Business Calculus

Exponentials and Logarithms

3.1 The Exponential Function

Know your facts for

1. Know the graph: A horizontal asymptote on the left at y = 0. Through the point (0,1)

Domain: (-∞, ∞) Range: (0, ∞)Increasing on the interval (-∞, ∞) .

2. Use the graph to find limits:

xexf )(

0lim

x

xe

x

xelim

3. Evaluate exponential functions by calculator.

4. Solve exponential functions using the logarithm.

5. Differentiate :

6. Differentiate exponential functions using the sum/difference,coefficient, product, quotient, or chain rule.

7. Find relative extrema, absolute extrema.

8. Use in marginal analysis or related rates, and interpret.

xedx

dyxey

3.2 Logarithmic Function

Know your facts for

1. Know the graph: A vertical asymptote below the x axis at x = 0. Through the point (1,0).

Domain: (0, ∞) Range: (-∞, ∞) Increasing on the interval (0, ∞) .

2. Use the graph to find limits:

xxf ln)(

x

xlnlim

0

xx

lnlim

3. Evaluate logarithmic functions by calculator.

4. Solve logarithmic functions using the exponential.

5. Properties of logarithms:

)ln()ln( MNM N

0)1ln(

)ln()ln()ln( NMNM

aea )ln( ae a ln

)ln()ln(ln NMN

M

6. Change of Base formula:

7. Differentiate :

8. Differentiate logarithmic functions using the sum/difference,coefficient, product, quotient, or chain rule.

9. Find relative extrema, absolute extrema.

10. Use in marginal analysis or related rates, and interpret.

xy lnxdx

dy 1

a

bba ln

lnlog

Logarithmic Differentiation

A new way to differentiate functions that are products andquotients involves the properties of logarithms.

If y = f (x) is a function which uses the product, quotient, or chainrules in combination, we can consider a new problem:

Take the natural log of both sidesln(y) = ln(f (x))

Rewrite ln(f (x)) using properties of logsDifferentiate both sides with respect to xSolve for dy/dx.

Note: when we take the natural log of both sides, the derivativebecomes implicit.

Uninhibited growth is a function that grows so that the rate ofchange of output with respect to input is proportional to theamount of output.

The formula for this is (for y output and x input).

This can only be true if the function is , k > 0.

In this exponential function, k represents the growth rate of y, andc represents the amount of y when x = 0.

kxcey

kydx

dy

3.3 & 3.4 Growth and Decay Models

Uninhibited Growth

Uninhibited decay is a function that declines so that the rate ofchange of output with respect to input is proportional to theamount of output.

The formula for this is (for y output and x input).

This is true if the function is , k < 0.

In this exponential function, k represents the decay rate of y, andc represents the amount of y when x = 0.

exponential exponential growth k > 0 decay k < 0

kxcey

kydx

dy

Uninhibited Decay

Logistic growth is an example of a limited growth model.

This function is a growth function if k > 0, and it isa decay function if k < 0.

k > 0 k < 0

ktbe

Lxf

1)(

Limited Growth/Decay

When analyzing information, we may be given data points instead of a function. We will make use of the regression capability of our calculator to find a function that approximates a set of data.

Print the Regression Equation handout on blackboard to finda list of steps to create this function.

Important Note: The exponential function used by thecalculator is not y = cekx . Instead, it uses y = abx.

Modeling Growth and Decay