§4.1 Exponential Functions - Directoryfaculty.ung.edu/.../MATH2040-U4-notes.pdf · Hartfield MATH 2040 | Unit 4 Page 1 §4.1 Exponential Functions Recall from algebra the formulas

Hartfield MATH 2040 | Unit 4 Page 1

§4.1 Exponential Functions Recall from algebra the formulas for Compound Interest: Formula 1 – For Discretely Compounded Interest

1

n tr

A t Pn

Formula 2 – Continuously Compounded Interest

r t

A t P e

t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t n = number of compounds per year

For Formula 1, the value of n is based on the frequency of compounding. Common frequencies include:

annually (n = 1), semiannually (n = 2), quarterly (n = 4), bimonthly (n = 6), monthly (n = 12), biweekly (n = 26), weekly (n = 52), daily (n = 360 or 365).

In economics, financial institutions have to report an annual percentage change even though their nominal rate of interest is for some other interval of time. A formula may ask for a future amount when given a present amount (given P, find A(t)) or it may ask for a present amount when given a future amount (given A(t), find P).


Ex. 1: Kelsey has $2000 to save for three years.

She goes to three different banks and finds she can lock in the following rates. Rounding down to the penny, how much would she have with each bank and thus which bank is the best option?

Bank A: 4.20% interest per year

compounded quarterly Bank B: 4.18% interest per year

compounded daily Bank C: 4.15% interest per year

compounded continuously


Ex. 2: To the nearest thousandth of a percent,

what is the Annual Percentage Yield on each bank?

Bank A: 4.20% interest per year

compounded quarterly

Bank B: 4.18% interest per year compounded daily

Bank C: 4.15% interest per year

compounded continuously


Ex. 3: A $24,000 automobile depreciates by

40% per year. Find its value in 6 months and then in 3 years.


§4.3a Differentiation of Exponential Functions Rule 8: Derivative of a Natural Exponential

x xde e

d x

If the exponential function has a function within the exponent, we can extend rule 8 using Chain Rule: Rule 8a: Generalized Derivative of a Natural

Exponential

f x f xde e f x

d x

Ex. 1: Find the derivative.

2

3xf x e


Ex. 2: Find the derivative. Then evaluate the

derivative at the given x-value and approximate it to three decimal places.

2

,x x

f x x e e f’(1)

Ex. 3: Find the second derivative.

3

xf x e


Applications Ex. 1: The percentage P(t) of people surviving

to age t years in ancient Rome can be approximated by P(t) = 92e –0.0277t.

Calculate P’(22) and explain what the

result indicates.

(Source: Finite Mathematics & Calculus Applied to the Real World (1996) p. 972, #73)

Ex. 2: A cup of coffee brewed at 200 degrees, if

left in a 70-degree room, will cool to T(t) = 70 + 130e –0.04t (°F) in t minutes.

Determine the temperature of the

coffee in 1 hour and the rate of change in the temperature at that time.

(Source: 5th edition p. 302, #76)


§4.2 Logarithms & Exponential Equations Review from algebra about solving exponential equations.

1. Isolate an exponential expression on one

side. 2. Take the natural logarithm (or common

logarithm) of both sides. 3. Use the laws of logarithms to rewrite the

exponential expression so that no variable remains in the exponent.

4. Apply basic algebraic and arithmetic manipulation to solve for x.

5. Use the laws of logarithms to simplify the solution and approximate the solution.

6. Check your solution.

Ex.: Solve 2 12 5 1 3

xe

.


Solving exercises with exponential equations Ex. 1: To the tenth of a year, how long would

it take an account to double if the interest rate is 9.9%

A: compounded monthly.

B: compounded continuously.


Ex. 2: To the tenth of a year, how long would

it take an account with an interest rate of 16.79% compounded daily (n=365) to

A: triple in value.

B: increase by 40%.


Ex. 3: Using information from a recent

Edmunds.com report, the resale value of a 2008 SUV is expected to depreciate up to 20% per year. To the tenth of a year, how long will it take value of an SUV to decrease by half?


