Deck 14: Further Integration Techniques and Applications of the Integral

Question 1

For the differential equation, find the particular solution if $y ( 0 ) = 1$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ( y + 5 )$

A) $y = e ^ { \frac { x } { 2 } } - 6$
B) $y = 6 e ^ { x ^ { 2 } } + 5$
C) $y = 6 e ^ { \frac { x ^ { 2 } } { 2 } } - 5$
D) $y = 12 e ^ { \frac { x ^ { 2 } } { 2 } } + 5$
E) $y = 6 e ^ { 2 x ^ { 2 } }$

Answer 1

$y = 6 e ^ { \frac { x ^ { 2 } } { 2 } } - 5$

Question 2

Your monthly sales of Tofu Ice Cream are falling at an instantaneous rate of 3% per month. If you currently sell $s = 1,070$ per month, solve the differential equation that describes your change in sales to predict your monthly sales. Assume that the current sales are given for time $t = 0$ .
NOTE: Enter your answer in the form $s = f ( t )$ , where f is some function of t.

A) $s = 1,070 + 0.03 t$
B) $s = 1,070 + t ^ { 0.03 }$
C) $s = 1,070 + t ^ { - 0.03 }$
D) $s = 1,070 e ^ { 0.03 t }$
E) $s = 1,070 e ^ { - 0.03 t }$

Answer 2

$s = 1,070 e ^ { - 0.03 t }$

Question 3

You have just introduced a new computer to the market. You predict that you will eventually sell 99,000 computers and that your monthly rate of sales will be 17% of the difference between the saturation value and the total number you have sold up to that point. Solve the differential equation for your total sales s (as a function of the month). Hint: What are your total sales at the moment when you first introduce the computer

A) $s = 99,000 e ^ { 0.17 t }$
B) $s = 99,000 \left( 1 - e ^ { - 0.17 t } \right)$
C) $s = 99,000 + 0.17 t$
D) $s = 99,000 \left( 1 - e ^ { - 1.7 t } \right)$
E) $s = 99,000 + t ^ { 0.17 }$

Answer 3

$s = 99,000 \left( 1 - e ^ { - 0.17 t } \right)$

Question 4

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

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 5

For the differential equation, find the particular solution if $y ( 8 ) = 51$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y + 5 } { x }$

A) $y = x - 5$
B) $y = 7 x - 5$
C) $y = 7 x$
D) $y = 7 x ^ { 2 } - 5 x$
E) $y = 7 x + 5$

Question 6

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = 27 x ^ { 2 } + 12 \sqrt { x }$
Solve for y as a function of x.

NOTE: Use the symbol C to enter the constant.

A) $y = 27 x ^ { 3 } + 8 x ^ { 1.5 } + C$
B) $y = 27 x ^ { 2 } + 8 x + C$
C) $y = 9 x ^ { 3 } + 8 x ^ { 1.5 } + C$
D) $y = 9 x - 8 x ^ { - 0.5 } + C$
E) $y = 27 x ^ { 3 } + 4 x ^ { 1.5 } + C$

Question 7

For the differential equation, find the particular solution $y = 0$ when $x = 0$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - 14 x$

A) $y = x ^ { 4 } - 14 x ^ { 2 }$
B) $y = \frac { x ^ { 4 } } { 4 } - 7 x ^ { 2 }$
C) $y = \frac { x ^ { 4 } } { 4 } + 7 x ^ { 2 }$
D) $y = 4 x ^ { 4 } + 7 x ^ { 2 }$
E) $y = \frac { x ^ { 4 } } { 2 } - 7 x$

Question 8

Find the general solution of the differential equation. Solve for y as a function of x. $x \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 y } \ln x$

A) $y = \sqrt { \frac { \ln ^ { 2 } | x | + C } { 2 } }$
B) $y = \frac { \sqrt { \ln ^ { 2 } | x | + C } } { 2 }$
C) $y = \sqrt { \frac { \ln | x | + C } { 2 } }$
D) $y = \sqrt { 2 \ln ^ { 2 } x + C }$
E) $y = \sqrt { \frac { \ln ^ { 2 } ( x + 2 ) } { C } }$

Question 9

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.

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 10

Your monthly profit p on sales of Avocado Ice Cream are rising at an instantaneous rate of 17% per month. If you currently make a profit of $14,500 per month, solve the differential equation that describes your change in profit to predict your monthly profits. Assume that the current sales are given for time $t = 0$ .

