1 §11.1 The Constant e and Continuous Compound Interest. ■ The student will be able to work with problems involving the irrational number e ■ The student.

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§11.1 The Constant e and Continuous

Compound Interest.

■ The student will be able to work with problems involving the irrational number e

■ The student will be able to solve problems involving continuous compound interest.

2

The Constant e.

Definition: e = 2.718 281 828 459 …

Do you remember how to find this on a calculator?

Reminder

e is also defined as one of the following limits:

nlime

0slime

n

n

11

s

1s1

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Compound Interest

Let P = principal, r = annual interest rate, t = time in years, n = number of compoundings per year, and A = amount realized at the end of the time period.

• Simple Interest A = P ( 1 + r )t

• Compound interest

• Continuous compounding A = P e rt.

nt

n

r1PA

4

Example: Generous Grandma

Your Grandma puts $1,000 in a bank at 5% for you. Calculate the amount after 5 and 20 years.

Simple interest

Compound interest (daily)

Continuous compounding

$1,276.28

$2,653.30

A = 1000 ( 1 + .05) 5 =

A = 1000 ( 1 + .05) 20 =

)5)(360(

360

05.11000A

)20)(360(

360

05.11000A

A = 1000 e (.05)(5) = A = 1000 e (.05)(20) =

$1,284.00

$2,718.09

$1,284.03$2718.28

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Example IRAAfter graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year. He plans to retire in 35 years.

a. What will be its value at the end of the time period?

A = P e rt = 3000 e (.12)(35) = $200,058.99

$177,436.41b. The second year he repeats the purchase of a Roth IRA. What will be its value in 34 years?

Show how to become a millionaire!!

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Example - Doubling Your Money

After graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year.

How long will it take for that investment to double?

A = P e rt OR 6000 = 3000 e 0.12t AND solve for t.

6000/3000 = e 0.12t or 2 = e 0.12t

But ln 2 = 0.12 t so t = ln 2/ .12 = 5.776 years

Remind the students of the “Rule of 72”.

Take the ln of both sides yielding -

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Summary.

• The constant e occurs in natural situations.

• There are three different interest formulas.

• These applications can be of interest.

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§11.2 Exponential Functions and Their

Derivatives

The student will learn about:

the composite functions,

the derivative of the exponential function, graphing strategies for these functions,

and applications.

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We now apply the four-step process from a previous section to the exponential function.

Step 1: Find f (x+h)

Step 2: Find f (x+h) – f (x+h)

The Derivative of ex

hxhx eeehxf )(

We will use (without proof) the fact that0

1lim 1

h

h

e

h

1)()( hxxhx eeeeexfhxf

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The Derivative of ex

(continued)

h

xfhxf )()(

h

xfhxfh

)()(lim

0

Step 3: Find

Step 4: Find

h

ee

h

xfhxf hx 1)()(

xh

h

x

he

h

ee

h

xfhxf

1lim

)()(lim

00

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The Derivative of ex (continued)

Result: The derivative of f (x) = ex is f ’(x) = ex.

This result can be combined with the power rule, product rule, quotient rule, and chain rule to find more complicated derivatives.

Caution: The derivative of ex is not x ex-1

The power rule cannot be used to differentiate the exponential function. The power rule applies to exponential forms xn, where the exponent is a constant and the base is a variable. In the exponential form ex, the base is a constant and the exponent is a variable.

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Examples

Find derivatives for

f (x) = ex/2

f (x) = ex/2

f (x) = 2ex +x2

f (x) = -7xe – 2ex + e2

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Examples (continued)

Find derivatives for

f (x) = ex/2 f ’(x) = ex/2

f (x) = ex/2 f ’(x) = (1/2) ex/2

f (x) = 2ex +x2 f ’(x) = 2ex + 2x

f (x) = -7xe – 2ex + e2 f ’(x) = -7exe-1 – 2ex

Remember that e is a real number, so the power rule is used to find the derivative of xe. The derivative of the exponential function ex, on the other hand, is ex. Note also that e2 7.389 is a constant, so its derivative is 0.

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The Natural Logarithm Function ln x

We summarize important facts about logarithmic functions from a previous section:

Recall that the inverse of an exponential function is called a logarithmic function. For b > 0 and b 1

Logarithmic form is equivalent to Exponential form

y = logb x x = by

Domain (0, ) Domain (- , )

Range (- , ) Range (0, )

The base we will be using is e. ln x = loge x

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We are now ready to use the definition of derivative and the four step process to find a formula for the derivative of ln x. Later we will extend this formula to include logb x for any base b. Let f (x) = ln x, x > 0.

Step 1: Find f (x+h)

Step 2: Find f (x+h) – f (x)

The Derivative of ln x

)ln()( hxhxf

h

hxxhxxfhxf

ln)ln()ln()()(

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Step 3: Find

Step 4: Find . Let s = x/h.

The Derivative of ln x (continued)

h

xfhxf )()(

h

xfhxfh

)()(lim

0

hx

x

h

xx

h

h

x

xx

hx

hh

xfhxf/

1ln1

1ln1

ln1)()(

x

ex

sxh

xfhxf s

sh

1ln

11lnlim

1)()(lim /1

00

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Examples

Find derivatives for

f (x) = 5 ln x

f (x) = x2 + 3 ln x

f (x) = 10 – ln x

f (x) = x4 – ln x4

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Examples (continued)

Find derivatives for

f (x) = 5 ln x f ’(x) = 5/x

f (x) = x2 + 3 ln x f ’(x) = 2x + 3/x

f (x) = 10 – ln x f ’(x) = – 1/x

f (x) = x4 – ln x4 f ’(x) = 4 x3 – 4/x

Before taking the last derivative, we rewrite f (x) using a property of logarithms:

ln x4 = 4 ln x

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Other Logarithmic and Exponential Functions

Logarithmic and exponential functions with bases other than e may also be differentiated.

xbx

dx

db

1

ln

1log

bbbdx

d xx ln

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Find derivatives for

f (x) = log5 x

f (x) = 2x – 3x

f (x) = log5 x4

Examples

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Find derivatives for

f (x) = log5 x f ’(x) =

f (x) = 2x – 3x f ’(x) = 2x ln 2 – 3x ln 3

f (x) = log5 x4 f ’(x) =

For the last example, use

log5 x4 = 4 log5 x

Examples (continued)

x

1

5ln

1

x

1

5ln

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Barnett/Ziegler/Byleen Business Calculus 11e

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Example

?)(' xf

12 2

8)( xxf

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Example (continued)

)4(8ln8

)12(8ln8)('

12

212

2

2

x

xdx

dxf

x

x

12 2

8)( xxf

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Summary

Exponential Rule

xx eedx

d

Log Rule

xx

dx

d 1ln

For b > 0, b 1

ln

1 1log ( )

ln

x x

b

db b b

dx

dx

dx b x