# Integrals : Average Value of a Function

Please see attached file for full problem description.

1. What is the average value of the function f in Figure 6.4 over the interval ?

From the graph, we can approximate:

The average value of f on the interval from 1 to 6 is

3. Find the average value of over the interval [0, 2].

The average value of g on the interval from 0 to 2 is

Using Quickmath.com to estimate the definite integral, we get

1

1. Supply and demand curves for a product are in Figure 6.23.

(a) Estimate the equilibrium price and quantity.

Equilibrium price:

Equilibrium quantity:

(b) Estimate the consumer and producer surplus. Shade them.

(c) What are the total gains from trade for this product?

4

4. A small business expects an income stream of $5000 per year for a four-year period.

S(t) = $5000/yr.

(a) Find the present value of the business if the annual interest rate, compounded continuously, is

Present Value = , where r = annual interest rate and M = the number of years in the future.

(i) 3%

(ii) 10%

(b) In each case, find the value of the business at the end of the four-year period.

Future Value = , r = annual interest rate and M = the number of years in the future.

(i) 3%

(ii) 10%

1, 3

1. Table 6.3 shows the cumulative number of AIDS deaths worldwide.

Year 1995 1996 1997 1998 1999 2000

Cases 8.2 10.6 13.2 15.9 18.8 21.8

Find the absolute increase in AIDS deaths between 1995 and 1996

Since the absolute rate of change is the derivative, we will approximate this value by finding the slope using a right-hand endpoint.

Absolute increase between 1995 and 1996: ??? million cases/yr.

and between 1999 and 2000.

Absolute increase between 1999 and 2000: ? million cases/yr.

Find the relative increase between 1995 and 1996

Since the relative increase is the absolute increase divided by the number of cases in 1995, we have

Relative increase between 1995 and 1996: ??%/yr

and between 1999 and 2000.

Relative increase between 1999 and 2000: ??%/yr

3. A town has a population of 1000. Fill in the table assuming that the town's population grows by

(a) 50 people per year Formula:

Year 0 1 2 3 4 ... 10

Population 1000

(b) 5% per year Formula:

Year 0 1 2 3 4 ... 10

Population 1000

#### Solution Summary

Integrals are investigated and the average values of functions are found. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.