Calculus

_______________________________

MATH 182 TEST #2

SHOW ALL WORK NEATLY WHERE APPROPRIATE

3 x 5 Notecard O.K.  Time Limit: 3 hrs

1. A. An inflection point of a function is a point where its ______________

changes.

B. You’re trying to find an interval where the graph of a function f

would be both increasing AND concave down. Which pair of

conditions would be used to determine the interval?

a. ( ) 0 ( ) 0x AND x   f f b. ( ) 0 ( ) 0x AND x   f f

c. ( ) 0 ( ) 0x AND x   f f d. ( ) 0 ( ) 0x AND x   f f

C. True/False _____ Knowing where a function is increasing and

decreasing gives information about any inflection point the function

might have.

2. Sketch a graph of a function whose domain is ℝ, and which is

differentiable (has a derivative) at all but four points on its domain.

3. Sketch a graph that satisfies ALL of the following properties:

a. Its domain is ℝ.

b. It’s increasing on ( 0).

c. It’s decreasing on (0, ).

d. It has no concavity on ( 0).

e. It’s concave up on (0, ).

f. It has a derivative everywhere except the origin.

4. Consider the graph of the function 232 10 28 4y x x x    , and

consider the point on the graph where x = 1. Use Calculus to answer

the following:

A. Is the graph increasing or decreasing at the point? ______________

B. Is the graph concave up or concave down at the point? ____________

5. Determine the intervals of concavity: 3 21 5 3

7y x x x     

Concave UP: _________________

Concave DOWN: _________________

6. Given the functions

S(v) = v 3/2

v(t) = t 2

Find: dS dt

____________________

7. Sketch a graph which satisfies ALL of the following properties:

a. x-intercept at (2, 3)

b. Increasing on (2, )

c. Decreasing on (, 2)

d. Concave down on ℝ

8. The quantity produced by a worker t hours after the beginning of their

shift is given by 3 21

6 ( ) 5 3 9Q t t t t    . For what value of t does the

production reach the point of diminishing returns? _______________

9. In an epidemic, after t weeks, N new cases will be reported, where

2

( ) 25

tN t t

 

At what time will the epidemic be at its worst? ________________

10. Consider the function: 6 3 2 5

xy x  

a. Domain: _______________ b. x-intercept: _____________

c. y-intercept: _____________ d. Vertical asymptote: ___________

e. Horizontal asymptote: ___________

11. Sketch a graph which satisfies ALL of the following properties:

a. Domain = ℝ

b. The derivative is 0 at the origin.

c. Increasing on (, 2) (0, 2)

d. Decreasing on (2, 0) (2, )

12. The Demand function as a function of the price, p, is given by:

10( )D p p



The Price function as a function of time, t, (in days), is given by

3( ) 5p t t At what rate will the demand be changing with respect to time 10 days

from now?

_______________

13. Consider the following Cost function:

2( ) 3 7 4C x x x  

a. Find the exact cost of producing the 5th item. _____________

b. Use the Marginal Cost function to approximate the cost of

producing the 5th item.

________________

14. Given the Demand function

p(x) = 3x 2 + 2x + 7

find the Marginal Revenue function. ____________________________

15. Find A and B so that the graph of the function

6 5

Ax y

Bx  

will have a vertical asymptote of x = 1 and an x-intercept of (3, 0).

___________________________

16. Each side of a cube is increasing at a rate of 19 m/s. How fast is the

volume of the cube increasing at the moment when the side is 2 m?

___________________

17. The surface area of a sphere is increasing at the rate of 7 m 2 /s. How fast

is the radius of the sphere increasing at the moment when r = 6? Leave

your answer in exact form.

[Surface Area = 4r 2 ]

Be sure to include the proper units in your answer. __________________

18. Find the equation of the tangent line to the curve x 2 + xy

3 + y = 69 at

the point (1, 4).

________________________________________

19. Consider the following demand function:

q = 4p 2 + 161

a. Find E(p), the price elasticity of demand. ______________________

b. Calculate E(p) when p = $6. _______________

c. Interpret the answer to b.

d. Which level of elasticity does this problem imply? ________________

e. What does the level of elasticity mean in terms of the sensitivity to a

price increase?

20. Find two positive numbers x and y whose sum is 105, and such that the

quantity xy 6 is as large as possible.

______________________

21. A farmer is building a rectangular horse corral using 1000 ft of fence.

One of the sides of the corral is bordered by a river, so no fence needs to

be built there. Find the dimensions of the corral that will produce the

maximum area.

_________________________

22 - 23. Consider the function: 2 4

12y x 



a. What is the domain of the function? ______________

b. What is the x-intercept of the function? ______________

c. What is the y-intercept of the function? ______________

d. Find the vertical asymptote(s) of the graph. ____________

e. Find the horizontal asymptote(s) of the graph. ____________

f. On what interval(s) is the graph increasing? ________________

[Don’t forget to show work.]

g. On what interval(s) is the graph decreasing? ________________

h. Find any extreme points, and state whether it’s a maximum or

a minimum.

______________________________

i. What is the range of the function? ________________

j. SKETCH the graph of the function.

24. In fencing a rectangular region, the unit cost on the length is $7/m, and

the unit cost on the width is $10/m. If the AREA must be 630 m 2 , find

the dimensions of the rectangle that will minimize the total cost.

_____________________________

25. Let Q(p) = 4p 2 − 6p + 8 represent the units of an air pollutant when the

population is p. If the population is currently 4300 people, and if it's

increasing at a rate of 90 people per year, at what rate is the level of

pollution increasing?

__________________________