Math 1100 Practice Exam 2
1. Be able to define/explain/identify in a picture/draw the following
(a) Absolute minimum/maximum
24 October, 2011
(b) Critical point
(c) Relative minimum/maximum
(d) Point of inflection
(e) Concave up/down
You should be able to deal with the geometry of circles, rectangles, right triangles, and boxes. (If any other shapes come up, I will provide the relevant formulas on the test.
2. Find the elasticity of the demand function p = 4 q = 80 at the point (10 , 40)

3. Find dy dx
.
(a)
(b)
(c)
(d)
(e) y = ln x y = 6 log
7 x
2 ln( x 3 + 6) ln
6 x 4 + 1 x − 3 ln(( x + 3)
2
( x
2
− 6)
4
)

(h)
(i)
(j)
(f)
(g) y = log
5 x
3 log
6 x − 1 x + 1 y = e
2 x − x y = e x
2 y = 2 x

(k)
(l)
(m)
(n)
(o) y =
1 e x
2 − 1 y = 4 x
2 − 2 x y = xe x y = ( x
2 − 1) ln x xy
2
− y
2
= 1

(p) x
2
+ 2 y
2 − 4 = 0
(q) x
4
+ 2 x
3 y
2
= x − y
3
4. A study showed that, on average, the productivity of a worker after t hours on the job can be modeled by the function P ( t ) = 27 t + 6 t 2 − t 3 for 0 ≤ t ≤ 8, where P is the number of units produced per hour.
(a) For which values of t is P increasing?
(b) When are workers most productive?

5. Find the absolute maximum of f ( x ) = x 3 − 2 x 2 − 4 x + 2 on the interval [ − 1 , 3]
6. The total cost function for a product is C ( x ) = 100 + x 2 units will result in a minimum average cost per unit?
dollars for x units. How many
7. Find all relative minima, maxima, and horizontal points of inflection of y =
1
4 x
4 −
2
3 x
3
+
1
2 x
2 − 6

8. Suppose the demand for a product is given by ( p + 1)
√ q + 1 = 1000
(a) Find the elasticity when p = $39.
(b) What type of elasticity is this?
(c) How would a price increase affect revenue?
9. Is the function f ( x ) = 2 x
3
+ 4 x − 8 concave up or down at x = − 1? At x = 4?
10. Find the critical values of y = 2 x 3 − 12 x 2 + 6

11. An agency charges $10 per person for a trip to a concert if 30 people travel in a group.
For each person above the 30, the charge per person will be reduced by $0.20. The agency cannot take more than fifty people. How many people will maximize revenue for the agency?
12. A 30-foot ladder is leaning against a wall. If the bottom of the ladder is pulled away from the wall at 1 ft/sec, at what rate is the top of the ladder sliding down the wall when the bottom is 18 ft from the wall?

13. A rectangular field with one side along a river is to be fenced. Suppose that no fence is needed along the river, the fence opposite the river costs $20 per foot and the fence on the other sides costs $5 per foot.
(a) If the field must contain 45,000 square feet, what dimensions will minimize the cost?
(b) How large an area can be fenced for $6,600?
14. If xy = x + 3, dx dt
= − 1, and x = 3, find dy
.
dt

15. When $700 is invested at 9% interest compounded continuously, its value after t years is S ( t ) = 700 e .
09 t . At what rate is the money in the account growing when t = 4? When t = 10?
16. At what points does the curve defined by x 2
Vertical tangents?
+ 4 y 2 − 4 = 0 have horizontal tangents?
17. Find the relative maxima and minima of f ( x ) = x ln x .

18. Make a sign diagram for y = x 3 minimum points of the function.
− 3 x 2 + 6 x + 1 and find all relative maximum and
19. Suppose that air is being pumped into a spherical balloon at a rate of 5 in
3
/min. At what rate is the radius of the balloon changing when the radius is 5 inches? (Recall that the volume of a sphere is given by V =
4
3
πr 3 )
20. Use the second derivative test to find all maxima and minima of f ( x ) = x
5 − 5 x
4
.

21. A rectangular box with a square base is to be formed from a square piece of cardboard with 24 inch sides. A square is cut out of each corner and the box is folded from the remainder of the material. (If this description isn’t clear, see problem 27 on page 740.)
How large a square should be cut from each corner to maximize the volume of the box?
22. A firm has total revenues given by R ( x ) = 2800 x − 8 x 2 − x 3 dollars for x units of a product. Find the maximum possible revenue for sale of this product.
23. Find the equation of the tangent line to the curve x 2 − 4 x + 2 y 2
(2 , 2).
− 4 = 0 at the point

24. Between the years 1960 and 2002 the percentage of women in the workforce can be modeled by
W ( x ) = 2 .
552 + 14 .
569 ln x when x is the number of years past 1950. If this model holds, at what rate will the percentage of women in the workforce be changing in 2015?
25. A firm can produce up to 100 units per week. If its total cost function is C ( x ) =
500 + 1500 x dollars and its total revenue is R ( x ) = 1600 x − x 2 dollars, how many units will maximize profit?
26. The amount of the radioactive isotope thorium-234 present after t years is given by
Q ( t ) = 100 e
− 0 .
2828 t . Find the function that describes the isotope is decaying