Calculus homework help
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.
f f ( ) 0 ( ) 0 x AND x
b.
f f ( ) 0 ( ) 0 x AND x
c.
f f ( ) 0 ( ) 0 x AND x
d.
f f ( ) 0 ( ) 0 x AND x
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
3 2
y x x x 2 10 28 4 , 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:
1 3 2 5
3
y x x x 7
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
1 3 2
6
Q t t t t ( ) 5 3 9
. 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
N t t
t
At what time will the epidemic be at its worst? ________________
10. Consider the function:
6 3
2 5
x
y
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 p t t ( ) 5
At what rate will the demand be changing with respect to time 10 days
from now?
_______________
13. Consider the following Cost function:
2 C x x x ( ) 3 7 4
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 = 4r
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
12 y
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?
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