In Exercises 1-9 determine the intervals on which each function is increasing and those where it is decreasing. Using a graphing software, such as Graphmatica, to plot the graph y = f(x) to see whether it agrees with your results.
1. f(x) = 3x + 2
2. f(x) = 8 - 2x2
3. f(x) = 6x - 2x2
4. f(x) = x4 - 2x2 + 1
5. f (x) = x/(x+1) [Note: f '(x) can change sign at x = -1. Why?]
6. f(x) = 3x4 +4x3 - 12x2
7. f(x) = x√(x2+1).............................. *
8. f(x) = 8x1/3 - x4/3................................................. *
9. f(x) = (x-1)2 / (x2-3)
In Exercise 10-17 determine the intervals on which each function is concave up (or convex) and those where it is concave down (or concave) and find any points of inflection.
Using a graphing software, such as Graphmatica, to plot the graph y = f(x) to see whether it agrees with your results.
10. f(x) = x2 - 4x +3
11. f(x) = x3 - 3x + 1
12. f(x) = 5 - 6x - x2
13. f(x) = x4................................*
14. f(x) = xs + 2x
15. *The total cost of producing x units of a certain commodity is C(x) thousand dollars, where
C(x) = x3- 20x2 + 179x + 242
a) Find A'(x), where A(x) = C(x) / x is the average cost minimized? What is the minimum average cost?
b) For what values of x is A (x) increasing? For what values is it decreasing?
c) For what level of production x is the average cost minimized? What is the minimum average cost?
16. The total cost of producing x units of a certain commodity is given by
C(x) = √(5x +2) + 3
Sketch the cost curve and find the marginal cost. Does the marginal cost increase or decrease with increasing production?
17. The cost of producing x units of a commodity per week is
C(x) = 0.3x3 - 5x2 +28x + 200
a) Find the marginal cost C'(x), and sketch its graph along with the graph of C(x) on the same coordinate plane.
b) Find all the values of x where C"(x) = 0. How are these levels of production related to the graph of the marginal cost?
18. *An efficiency study of the morning shift (from 8:00A.M to 12:00 noon) at a factory indicates that an average worker who arrives on the job at 8:00A.M will have produced Q units t hours later, where
Q (t) = -t3+ (9/2)t2 + 15t
a) At what time during the morning is the worker performing most efficiently?
b) At what time during the morning is the worker performing least efficiently?
19. * During a recession, Congress decides to stimulate the economy by providing funds to hire unemployed workers for government projects. Suppose that t months after the stimulus program begins, there are N (t) thousand people unemployed, where
N(t) = -t3+ 45t2 + 408t + 3078
a) What is the maximum number of unemployed workers? When does the maximum level of unemployment occur?
b) In order to avoid overstimulating the economy and inducing inflation, a decision is made to terminate the stimulus program as soon as the rate of unemployment begins to decline. When does this occur? At this time, how many people are unemployed?
20. * Suppose that in a certain community, there will be M(r) thousand new houses built when the 30-year fixed mortgage rate is r%, where
M(r) = (1 + 0.02r) / (1 + 0.009r2)
a) Find M'(r) and M"(r).
b) Sketch the graph of the construction function M(r).
c) At what rate of interest r is the rate of construction of new houses minimized?