A) $s = 14,500 e ^ { - 0.17 t }$ .
B) $s = 14,500 e ^ { - 1.7 t }$
C) $s = 14,500 e ^ { - 0.17 t }$
D) $s = 14,500 + e ^ { - 0.17 t }$
E) $s = 14,500 + t ^ { 0.17 }$

Question 11

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

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 12

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( x + 3 ) y ^ { 2 } }$
Solve for y as a function of x.

A) $y = \sqrt [ 3 ] { 3 \ln x + C }$
B) $y = 3 \sqrt [ 3 ] { \ln | x + 3 | + C }$
C) $y = \sqrt [ 3 ] { 3 \ln | x + 3 | + C }$
D) $y = \sqrt [ 3 ] { 3 \ln ( x + 3 ) + C }$
E) $y = ( 3 \ln | x + 3 | + C ) ^ { 3 }$

Question 13

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } + 7$
Solve for y as a function of x.

A) $y = \ln ( x + 7 ) + 7 x + C$
B) $y = \ln | x | + 7 x + C$
C) $y = \ln | x | + C$
D) $y = - 2 x + C$
E) $y = x + C$

Question 14

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 x y$
Solve for y as a function of x.

A) $y = C e ^ { x ^ { 2 } } + 6$
B) $y = C e ^ { 6 x ^ { 2 } }$
C) $y = C e ^ { 12 x }$
D) $y = C e ^ { 6 x }$
E) $y = e ^ { 6 x ^ { 2 } } + C$

Question 15

For the given differential equation, find the particular solution if $y = 3$ when $x = 2$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } } { y ^ { 2 } }$

A) $y = \sqrt [ 3 ] { x ^ { 3 } - 19 }$
B) $y = \sqrt [ 3 ] { x - 19 }$
C) $y = \sqrt { x ^ { 2 } + 19 }$
D) $y = x + 19$
E) $y = \sqrt [ 3 ] { x ^ { 3 } + 19 }$

Question 16

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y + 4 } { x }$
Solve for y as a function of x.

A) $y = \ln ( x ) - 3 + C$
B) $y = C x - 4$
C) $y = \ln ( x ) + 4 x + C$
D) $y = 4 x + C$
E) $y = x + C$

Question 17

For the differential equation, find the particular solution if $y ( 5 ) = \frac { 1 } { 5 }$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x y ^ { 2 } } { x ^ { 2 } + 8 }$

A) $y = - \ln \left( x ^ { 2 } + 8 \right) - 5 - \frac { \ln 33 } { 2 }$
B) $y = - \ln \left( x ^ { 2 } - 8 \right) - 5 - \frac { \ln 33 } { 2 }$
C) $y = \frac { 2 } { \ln \left( 2 x ^ { 2 } + 8 \right) - 5 - \ln 33 }$
D) $y = - \frac { 2 } { \ln \left( x ^ { 2 } + 8 \right) - 5 }$
E) $y = - \frac { 2 } { \ln \left( x ^ { 2 } + 8 \right) - 10 - \ln 33 }$

Question 18

For the differential equation, find the particular solution if $y \left( \frac { 1 } { 7 } \right) = e ^ { 7 }$ $x ^ { 2 } \frac { \mathrm { d } y } { \mathrm {~d} x } = y$

A) $y = e ^ { 14 - \frac { 1 } { x } }$
B) $y = e ^ { 14 - x }$
C) $y = e ^ { 14 + \frac { 1 } { x } }$
D) $y = 14 e ^ { - \frac { 1 } { x } }$
E) $y = e ^ { 1 - \frac { 14 } { x } }$

Question 19

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 1 ) y ^ { 2 }$
Solve for y as a function of x.

