Average Value & Average Rate of Change
Improper Integrals
Streams in Business
Supply & Demand
Probability
100
Find the average value of the function f(x) = 3x^2 in the interval -1 < x < 4.
13
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
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
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
100
Is the function a PDF? Explain: f(x) = 1.5(1-x^2) when 0
Yes
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
200
Integrate 9.6x^(-.432) from .36 to ∞
DIVERGES
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
200
Determine the price at which demand is zero (Pmax): D(p) = 50-2p
$25
200
The PDF of frozen yogurt sales is approximate by f(x) = .32x for 0
.16
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
300
Integrate 3x^-2 from 10 to ∞
.3
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
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
300
At a grocery store check out counter, the average wait time is 2.5 minutes. Suppose the waiting time follows an exponential density function. What is the probability of waiting less than 2 minutes to check out?
55.1%
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
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
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
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
400
Suppose the weight of passenger's luggage follows a normal distribution with a mean of 40 pounds and a standard deviation of 10.63 pounds. Calculate the probability that a piece of luggage weighs less than 45 pounds:
68.1%
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
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
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
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
500
Scores on an exam are normally distributed with a mean of 72.3 and a standard deviation of 28.65. What % of students had scores between 60 and 80? What % had at least a 90?
27.2% 26.8%
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