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5.2 "Exponential Functions"
Before: f(x) = x2 , f(x) = 5x3/4 , ...
Now: f(x) = 3x (variable exponent) called "Exponential Functions"
(ex) Sketch f(x) = 3x x Since it's increasing this is sometimes called a "growth function."
f(x) (ex) Sketch f(x) = ( )x x f(x) Since it's decreasing, this is sometimes called a "decay function."
* Functions that increase or decrease are one­to­one.
Solve the equation.
(ex) 6 7­x = 6 2x+1
(ex)
(ex) (ex) 27 x­1 = 9 2x­3
(ex)
.
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Sketch the graph of f.
(ex)
(ex)
(ex)
(ex)
(ex)
(ex)
*Look at the example "application" problems beginning on the bottom of p.335
Compound Interest Formula:
P=Principal
r=annual interest rate (as a decimal)
n=number of interest periods per year
t=number of years P is invested
A=amount of interest after t years
Drug Dosage. A drug is eliminated from the body through urine. Suppose that for an initial dose of 10 mg, the amount A(t) in the body t hours later is given by
A(t) = 10(0.8)t .
(a) Estimate the amount of the drug
in the body 8 hours after the initial dose.
(b) What percentage of the drug
still in the body is eliminated each hour?
According to Newton's Law of Cooling, the rate at which an object cools is directly proportional to the difference in temperature between the object and the surrounding medium. The face of a household iron cools from 125o to 100o in 30 minutes in a room that remains at a constant temperature of 75o. From calculus, the temperature f(t) of the face after t hours of cooling is given by f(t) = 50(2)­2t + 75.
(a) Assuming t=0 corresponds to 1:00 pm,
approximate to the nearest tenth of a degree the temperature of the face at
2:00 pm, 3:30 pm, and 4:00 pm.
temp
200
100
(b) Sketch the graph of f for 1
2
3
4
Hours
2
An important problem in oceanography is to determine the amount of light that can penetrate to various ocean depths. The Beer­Lambert Law asserts that the exponential function given by I(x) = I0cx is a model for this phenomenon. For a certain location, I(x) = 10(0.4)x is the amount of light (in calories/cm2/sec) reaching a depth of x meters.
I (cal/cm2/sec)
10
(a) Find the amount of light at a depth of 2 meters.
5
(b) Sketch the graph of I for 1
2
3
4
5
x (meters)
q (grams)
Dissolving salt in water. If 10 grams of salt is added to a quantity of water, then the amount q(t) that is undissolved after t minutes is given by q(t) = 10
5
Sketch a graph that shows the value q(t) at any time from t=0 to t=10
5
Compound Interest. If a savings fund pays interest at a rate of 6% per year compounded semiannually, how much money invested now will amount to $5000 after 1 year?
t (min)
10
Credit­card interest. A certain department store requires its credit card customers to pay 18% interest per year compounded monthly on unpaid bills. If a customer buys a $500 TV on credit and makes no payments for one year, how much is owed at the end of the year?
(Look at problems #44 & #64 on p.341)
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