§4.3b Differentiation of Logarithmic Functions Rule 9: Derivative of a Natural Logarithm

1ln

dx

d x x

If the logarithmic function has a function within the logarithm, we can extend rule 9 using Chain Rule: Rule 9a: Generalized Derivative of a Natural Logarithm

ln

f xdf x

d x f x

Derivatives of Exponential & Logarithmic Functions of other bases Rule 10: Derivative of Exponential (base a)

lnx xd

a a ad x

Rule 11: Derivative of a Logarithm (base a)

1lo g

lna

dx

d x a x



2

( ) ln 2f x x x


5

3( ) ln 1f x x



( ) ln 1x

f x e x

Ex. 4: Find the derivative. Then evaluate the

derivative at the given x-value.

2

ln( )

xf x

x f’(1)


§4.3-c & 4.4 Economic Applications Definition: Let D(p) be the consumer demand

at price p. Then the consumer expenditure E is

E(p) = p · D(p) Ex. 1: If consumer demand for a commodity

is given by the function D(p) = 8000e –0.05p, where p is the

selling price in dollars, find the price that maximizes consumer expenditure.



A similar exercise for maximizing revenue can be approached where the independent variable is the quantity x sold and price is a function of x (as opposed to the inverse relationship in consumer expenditure): R(x) = p(x) · x Ex. 2: Find the quantity x (in thousands) and

the price p (in dollars) where revenue is maximized if p(x) = 200e –0.25x.


Relative Rates of Change While all derivatives are rates of change, not all derivatives consider the relative values of a function at a given point of time. In absolute terms a $10,000 product increasing at a rate of $100 a year has a higher rate of change than a $100 product increasing at $20 a year. Taking into respect the current value of each product however tells us that the $100 object has a higher relative rate of change. Definition: The relative rate of change in a

function is the derivative of the natural log of the function; that is,

( )

ln ( ) .( )

d f tf t

d t f t

Ex. 1: Find the absolute rate of change and

the relative rate of change in f. Then evaluate the relative rate of

change at the given value of t. f(t) = t3, t = 10


Ex. 2: Find the absolute rate of change and

the relative rate of change in f. Then evaluate the relative rate of

change at the given value of t. f(t) = e –t³, t = 5

Applications of Relative Rates of Change Ex 1: The gross domestic product of a

developing country is forecast to be G(t) = 5 + 2e0.01t million dollars t years from now. Find the relative rate of change in the GDP 20 years from now.



Ex 2: The population (in millions) of a city t

years from now is given by the function P(t) = 6 + 1.7e0.05t.


a. Find the relative rate of change of

the population 8 years from now. b. Will the relative rate of change ever

reach 1.5%?


Demand Functions & the Elasticity of Demand Definition: The demand function x = D(p)

gives the quantity x of an item that will be demanded by consumers at price p.

Statement: The Law of Downward-Sloping

Demand states that since demand generally falls as prices rise, the slope of the demand function is negative.

Since revenues are a product of prices and demand, a balance point should exist between increasing prices and increasing demand so that revenues are maximized.

Definition: For a demand function D(p), the

elasticity of demand is

( )

( ) .( )

p D pE p

D p

If E(p) > 1, then demand is elastic

and prices should be lowered to increase revenue. If E(p) < 1, then demand is inelastic and prices should be raised to increase revenue. At maximum revenue, E(p) = 1 and demand is unitary.


Ex. 1: For the demand function

D(p) = 100 – p2, determine where demand is elastic, inelastic, or unitary.

Ex. 2: A liquor distributor wants to increase

its revenues by discounting its best-selling liquor. If the demand function for this liquor is D(p) = 60 – 3p, where p is the price per bottle, and the current price is $15, will the discount succeed?



Ex. 3: At a market level, the demand function

for distilled spirits is defined by D(p) = 3.509p–0.859. Determine where demand is elastic, inelastic, or unitary and analyze the relationship implied between taxes, liquor consumption, and government revenues.

(Source: Journal of Consumer Research 12)

Documents

§4.1 Exponential Functions - Directoryfaculty.ung.edu/.../MATH2040-U4-notes.pdf · Hartfield MATH 2040 | Unit 4 Page 1 §4.1 Exponential Functions Recall from algebra the formulas