Unit 2

Average Value and Average Rate of Change

100

Find the average value of the function f(x) = 3x^2 in the interval -1 < x < 4.

13

200

Let H(x) be a quantity function. H(x)= 2 - x^2 dollars where x is the number of days after January 1. Find the average rate of change in H(x) over the interval -1 ≤ x ≤ 5

-4 dollars per day

300

U.S factory sales of electronic goods to dealers from 1990 to 2001 can be modeled as: s(t) = .0388t^3 - 0.495t^2 + 5.698t +43.6 billion dollars, t years since 1990 Calculate the average value of U.S factory sales from 1990 to 2001

$67.885 billion

400

The circulation of newspapers in the U.S between 1986 and 2000 can be modeled as n(x) = 0.00792x^3 - 0.32x^2 + 3.457x + 51.588 million newspapers where x is the number of years since 1980 What was the average newspaper circulation from 1986 to 2000?

59.668 million newspapers

500

d(t) = 0.024t^2 - 1.72t + 22.58 units / year Find the average VALUE and the average ROC between 0 and 10:

a.) 14.78 units / year b.) -1.48 units / year

Improper Integrals

100

It is estimated that a sunken oil tanker will leak oil into the surrounding waters at a rate of r(t) = (19t + 30) / (1+t^3) million gallons per year, t years from the date the vessel sank. a.) How much oil will leak over the first 5 years? b.) How much will leak eventually?

a.) 54.860 million gallons b.) 59.251 million gallons

200

Integrate 9.6x^(-.432) from .36 to ∞

DIVERGES

300

Integrate 3x^-2 from 10 to ∞

0.3

400

A substance will decay at a rate of r(t) = -0.027205(0.998188^t) grams per year How much will eventually decay?

15 grams

500

The rate of change of the value of an antique chair is r(t) = 2500/(x^1.5) dollars per year where x is the age of the chair in years The chair was valued at $300 25 years after it was crafted. How much will the value increase between 25 and 100 years after being crafted? How much will the chair be worth 100 years after it was crafted?

$500, $800

Streams in Business

100

A company will invest 4% of its profit into a fund to fund capital improvements. This account bears 3.8% interest, compounded continuously. The current level of profit is 6 million dollars per year, but the company believes that this amount will increase by 3% per year. What is the flow rate equation?

R(t) = .04(6(1.03^t)) million dollars per year

200

A firm has an annual profit of $4.2 million and allocates 5% of its profits into as a continuous stream into investments. If the profits remain constant, what is the flow rate equation?

R(t) = .05(4.2) million dollars per year

300

A company showed a profit of $1.8 million last year. The company expects their profit to decrease by 7% each year over the next 5 years and will be continuously invested into an account with an interest rate of 4.75%. Calculate the 5 year present value.

$6.767 million

400

Last year, a company had a profit of 1.8 million dollars. They expect their profits to decrease by .04 million dollars per year over the next 5 years and will be continuously invested into an account with an interest rate of 4.75%. Find the future value.

$9.617 million

500

A firm has a flow rate equation of R(t) = .03(2(.95^t)) billion dollars per year. How much will the corporation invest over the next 20 years?

$0.750 billion

Supply & Demand

100

The demand for ceiling fans is given by D(p) = 25.92(0.996^p) thousand ceiling fans where p is the price in dollars of a ceiling fan. Compute the price of unit elasticity.

$250 per ceiling fan

200

Determine the price at which demand is zero (Pmax): D(p) = 50-2p

$25

300

Given the following supply function, calculate producer revenue and producer surplus at a price of $19.99 s(p) = .2p when p > 5 s(p) = 0 elsewhere

Producer revenue = $79.920 million Producer surplus = $37.460

400

D(p) = (40.007) / (1+.033e^.354p) million pounds S(p) = (51) / (1 + 53.98e^-.395p) million pounds when p > .5 Find market equilibrium and total social gain.

Market equilibrium: $9.26 per pound, 21.318 million pounds TSG: $153.123 million

500

D(p) = 38.301 / (1 + .003e^.050p) million calculators S(p) = .747p - 35.467 million calculators when p > 47.5 Calculate producer surplus, consumer surplus, and total social gain at market equilibrium

PS: 606.026 million dollars CS: 1184.337 million dollars TSG: 1790.363 million dollars

Differences in Accumulated Change

100

f(x)=0.17x^3 - 2.75x^2 + 9x + 30 g(x)=0.15x^2 - x + 3 Find the area between the two curves from a=1 to b=9.

191.067

200

f(x)=2x g(x)= -5x+12 Find the area from a=0 to the value of the x-intercept on g(x).

11.93

300

f(x)=e^(0.5x) g(x)=2/x Find the total area from a=0.5 to b=3

4.17

400

r(x)=.125x^2 - .25x + 2 c(x)=-.3x^2 + 1.3x + 8 r(x) is a revenue curve and c(x) is a cost curve. X is the number of years since 2000. Monetary unit is in thousand dollars. Find the area between the curves from a=4 to b=6. Is this revenue or costs?

$5.967 thousand: these are costs

500

r(x)=.125x^2 - .25x + 2 c(x)=-.3x^2 + 1.3x + 8 r(x) is a revenue curve and c(x) is a cost curve. X is the number of years since 2000. Monetary unit is in thousand dollars. If the profit in 2004 is $15 thousand, what is the profit at the end of 2007?

$19.05 thousand