A) $y = - ( x + 1 ) ^ { 2 } + C$
B) $y = - \frac { 2 } { x ^ { 2 } + C }$
C) $y = - \frac { 2 } { x ( x + 2 ) + C }$
D) $y = \frac { C } { ( x + 1 ) ^ { 2 } }$
E) $y = - \frac { 2 } { x + C }$

Question 20

For the given differential equation, find the particular solution if $y ( 3 ) = 12$ . $x \frac { \mathrm { d } y } { \mathrm {~d} x } = y$

A) $y = 4 x ^ { 2 }$
B) $y = 4 + x$
C) $y = - 4 x + 4$
D) $y = 4 x$
E) $y = 24 x$

Question 21

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - \infty } ^ { \infty } 9 x e ^ { - 6 x ^ { 2 } } \mathrm {~d} x$

A)6
B) 9
C) 0
D) 18
E) diverges

Question 22

The Laplace transform F(x) of a function f (t ) is given by the formula. $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$
Find $F ( x )$ if $f ( t ) = 8$ .

A) $\frac { 1 } { 8 x }$
B) $\frac { 8 } { x ^ { 2 } }$
C) $\frac { x } { 8 }$
D) $- \frac { x } { 8 }$
E) $\frac { 8 } { x }$

Question 23

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

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 24

The Laplace transform F(x) of a function f(t) is given by the formula. $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$
Find F(x) if $f ( t ) = e ^ { n t }$ (n constant).

A) $\frac { 1 } { x }$
B) $\frac { 1 } { x - n }$
C) $\frac { n } { x }$
D) $\frac { n } { x - \mathrm { n } }$
E) $n x - n$

Question 25

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - \infty } ^ { - 8 } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$

A) $- \frac { 1 } { 8 }$
B) $\frac { 1 } { 64 }$
C) $- \frac { 1 } { 64 }$
D) $\frac { 1 } { 8 }$
E) diverges

Question 26

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 4 } ^ { + \infty } x \mathrm {~d} x$

A)converges to $\frac { 1 } { 4 }$
B) converges to $\frac { 1 } { 8 }$
C) converges to 8
D) converges to 0
E) diverges

Question 27

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - 4 } ^ { + \infty } e ^ { - 0.25 x } \mathrm {~d} x$

A) $0.25 e$
B) $4 e ^ { - 1 }$
C) 0
D) $4 e$
E) diverges

Question 28

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 1 } ^ { + \infty } \frac { 1 } { x ^ { 1.5 } } \mathrm {~d} x$

A)2
B) 1
C) $\frac { 1 } { 2 }$
D) -2
E) diverges

Question 29

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - \infty } ^ { 1 } \frac { 1 } { x ^ { \frac { 1 } { 7 } } } d x$

A)converges to 0
B) converges to 7
C) converges to 1
D) diverges
E) converges to 4

Question 30

If the integral converges, compute its value. $\int _ { - \infty } ^ { 2 } e ^ { x } \mathrm {~d} x$

A) $e ^ { - 2 }$
B) $e ^ { 2 }$
C) $\ln 2$
D) $\frac { 1 } { 2 }$
E) diverges

Question 31

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - 9 } ^ { 9 } \left( \frac { 1 } { ( x + 8 ) ^ { \frac { 1 } { 5 } } } \right) d x$

A) $\left( 17 ^ { \frac { 1 } { 5 } } - 1 \right)$
B) $\frac { 5 } { 4 } \left( 17 ^ { \frac { 4 } { 5 } } - 1 \right)$
C) $\left( 17 ^ { \frac { 2 } { 5 } } - 1 \right)$
D) $- \frac { 5 } { 4 } \left( 17 ^ { - \frac { 4 } { 5 } } + 1 \right)$
E) diverges

Question 32

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - 4 } ^ { 10 } \frac { 5 } { ( x + 4 ) ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$

A) $5 \sqrt { 14 }$
B) $- 5 \sqrt { 14 }$
C) $- 20 \sqrt { 14 }$
D) $10 \sqrt { 14 }$
E) diverges

Question 33

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { 11 } \frac { 2 } { x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x$

A) $2 \left( 11 ^ { \frac { 2 } { 3 } } \right)$
B) 0
C) $11 ^ { \frac { 2 } { 3 } }$
D) $3 \left( 11 ^ { \frac { 2 } { 3 } } \right)$
E) diverges

Question 34

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { + \infty } x ^ { 2 } e ^ { - 10 x } \mathrm {~d} x$

A) $- \frac { 1 } { 500 }$
B) $\frac { 1 } { 10 }$
C) $\frac { 1 } { 500 }$
D) 0
E) diverges

Question 35

Decide whether the integral converges. If the integral converges, compute its value.
​
If the integral diverges, enter your answer as diverges .
​

Question 36

The weekly demand for your company's Lo-Cal Mousse is modeled by the equation $q ( t ) = \frac { 18 e ^ { 2 t - 2 } } { 1 + e ^ { 2 t - 2 } }$ where t is time from now in weeks and q(t) is the number of gallons sold per week. Evaluate the integral $\int _ { - \infty } ^ { 0 } \frac { 18 e ^ { 2 t - 2 } } { 1 + e ^ { 2 t - 2 } } \mathrm {~d} t$ .

