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Hughes-Hallett et al-Calculus: Single & Multivar 5 (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 24 / 30

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving...  ()

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24/30 (80.0%)

  • Instructions

    Here are some textbook questions from Calculus: Single and Multivariable 5/e by Deborah Hughes-Hallett, Andrew M. Gleason, and William G. McCallum published by John Wiley & Sons, Inc. Click here for a list of all of the questions coded in WebAssign. This demo assignment allows many submissions and allows you to try another version of the same question for practice.

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1. –/3 points HGMCalc5 1.1.033. My Notes
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The value of a car, V = f(a), in thousands of dollars, is a function of the age of the car, a, in years.
(a) Interpret the statement f(3) = 11.
    
(b) What does the following statement represent in mathematical terms?
V0 is the value of the car when it is new.
    
(c) What does the following statement represent in mathematical terms?
an is the age of the car when it is worth nothing.
    
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2. –/4 points HGMCalc5 1.4.049. My Notes
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–/4
 
In 2000, there were about 208 million vehicles and about 281 million people in a certain country. The number of vehicles has been growing at 4.5% a year, while the population has been growing at 1% a year.
(a) Write a formula for the number of vehicles (in millions) as a function of t, the number of years since 2000.
V(t) =

(b) Write a formula for the number of people (in millions) as a function of t, the number of years since 2000.
P(t) =

(c) If the growth rates remain constant, when is there, on average, one vehicle per person? Give your answer in exact form and decimal form.
exact form
years since 2000
decimal form (nearest tenth)  years since 2000
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3. –/7 points HGMCalc5 3.3.051. My Notes
Question Part
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–/7
 
The quantity, q, of a certain skateboard sold depends on the selling price, p, in dollars, so we write q = f(p). You are given that f(145) = 15,600 and f '(145) = −80.
(a) What does f(145) = 15,600 tell you about the sales of skateboards?
When the price of the skateboard is $ , then skateboards will be sold.

What does f '(145) = −80 tell you about the sales of skateboards?
If the price increases from $145 to $146, the number of skateboards sold would by roughly skateboards.

(b) The total revenue, R, earned by the sale of skateboards is given by R = pq. Find R'(p).
    

(c) If the skateboards are currently selling for $145, what happens to revenue if the price is increased to $146?
The revenue by roughly $ .
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4. –/3 points HGMCalc5 4.3.012. My Notes
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Investigate the two parameter family of functions. Assume a, b, and x are all positive.
f(x) = a/x + bx
(a) Graph f(x) using b = 1 and three different values of a. How does the critical point of f appear to move as a increases?
    
(b) Graph f(x) using a = 1 and three different values of b. How does the critical point of f appear to move as b increases?
    

(c) Find a formula for the x-coordinate of the critical point of f in terms of a and b.
x =
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5. –/3 points HGMCalc5 4.4.034. My Notes
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–/3
 
A rectangular swimming pool is to be built with an area of 1250 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. In this problem you will find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions.
(a) If the dimensions of the swimming pool are denoted by x and y, which of the following functions could be used to represent the area of the piece of property needed?
    

(b) What are the dimensions of the smallest piece of property on which the pool can be built?
shorter side   ft
longer side   ft
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6. –/3 points HGMCalc5 4.4.042. My Notes
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The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed v. A certain ferry boat uses $196 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor, maintenance, and so on) is $1323 per hour.
(a) Find the proportionality constant for the cost of the fuel per hour.
k =

(b) Express the cost per mile in terms of v only.
C(v) =


(c) At what speed should the boat travel so as to minimize the cost per mile traveled?
mph
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7. –/2 points HGMCalc5 7.4.048. My Notes
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(a) If we use the substitution t = 5 tan(θ), then which of the following integrals is equivalent to int (1/(t^2*sqrt(25+t^2))) text( ) dt ?
    

(b) Use the integral in part (a) to evaluate int 1/(t^2*sqrt(25+t^2)) text( ) dt
+ C
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8. –/2 points HGMCalc5 7.8.020. My Notes
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Consider the improper integral int_0^1 1 / sqrt(theta^3 + theta) text( ) dtheta. Determine if the improper integral converges or diverges.
    
Which of the following inequalities can be used to help prove your conclusion?
    
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9. –/3 points HGMCalc5 8.2.006. My Notes
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Consider the region bounded by y = cos(x), y = 0, x = 0, and x = π/2. Rotate the region around the x-axis. Write an integral giving the volume of the solid. Give your answer using the form below.
integral_A^B R(x)text( )dx
R(x) = 
A = 0
B = 


Evaluate this integral.
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