Revenue, Supply and Demand Functions : Derivatives and Integrals

5. A man was sentenced to 50 years in prison when he was 20 years old. While in prison he reflected on his life and decided that he should turn his life around and do something good for his society. He then became a model prisoner and his good behavior earned him the privilege to pursue a career in law. When he became 39 years, he was released on parole. He then spent one year doing an internship as he completed his career in law.

At 40, he was hired by a huge law firm, which promised him promotions if he proved to be a good attorney. He was offered a starting salary of $75,000 per year. This left him with 25 years before retirement.

He contributed (at the end of each year) 10% of his annual salary to his company-provided 401(k) plan. He had chosen an investment option with a fixed rate of return of 7 % compounded continuously. His annual salary increased at the rate of approximately 8 percent per year.

Thus his salary was determined by the function y = S(t), where t is the number of years since the time he became 40.

a. According to the function S(t), how much money did he make over that 25-year period? This amount can be found by computing the integral of the function that defined his income.
= __________________
(b) Assuming a continuous reinvestment of interest, what would his 401(k) retirement investment be worth at the end of the 25 years? (Hints: (i) Modify the integral from part (a) to find 10% of his annual income each year, and (ii) include a continuously compounded interest term at the rate of 8% per year. Evaluate it either by hand or numerically using your calculator.)

= ________________________

In problems 6 - 7, circle the best answer. SHOW ALL WORK IN A NEAT AND ORDERLY MANNER.
6. Find the derivative of the following function:

a.
(i) (ii)
(iii) (iv)
(v) none of these

Show work:

Find the derivative of the following function:

6b.

(i) (ii)
(iii) (iv) (v) none of these

Show work:

Find the derivative of the following function:
c.

(i) (ii)
(iii (iv)
(v) none of these

Show work:

Find the derivative of the following function:
d.

(i) (ii)
(iii) (iv)
(v) none of these

Show work:

Find the derivative of the following function:

6e.

(i) (ii)
(iii) (iv)

(v) none of these

Show work:

7. Evaluate the integrals.

a.

(i) (ii) (iii)
(iv) (v) none of these

Show work:

7b. (hint: try substitution)

(i) (ii) (iii)
(iv) (v) none of these

Show work:
c. (hint: try substitution)

(i) (ii) (iii) (iv) (v) none of these

Show work:

d. (Use your graphing calculator to solve this problem. Approximate your solution to 2 decimal places.)

(i) 2.98 (ii) 3.85 (iii) 4.57 (iv) 6.28 (v) none of these

e. (Hint: integration by parts.)
(i) (ii) (iii) (iv) (v) none of these
Show work:

Consumer Surplus

The Consumers' surplus is the area between the demand curve and the market price curve (line) for a given commodity. That is,
Consumers' Surplus = where D(x) is the demand and M(x) is the market price evaluated at a particular quantity x.
Getting at the concept: Recall that the demand curve can be viewed as a willingness-to-pay curve. It shows the value that consumers place on extra units of the good. Consumer surplus is the difference between the amount that consumers actually pay and the amount that they would have been willing to pay. On a graph, consumer surplus can be shown as the area under the demand curve and above the prevailing market price. The shaded area "c" in the graph represents the amount of consumer surplus.
If the demand function for electricity at SRP (Salt River Project, a local utility) is D(x) = 1100 - 10x dollars (where x is in millions of kilowatt-hours, ), find the consumers' surplus at the demand level x = 80. The following will assist you in finding the solution:

a) Find the market price, which is the demand curve evaluated at the specified demand level.

b) Substitute and evaluate the integral given in the definition of consumer surplus.

keywords: integration, integrates, integrals, integrating, double, triple, multiple