Area Between Curves
f(x) = 0.29x − 3; g(x) = 13(0.93^x); a = 15; b = 50
Solve for the input value(s) at which the graphs of f and g intersect
20.483
What is the formula for change in f when given f'(x)
(integral from a to b)f'(x)dx
Integrate 9.6x^(-.432) from .36 to ∞
Diverges
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)=0.05(4.2) million dollars per year
Determine the price at which demand is zero (Pmax): D(p) = 50-2p
$25
f(x)=3x^2
g(x)=1-x^2
Find the Area Between Curves from 0>x>1
1
Find the average value of the function
f(x)=3x^2 in the interval -1 < x < 4.
13
Integrate 3x^-2 from 10 to ∞
.3
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
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
f(x) = 0.29x − 3; g(x) = 13(0.93^x); a = 15; b = 50
Calculate the area of the region between the graph of f and the horizontal axis from a to b minus the area of the region between the graph of g and the horizontal axis from a to b.
169.317
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
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
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
Given the following supply function, calculate producer revenue and producer surplus at a price of $19.99 s(p) = .2p when p > 5
Producer revenue = $79.920 million Producer surplus = $37.460
r(t)=3x^2 thousand dollars per year is the rate of change of revenue for a business, where t is the number of years since 2012, 0 < t < 1.
c(t)=2-x^2 thousand dollars per year is the rate of change of cost for a business, where t is the number of years since 2012, 0 < t < 1.
What is the total change in profit?
Profit decreased by $666 in 2013
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
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. How much will leak eventually?
59.251 million gallons
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%. How much is the account worth today?
$9.617 million
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
f(x)=5ln(x)+5; g(x)=e^x; a=1, b=3
Find the TOTAL area in between the curves
5.622
d(t) = 0.024t^2 - 1.72t + 22.58 units per year Find the average VALUE and the average ROC between 0 and 10:
a.) 14.78 units per year b.) -1.48 units per year
The value of an artifact increases with age. Thirty-six years after discovery it was appraised at $1200. The rate of change in the value of the artifact can be modeled as: c(x)=4530/x^1.5 dollars per year
If the artifact is preserved forever, what will it eventually be worth?
$2710
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?
.75 billion dollars
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