepted Answer

A)  $\frac { 1 } { 14 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  1 
D)  $\frac { 1 } { 7 }$ 
E)  $\frac { 1 } { 10 }$ 
A)  $\frac { 1 } { 14 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  1 
D)  $\frac { 1 } { 7 }$ 
E)  $\frac { 1 } { 10 }$

Question 16

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate.   $R ( t ) = 90,000 e ^ { 0.03 t }$ , $0 \leq t \leq 5$ , at 6%

Accepted Answer

A)  $T V = \$ 14,565.08$ , $P V = \$ 417,876.07$ 
B)  $T V = \$ 5,582,123.93$ , $P V = \$ 12,536.28$ 
C)  $T V = \$ 417,876.07$ , $P V = \$ 524,788.21$ 
D)  $T V = \$ 485,502.73$ , $P V = \$ 417,876.07$ 
E)  $T V = \$ 0$ , $P V = \$ 2,568,312.19$ 
A)  $T V = \$ 14,565.08$ , $P V = \$ 417,876.07$ 
B)  $T V = \$ 5,582,123.93$ , $P V = \$ 12,536.28$ 
C)  $T V = \$ 417,876.07$ , $P V = \$ 524,788.21$ 
D)  $T V = \$ 485,502.73$ , $P V = \$ 417,876.07$ 
E)  $T V = \$ 0$ , $P V = \$ 2,568,312.19$

Question 17

The annual net sales (revenue) earned by Nintendo Co., Ltd., in the fiscal years ending March 31, 1995 to March 31, 2003, can be approximated by $R ( t ) = - 4 t ^ { 2 } + 10 t + 530$ billion yen per year $( - 6 \leq t \leq 3 )$ where t is time in years ( $t = 0$ represents March 31, 2000). Suppose that, from April 1994 on, Nintendo invested its revenue in an investment yielding 4% compounded continuously. What, to the nearest 100 billion, would the total value of Nintendo's revenue have been by the end of March 2003

Accepted Answer

A) Total revenue   500 billion yen 
B)  Total revenue   2,600 billion yen 
C)  Total revenue   1,000 billion yen 
D)  Total revenue   5,100 billion yen 
E)  Total revenue   51,200 billion yen 
A) Total revenue   500 billion yen 
B)  Total revenue   2,600 billion yen 
C)  Total revenue   1,000 billion yen 
D)  Total revenue   5,100 billion yen 
E)  Total revenue   51,200 billion yen

Question 18

Find the area of the region enclosed by $y = e ^ { x }$ , $y = 2$ , and the y-axis.

Accepted Answer

A)  $2 \ln 2 + 2$ 
B)  $2 \ln 2 - 2$ 
C)  $2 \ln 2 - 1$ 
D)  $2 \ln 2 + 1$ 
E)  $\ln 2 - 1$ 
A)  $2 \ln 2 + 2$ 
B)  $2 \ln 2 - 2$ 
C)  $2 \ln 2 - 1$ 
D)  $2 \ln 2 + 1$ 
E)  $\ln 2 - 1$

Question 19

Evaluate the integral.   $\int \left( x ^ { 2 } + 1 \right) e ^ { 8 x + 2 } \mathrm {~d} x$

Accepted Answer

A)  $\frac { e ^ { 8 x + 2 } } { 512 } \left( 64 x ^ { 2 } - 16 x - 66 \right) + C$ 
B)  $\frac { e ^ { 8 x + 2 } } { 64 } \left( 64 x ^ { 2 } - 16 x + 66 \right) + C$ 
C)  $\frac { e ^ { 8 x + 2 } } { 512 } \left( 64 x ^ { 2 } + 16 x - 66 \right) + C$ 
D)  $\frac { e ^ { 8 x + 2 } } { 8 } \left( 64 x ^ { 2 } + 16 x - 66 \right) + C$ 
E)  $\frac { e ^ { 8 x + 2 } } { 512 } \left( 64 x ^ { 2 } - 16 x + 66 \right) + C$ 
A)  $\frac { e ^ { 8 x + 2 } } { 512 } \left( 64 x ^ { 2 } - 16 x - 66 \right) + C$ 
B)  $\frac { e ^ { 8 x + 2 } } { 64 } \left( 64 x ^ { 2 } - 16 x + 66 \right) + C$ 
C)  $\frac { e ^ { 8 x + 2 } } { 512 } \left( 64 x ^ { 2 } + 16 x - 66 \right) + C$ 
D)  $\frac { e ^ { 8 x + 2 } } { 8 } \left( 64 x ^ { 2 } + 16 x - 66 \right) + C$ 
E)  $\frac { e ^ { 8 x + 2 } } { 512 } \left( 64 x ^ { 2 } - 16 x + 66 \right) + C$