A)0.254 gallons
B) 5.639 gallons
C) 2.285 gallons
D) 1.142 gallons
E) 2.819 gallons

Question 37

Calculate the producers' surplus at the unit price $\bar { p } = 15$ for the following supply equation. $p = 3 + 4 q ^ { \frac { 1 } { 3 } }$

A)$80.00
B) $81.00
C) $86.00
D) $89.00
E) $83.00

Question 38

Calculate the producers' surplus at the unit price $\bar { p } = 42$ for the following supply equation. $q = 6 p - 240$

A)$18.00
B) $11.00
C) $20.00
D) $12.00
E) $15.00

Question 39

Sales of the text "Calculus and You" have been declining continuously at an annual rate of 8%. Assuming that "Calculus and You" currently sell 800 copies per year and that sales will continue this pattern of decline, calculate total future sales of the text. Use the formula for continuously compounded interest with a negative rate. ​

A)100,000 copies
B) 80,000 copies
C) 10,000 copies
D) 250,000 copies
E) 500,000 copies

Question 40

The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function $p ( x ) = \frac { 1 } { \sqrt { 2 \pi } \sigma } e ^ { \frac { - ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }$ , $\pi = 3.14159265 \ldots$ and and are constants called the standard deviation and the mean, respectively. With $\sigma = 3$ and $\mu = 5$ , approximate $\int _ { - \infty } ^ { \infty } p ( x ) \mathrm { d } x$ .

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

Question 41

The annual revenue earned by Target for fiscal years 1998 through 2006 can be approximated by $R ( t ) = 37 e ^ { 0.09 t }$ billion dollars per year $( - 2 \leq t \leq 6 )$ where t is time in years ( $t = 0$ represents the beginning of fiscal year 2000). Suppose that, from fiscal year 1998 on, Target invested its revenue in an investment that depreciated continuously at a rate of 5% per year. What, to the nearest $10 billion, would the total value of Target's revenue have been by the end of fiscal year 2004

A)Total revenue 40 billion dollars
B) Total revenue 220 billion dollars
C) Total revenue 2,200 billion dollars
D) Total revenue 110 billion dollars
E) Total revenue 20 billion dollars

Question 42

The Megabucks Corporation is issuing a 20-year bond paying 7% interest. An investor wishes to buy the rights to receive the interest on $30,000 worth of these bonds and seeks a 12% rate of return. If the investor payments are made continuously, what will the investor pay Round your answer to the nearest cent. Round your answer to the nearest cent.

A)$159.12
B) $17,500.00
C) $210,405.59
D) $1,909.49
E) $15,912.44

Question 43

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

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 44

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate. $R ( t ) = 70,000$ , $0 \leq t \leq 15$ , at 6%

A) $T V = \$ 1,050,000$ , $F V = \$ 1,702,870.30$
B) $T V = \$ 1,050,000$ , $F V = \$ 4,036,203.63$
C) $T V = \$ 1,050,000$ , $F V = \$ 2,869,536.96$
D) $T V = \$ 2,100,000$ , $F V = \$ 102,172.22$
E) $T V = \$ 525,000$ , $F V = \$ 1,640,997.94$

Question 45

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%

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 46

The annual revenue earned by Wal-Mart in the fiscal years ending January 31, 1994 to January 31, 2002 can be approximated by $R ( t ) = 150 e ^ { 0.14 t }$ billion dollars per year $( - 7 \leq t \leq 2 )$ where t is time in years ( $t = 0$ represents January 31, 2000). Suppose that, from January 1999 on, Wal-Mart invested its revenue in an investment that depreciated continuously at a rate of 4% per year. What, to the nearest $10 billion, would the total value of Wal-Mart's revenues have been by the end of January 2002

A)Total revenue 460 billion dollars
B) Total revenue 4,600 billion dollars
C) Total revenue 50 billion dollars
D) Total revenue 230 billion dollars
E) Total revenue 90 billion dollars

Question 47

The U. S. Treasury issued a 20-year bond on November 16, 1998, paying 6.07% interest. Thus, if you bought $100,000 worth of these bonds, you would receive $6,070 per year in interest for 20 years. At investor wishes to buy the rights to receive the interest on $100,000 worth of these bonds. The amount the investor is willing to pay is the present value of the interest payments, assuming a 6% rate of return. If we assume (incorrectly, but approximately) that the interest payments are made continuously, what will the investor pay Round your answer to the nearest cent.

