Survey of Calculus—Departmental Review SheetFall 2005
(Brief Applied Calculus, 3rd Edition. Berresford, Rockett. Houghton Mifflin, 2004.)
All work should be shown and all answers should be exact unless stated otherwise. It is fine to leave radicals in the denominator of a fraction, but they should be reduced. The departmental formula sheet and an acceptable graphing calculator are the only resources allowed for use during the final exam. Each problem on this review sheet is matched with its corresponding objective in the course outline and the section of the textbook in which it appears.
[Objective #1, Section 2.1]
- Evaluate the following limits without using a graphing calculator or making tables.
a. b. c.
[Objective #1, Sections 2.1, 2.7]
2.Use the given graph to answer the following:
a. b. c.
d. x-values where the function is discontinuous
e.x-values where the function is nondifferentiable
f.x-values where the slope of the tangent line is zero.
g.x-values where the slope of the tangent line is undefined
[Objective #2, Section 2.2]
3.Use the definition of derivative to find .
a. b.
[Objective #2, Sections 2.3, 2.4, 2.6]
4.Find the equation for the tangent line to the curve of at the point (0,2). Write your answer in slope-intercept form.
5.Find for each of the following:
a. b.
c. d. y=
e. f. g.
h. i.
[Objective #4, Sections 3.1, 3.2]
6.For each of the following find:
a. critical points
b.intervals where the function is increasing and decreasing
c.relative extrema
d.possible points of inflection
e. intervals where the function is concave up and concave down
f.graph by hand
i) ii) iii)
[Objective #5, Sections 5.1, 5.2, 5.6]
7.Integrate the following:
a. b. c.
d. e. f.
g. h. i.
j. k. l.
[Objective #9, Section 7.2]
8.Find , , , , and for each of the following:
a. b. c.
d.
[Objective #3, Section 2.5]
9.After t hours a car is a distance miles from its starting point. Find the velocity after 2 hours.
[Objective #3, Section 2.3]
10.A company’s cost function is dollars (for . Find the marginal cost function.
Find the marginal cost at x=9 and interpret your answer.
[Objective #3, Section 2.6]
11. Blood flowing through an artery encounters a resistance of , where x is the distance
(in centimeters from the heart. Find the instantaneous rate of change of the resistance 100 centimeters from the heart.
[Objective #3, Sections 3.3, 3.4]
12.A homeowner wants to enclose three adjacent rectangular pens of equal size, as in the diagram below. What is the largest total area that can be enclosed using only 240 feet of fence?
13.A open-top box with a square base is to have a volume of exactly 500 cubic inches. Find the dimensions of the box that can be made with the smallest amount of materials.
14.A computer dealer can sell 12 personal computers per week at a price of $2000 each. He estimates that each $400 price decrease will result in three more sales per week. If the computers cost him $1200 each, what price should he charge to maximize his profit? How many will he sell at that price?
15.A store can sell 12 telephone answering machines per day at a price of $200 each. The manger estimates that for each $10 price reduction she can sell 2 more per day. The answering machines cost the store $80 each. Find the price and quantity sold per day to maximize the company’s profit.
[Objective #3, Section 3.6]
16.A cube of ice is melting so that each edge is decreasing at the rate of 2 inches per hour. Find how fast the volume of the ice is decreasing at the moment when each edge is 10 inches long.
[Objective #3, Section 4.3]
17.A patient receives an injection of 1.2 cubic centimeters of a drug, and the amount remaining in the bloodstream t hours later is . Find the rate of change of this amount just after the injection (at t=0) and after 2 hours.
[Objective #6, Section 4.4]
18.A videotape store works out a demand function for its videotape rentals and finds it to be where p is the price per rental. Find the elasticity of demand at the price of $2. Should the store raise or lower the price to increase revenue?
19.The demand function for a newspaper is , where p is the price in cents. The publisher currently charges 50 cents, and it plans to raise the price to increase revenues. Will this strategy succeed? Why?
[Objective #7, Section 5.3]
20.Approximate the area below from a=1 to b=3 by calculating a Riemann sum with the 4 rectangles. Then use integration to find the exact area below the curve from a to b.
21.Find the area under each curve between the given x-values.
a. from x=0 to x=4b. from x=0 to x=1
[Objective #5, Section 5.4]
22.A student can memorize words at the rate of words per minute after t minutes. Find the total number of words that the student can memorize in the first 15 minutes.
23.Find the average value of on the interval .
24. A company’s marginal cost function is dollars, where x is the number of units. Find the total cost of the first 400 units.
[Objective #8, Section 5.5]
25.For the demand function and the supply function Find the market demand. Find the consumers’ surplus at the market demand. Find the producers’ surplus at the market demand.
[Objective #7, Section 5.4]
26.Find the area bounded by each pair of curves
a. and b. and c. and
[Objective #11, Section 7.3]
27.A firm that produces two kinds of baseball glove, one that sells for $18 each and the other for $25 each. The total revenue from the sale of x thousand gloves at $18 each and y thousand at $25 each is given by . The company determines that the total cost, in thousands of dollars to produce these gloves is . Find the amount of each type of glove that must be produced and sold in order to maximize profit.
28.The number of office workers near a beach resort who call in “sick” on a warm summer day is where x is the air temperature and y is the water temperature . Find the air and water temperatures that maximize the number of absentees.
[Objective #10, Section 7.5]
29.Find the maximum value of subject to the constraint
30.Find the minimum value of subject to the constraint
31.A metal box with a square base is to have a volume of 45 cubic inches. If the top and bottom cost 50 cents per square inch and the sides cost 30 cents per square inch, find the dimensions that minimize cost.