Question 20

Calculate the 11-unit moving average of the function.   $f ( x ) = \sqrt { x }$

Accepted Answer

A)  $\bar { f } ( x ) = \frac { 2 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x + 11 ) ^ { 3 } } \right)$ 
B)  $\bar { f } ( x ) = \frac { 2 } { 11 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
C)  $\bar { f } ( x ) = \frac { 1 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
D)  $\bar { f } ( x ) = \frac { 2 } { 11 } \left( \sqrt { x ^ { 3 } } + \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
E)  $\bar { f } ( x ) = \frac { 2 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
A)  $\bar { f } ( x ) = \frac { 2 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x + 11 ) ^ { 3 } } \right)$ 
B)  $\bar { f } ( x ) = \frac { 2 } { 11 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
C)  $\bar { f } ( x ) = \frac { 1 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
D)  $\bar { f } ( x ) = \frac { 2 } { 11 } \left( \sqrt { x ^ { 3 } } + \sqrt { ( x - 11 ) ^ { 3 } } \right)$ 
E)  $\bar { f } ( x ) = \frac { 2 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$

Evaluate the integral. $\int _ { 0 } ^ { 1 } x ^ { 2 } ( x + 2 ) ^ { 2 } \mathrm {~d} x$

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. $R ( t ) = 50,000$ , $0 \leq t \leq 15$ , at 9%

Evaluate the integral. $\int \left( 5 z ^ { 2 } e ^ { z } - 5 z e ^ { z ^ { 2 } } \right) \mathrm { d } z$

Evaluate the integral. $\int x ^ { 2 } \ln ( 2 x ) \mathrm { d } x$

Calculate the 7-unit moving average of the function. $f ( x ) = e ^ { 0.5 x }$

Calculate the 12-unit moving average of the given function. $f ( x ) = x ^ { 3 } - 9 x$

Find the area of the region enclosed by $y = 2 x ^ { 3 }$ and $y = 2 x ^ { 4 }$ .

For the given differential equation, find the particular solution $y = - 9 \ln 2$ when $x = - \ln 2$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = 9 - e ^ { - x }$

For the differential equation, find the particular solution if $y = \frac { 1 } { 2 }$ when $x = \frac { 1 } { 5 }$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { x ^ { 2 } }$

The rate of U.S. per capita sales of bottled water for the period 1993-2003 could be approximated by $Q ( t ) = 0.05 t ^ { 2 } + 0.4 t + 9$ where t is the time in years since 1990. Compute the average annual sales of bottled water over the period 1996-2003, to the nearest gallon per year.

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 1 } ^ { 5 } \frac { 9 } { x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x$

Find the area of the region enclosed by $y = 6 \ln x$ , $y = 7 - \ln x$ , and the line $x = 6$ .

Find the area of the region between $y = \frac { 1 } { 7 } x$ and $y = \frac { 1 } { 7 } x ^ { 2 }$ for x in $[ - 1,1 ]$ .

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. $R ( t ) = 90,000 e ^ { 0.03 t }$ , $0 \leq t \leq 5$ , at 6%

Find the area of the region enclosed by $y = e ^ { x }$ , $y = 2$ , and the y-axis.

Evaluate the integral. $\int \left( x ^ { 2 } + 1 \right) e ^ { 8 x + 2 } \mathrm {~d} x$

Calculate the 11-unit moving average of the function. $f ( x ) = \sqrt { x }$

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Exam 14: Further Integration Techniques and Applications of the Integral

Evaluate the integral. ∫01x2(x+2)2 dx\int _ { 0 } ^ { 1 } x ^ { 2 } ( x + 2 ) ^ { 2 } \mathrm {~d} x∫01​x2(x+2)2 dx

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. R(t)=50,000R ( t ) = 50,000R(t)=50,000 , 0≤t≤150 \leq t \leq 150≤t≤15 , at 9%

Evaluate the integral. ∫(5z2ez−5zez2)dz\int \left( 5 z ^ { 2 } e ^ { z } - 5 z e ^ { z ^ { 2 } } \right) \mathrm { d } z∫(5z2ez−5zez2)dz

Evaluate the integral. ∫x2ln⁡(2x)dx\int x ^ { 2 } \ln ( 2 x ) \mathrm { d } x∫x2ln(2x)dx

Calculate the 7-unit moving average of the function. f(x)=e0.5xf ( x ) = e ^ { 0.5 x }f(x)=e0.5x

Calculate the 12-unit moving average of the given function. ​ f(x)=x3−9xf ( x ) = x ^ { 3 } - 9 xf(x)=x3−9x

Find the area of the region enclosed by y=2x3y = 2 x ^ { 3 }y=2x3 and y=2x4y = 2 x ^ { 4 }y=2x4 .