A)$70,695.85
B) $437,051.83
C) $335,885.16
D) $69,880.58
E) $706.96

Question 48

The annual net sales (revenue) earned by the Finnish cell phone company Nokia from January 1999 to January 2003 can be approximated by $R ( t ) = - 1.7 t ^ { 2 } + 5 t + 28$ billion euros per year $( - 1 \leq t \leq 3 )$ where t is time in years ( $t = 0$ represents January 2000). Estimate, to the nearest billion, Nokia's total revenue from January 1999 through December 2002.

A)Total revenue 116 billion euros
B) Total revenue 120 billion euros
C) Total revenue 91 billion euros
D) Total revenue 4 billion euros
E) Total revenue 20 billion euros

Question 49

You are saving for your retirement by investing $1000 per month in an annuity with a guaranteed interest rate of 6% per year. With a continuous stream of investment and continuous compounding, how much will you have accumulated in the annuity by the time you retire in 15 years Round your answer to the nearest cent.

A)$180,000.00
B) $1,459.60
C) $691,920.62
D) $281,313.93
E) $291,920.62

Question 50

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%

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 51

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate.
​ , , at 11%
​
Please enter your answer in the form TV = , FV = . Round FV to the nearest cent.
​
TV = $__________ FV = $__________

Question 52

When your first child is born, you begin to save for college by depositing $600 per month in an account paying 6% interest per year. With a continuous stream of investment and continuous compounding, how much will you have accumulated in the account by the time your child enters college 18 years later Round your answer to the nearest cent.

A)$233,361.55
B) $353,361.55
C) $1,166.81
D) $473,361.55
E) $79,248.54

Question 53

Calculate the producers' surplus at the unit price for the following supply equation.
​ ​
$ __________

Question 54

Find the total value or the given income stream and also find its future value (at the end of the given interval) using the given interest rate. $R ( t ) = 50,000 e ^ { 0.04 t }$ , $0 \leq t \leq 5$ , at 7%

A) $T V = \$ 1,526,753.45$ , $F V = \$ 391,230.90$
B) $T V = \$ 276,753.45$ , $F V = \$ 555,183.07$
C) $T V = \$ 553,506.90$ , $F V = \$ 2,185,364,58$
D) $T V = \$ 276,753.45$ , $F V = \$ 329,441.32$
E) $T V = \$ 11,070.14$ , $F V = \$ 22,158.93$

Question 55

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.

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

Question 56

The annual net sales (revenue) earned by the Finnish cell phone company Nokia from January 1999 to January 2004 can be approximated by $R ( t ) = - 1.7 t ^ { 2 } + 5 t + 28$ billion euros per year $( - 1 \leq t \leq 4 )$ where t is time in years ( $t = 0$ represents January 2000). Suppose that, from January 1999 on, Nokia invested its revenue in an investment yielding 7% compounded continuously. What, to the nearest €10 billion, would the total value of Nokia's revenues have been at the end of 2003

A)Total revenue 150 billion euros
B) Total revenue 180 billion euros
C) Total revenue 160 billion euros
D) Total revenue 170 billion euros
E) Total revenue 190 billion euros

Question 57

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate. $R ( t ) = 80,000 + 2,000 t$ , $0 \leq t \leq 10$ , at 8%

A) $T V = \$ 900,000$ , $F V = \$ 1,358,522.47$
B) $T V = \$ 850,000$ , $F V = \$ - 977,992.01$
C) $T V = \$ 700,000$ , $F V = \$ 1,092,559.39$
D) $T V = \$ 2,500,000$ , $F V = \$ 2,152,244.27$
E) $T V = \$ 900,000$ , $F V = \$ 2,358,522.47$

Question 58

When your first child is born, you begin to save for college by depositing $500 per month in an account paying 12% interest per year. You increase the amount you save by 2% per year. With continuous investment and compounding, how much will you have accumulated in the account by the time your child enters college 18 years later Round your answer to the nearest cent.