For the given differential equation, find the particular solution y=−9ln⁡2y = - 9 \ln 2y=−9ln2 when x=−ln⁡2x = - \ln 2x=−ln2 . dy dx=9−e−x\frac { \mathrm { d } y } { \mathrm {~d} x } = 9 - e ^ { - x } dxdy​=9−e−x

For the differential equation, find the particular solution if y=12y = \frac { 1 } { 2 }y=21​ when x=15x = \frac { 1 } { 5 }x=51​ . dy dx=y2x2\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { x ^ { 2 } } dxdy​=x2y2​

Decide whether the integral converges. If the integral converges, compute its value. ∫159x13 dx\int _ { 1 } ^ { 5 } \frac { 9 } { x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x∫15​x31​9​ dx

Find the area of the region enclosed by y=6ln⁡xy = 6 \ln xy=6lnx , y=7−ln⁡xy = 7 - \ln xy=7−lnx , and the line x=6x = 6x=6 .

Find the area of the region between y=17xy = \frac { 1 } { 7 } xy=71​x and y=17x2y = \frac { 1 } { 7 } x ^ { 2 }y=71​x2 for x in [−1,1][ - 1,1 ][−1,1] .

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. R(t)=90,000e0.03tR ( t ) = 90,000 e ^ { 0.03 t }R(t)=90,000e0.03t , 0≤t≤50 \leq t \leq 50≤t≤5 , at 6%

Find the area of the region enclosed by y=exy = e ^ { x }y=ex , y=2y = 2y=2 , and the y-axis.

Evaluate the integral. ∫(x2+1)e8x+2 dx\int \left( x ^ { 2 } + 1 \right) e ^ { 8 x + 2 } \mathrm {~d} x∫(x2+1)e8x+2 dx

Calculate the 11-unit moving average of the function. f(x)=xf ( x ) = \sqrt { x }f(x)=x​

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Functions of Several Variables

Trigonometric Models

Filters

Evaluate the integral. $\int _ { 0 } ^ { 1 } x ^ { 2 } ( x + 2 ) ^ { 2 } \mathrm {~d} x$

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. $R ( t ) = 50,000$ , $0 \leq t \leq 15$ , at 9%

Evaluate the integral. $\int \left( 5 z ^ { 2 } e ^ { z } - 5 z e ^ { z ^ { 2 } } \right) \mathrm { d } z$

Evaluate the integral. $\int x ^ { 2 } \ln ( 2 x ) \mathrm { d } x$

Calculate the 7-unit moving average of the function. $f ( x ) = e ^ { 0.5 x }$

Calculate the 12-unit moving average of the given function. $f ( x ) = x ^ { 3 } - 9 x$

Find the area of the region enclosed by $y = 2 x ^ { 3 }$ and $y = 2 x ^ { 4 }$ .

For the given differential equation, find the particular solution $y = - 9 \ln 2$ when $x = - \ln 2$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = 9 - e ^ { - x }$

For the differential equation, find the particular solution if $y = \frac { 1 } { 2 }$ when $x = \frac { 1 } { 5 }$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { x ^ { 2 } }$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 1 } ^ { 5 } \frac { 9 } { x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x$

Find the area of the region enclosed by $y = 6 \ln x$ , $y = 7 - \ln x$ , and the line $x = 6$ .

Find the area of the region between $y = \frac { 1 } { 7 } x$ and $y = \frac { 1 } { 7 } x ^ { 2 }$ for x in $[ - 1,1 ]$ .

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. $R ( t ) = 90,000 e ^ { 0.03 t }$ , $0 \leq t \leq 5$ , at 6%

Find the area of the region enclosed by $y = e ^ { x }$ , $y = 2$ , and the y-axis.

Evaluate the integral. $\int \left( x ^ { 2 } + 1 \right) e ^ { 8 x + 2 } \mathrm {~d} x$

Calculate the 11-unit moving average of the function. $f ( x ) = \sqrt { x }$