A)$434,268.49
B) $4,342.68
C) $4,247,097.21
D) $5,052.23
E) $50,082.07

Question 59

You are saving for your retirement by investing $500 per month in an annuity with a guaranteed interest rate of 6% per year. You increase the amount you invest at the rate of 3% per year. With continuous investment and compounding, how much will you have accumulated in the annuity by the time you retire in 35 years Round your answer to the nearest cent.

A)$1,061,703.76
B) $130,012.45
C) $95,283.13
D) $10,617.04
E) $12,160,006.57

Question 60

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. $R ( t ) = 60,000 + 4,000 t$ , $0 \leq t \leq 5$ , at 11%

A) $T V = \$ 240,025,000$ , $P V = \$ 269,133.35$
B) $T V = \$ 250,000$ , $P V = \$ 2,079,283.09$
C) $T V = \$ 354,000$ , $P V = \$ 265,705.99$
D) $T V = \$ 80,000$ , $P V = \$ 195,803.31$
E) $T V = \$ 350,000$ , $P V = \$ 265,705.99$

Question 61

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

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) $\bar { f } ( x ) = \frac { 2 } { 33 } \left( \sqrt { x ^ { 3 } } - \sqrt { ( x - 11 ) ^ { 3 } } \right)$

Question 62

Calculate the 5-unit moving average of the given function. $$f ( x ) = 8 x ^ { 3 }$$

A) $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x + 250$
B) $\bar { f } ( x ) = 8 x ^ { 3 } + 60 x ^ { 2 } + 200 x - 250$
C) $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x - 250$
D) $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } - 250 x + 200$
E) $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x$

Question 63

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.
​ , , at 10%
​
Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary.
​
TV = $__________ PV = $__________

Question 64

Some values of a function and its 3-unit moving average are given. Supply the missing information: $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 3 & & 14 & & 4 & 20 \\\hline \text { average } & \text { undefined } & \text { undefined } & 5 & & 11 & & \\\hline\end{array}$$

A) $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline r ( x ) & 2 & 3 & 10 & 14 & 9 & 4 & 20 \\\hline \text { average } & \text { undefined } & \text { undefined } & 5 & 9 & 11 & 9 & 11 \\\hline\end{array}$
B) $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 3 & 10 & 14 & 9 & 4 & 20 \\\hline \text { average } & \text { undefined } & \text { undefined } & 5 & 9 & 11 & 9 & 11 \\\hline\end{array}$
C) $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 3 & 10 & 14 & 9 & 4 & 20 \\\hline \text { average } & \text { undefined } & \text { undefined } & 5 & 9 & 11 & 12 & 14 \\\hline\end{array}$

Question 65

If you invest $20,000 at 7% interest compounded continuously, what is the average amount in your account over 1 year
Please round your answer to the nearest cent.

A)$20,700.00
B) $5,350.00
C) $10,700.00
D) $21,400.00
E) $20,716.62

Question 66

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 6 x$ over $[ 0,2 ]$ .

A)Average = 5
B) Average = 6
C) Average = 12
D) Average = 10
E) Average = 7

Question 67

Find the formula for the a-unit moving average of a general exponential function. $f ( x ) = K e ^ { p x }$

A) $\bar { f } ( x ) = \frac { K } { p a } \left( e ^ { p x } + e ^ { p ( x - a ) } \right)$
B) $\bar { f } ( x ) = \frac { K } { a } \left( e ^ { p x } - e ^ { p ( x - p ) } \right)$
C) $\bar { f } ( x ) = \frac { K } { p a } \left( e ^ { p x } - e ^ { p ( x + a ) } \right)$
D) $\bar { f } ( x ) = \frac { K } { p } \left( e ^ { p x } - e ^ { a } \right)$
E) $\bar { f } ( x ) = \frac { K } { p a } \left( e ^ { p x } - e ^ { p ( x - a ) } \right)$

Question 68

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.
​ , , at 5%
​
Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary.
​
TV = $__________ PV = $__________

Question 69

Find the average of the function $f ( x ) = 2 x ^ { 3 }$ over $[ 0,8 ]$ .

A)Average = 32
B) Average = 64
C) Average = 256
D) Average = 1,024
E) Average = 1,028

Question 70

Your monthly salary has been increasing steadily for the past year, and your average monthly salary over the past year was x dollars. Would you have earned more money if you had been paid ?x dollars per month ?

A)No. You would have earned exactly the same amount of money.
B) No. You would have earned less money.
C) Yes. You would have earned more money.
D) More information is needed to answer the question.

Question 71

If you invest $12,000 at 9% interest compounded continuously, what is the average amount in your account over one year Round your answer to the nearest cent.

A)$12,556.57
B) $12,460.54
C) $12,844.66
D) $12,652.60
E) $12,748.63

Question 72

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.
​ , , at 11%
​
Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary.
​
TV = $__________ PV = $__________

Question 73

Find the total value or the given income stream and also find its future value (at the end of the given interval) using the given interest rate.
​ , , at 7%
​
Please enter your answer in the form TV = , FV = . Give the answers to the nearest cent.
​
TV = $__________ FV = $__________

Question 74

Complete the table with the values of the 3-unit moving average of the given function: $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 1 & 6 & 8 & 1 & 3 & 8 \\\hline \text { average } & \text { undefined } & \text { undefined } & & & & & \\\hline\end{array}$$

A) $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 1 & 6 & 8 & 1 & 3 & 8 \\\hline \text { average } & \text { undefined } & \text { undefined } & 4 & 3 & 5 & 5 & 4 \\\hline\end{array}$

B) $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 1 & 6 & 8 & 1 & 3 & 8 \\\hline \text { average } & \text { undefined } & \text { undefined } & 3 & 5 & 5 & 4 & 4 \\\hline\end{array}$

C) $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline \boldsymbol { r } ( \boldsymbol { x } ) & 2 & 1 & 6 & 8 & 1 & 3 & 8 \\\hline \text { average } & \text { undefined } & \text { undefined } & 5 & 5 & 4 & 4 & 3 \\\hline\end{array}$

Question 75

Calculate the 5-unit moving average of the given function. $f ( x ) = 2 x ^ { \frac { 2 } { 3 } }$

A) $\bar { f } ( x ) = \frac { 6 } { 25 } \left( x ^ { \frac { 5 } { 3 } } - ( x - 5 ) ^ { \frac { 5 } { 3 } } \right)$
B) $\bar { f } ( x ) = \frac { 6 } { 25 } \left( x ^ { \frac { 5 } { 3 } } + ( x - 5 ) ^ { \frac { 5 } { 3 } } \right)$
C) $\bar { f } ( x ) = \frac { 2 } { 25 } \left( x ^ { \frac { 5 } { 3 } } - ( x - 5 ) ^ { \frac { 5 } { 3 } } \right)$
D) $\bar { f } ( x ) = \frac { 12 } { 25 } \left( x ^ { \frac { 5 } { 3 } } - ( x - 5 ) ^ { \frac { 5 } { 3 } } \right)$
E) $\bar { f } ( x ) = \frac { 2 } { 25 } \left( x ^ { \frac { 2 } { 3 } } - ( x - 5 ) ^ { \frac { 2 } { 3 } } \right)$

Question 76

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

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 77

Find the average of the function $f ( x ) = e ^ { x }$ over $[ 0,8 ]$ .

A)Average = $1 - e ^ { 8 }$
B) Average = $\frac { e ^ { - 8 } } { 8 }$
C) Average = $\frac { 1 + e ^ { - 8 } } { 8 }$
D) Average = $\frac { e ^ { 8 } - 1 } { 8 }$
E) Average = $1 - e ^ { - 8 }$

Question 78

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.

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 79

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate.
​ , , at 10%
​
Please enter your answer in the form TV = , FV = . Round FV to the nearest cent.
​
TV = $__________ FV = $__________

Question 80

Find the formula for the c-unit moving average of a general linear function. $f ( x ) = n x + q$

A) $f ( x ) = n x + q + n c$
B) $\bar { f } ( x ) = n x - q + \frac { n c } { 2 }$
C) $\bar { f } ( x ) = n x + q + \frac { n c } { 2 }$
D) $\bar { f } ( x ) = n x + q - \frac { n c } { 2 }$
E) $\bar { f } ( x ) = n x - q - \frac { n c } { 2 }$