Aim: Differentiating & Integrating Expo Functions Course: Calculus Do Now: Aim: How do we...

Aim: Differentiating & Integrating Expo Functions

Course: Calculus

Do Now:

Aim: How do we differentiate and integrate the exponential function?

1'( ) , '( ( )) 0

'( ( ))g x f g x

f g x

1 11

1'( ) , '( ( )) 0

'( ( ))f x f f x

f f x

1 3Find ' 2 for ( ) 2 1f f x x x


Course: Calculus

Do Now

1 3Find 2 for ( ) 2 1f f x x x


Course: Calculus

The Natural Exponential Function

4

2

-2

-4

-5 5

f x = ln x

Characteristics of Natural Log Function

Monotonic - increasingDomain – (0, )Range – all reals

Has an inverse f -1

f -1(x) = ex

Natural Exponential

Function


Course: Calculus

4

2

-2

-4

-5 5

f x = ln x

Definition of Natural Exponential Function

Natural Log Function

f -1(x) = ex

Natural Exponential

Function 1 1lnf f x f x x

The inverse of the natural logarithmic function f(x) = ln x is called the natural

exponential function and is denoted by f -1(x) = ex. That is, y = ex x = ln y

ln(ex) = x ln e = x(1) = x

if x is rational

ln ex = x


Course: Calculus

4

2

-2

-4

-5 5

f x = ln x

Properties of Natural Exponential Function

Natural Log Function

f -1(x) = ex

Natural Exponential

Function

1. domain – (-, ); range – (0, )

2. continuous, increasing, and 1-to-1

3. concave up on its entire domain

4. lim 0 and limx x

x xe e


Course: Calculus

Problems

Solve 4e2x = 5 to 3 decimal places

x 1

2ln

5

4

x 1

2.2231435513 0.112

ln e2x = ln 5/4 Property of Equality for Ln functions

2x = ln 5/4Inverse Property of Logs & Expos

e2x = 5/4 Divide both sides by 4

Check: 4e2(0.112) = 5


Course: Calculus

Solving Exponential Equations

1Solve 7 xe

1ln7 ln xe take ln of both sides

ln7 1x

ln7 1 lnx e

ln7 1 x

0.946x

solve for x

apply inverse propertyln xe x

ln e = 1


Course: Calculus

Solving Log Equations

Solve ln 2 3 5x

ln 2 3 5xe e expo both sides

52 3x e

513

2x e

75.707x solve for x

apply inverse propertyln xe x


Course: Calculus

Complicated Problem

Solve e2x – 3ex + 2 = 0

Quadratic Form(ex)2 – 3ex + 2 = 0

Factor(ex – 2)(ex – 1) = 0

Set factors equal to zero(ex – 2) = 0 (ex – 1) = 0

ex = 2 ex = 1

x = ln 2 x = 0

x = 0.693 x = 0

Graph to verify


Course: Calculus

Derivatives of Exponential Functions

Let be a differentiable function of

1. 2.x x u u

u x

d d due e e e

dx dx dx

2 1. xda e

dx

u due

dx

2 1 2 12 2x xe e

u = 2x - 1u’ = 2

3/. xdb e

dx

u due

dx

3/3/

2 2

3 3 xx e

ex x

u = -3/xu’ = 3/x2


Course: Calculus

Model Problem

Find the relative extrema of f(x) = xex

'( ) 1x xf x x e e

1 1 0x x xx e e e x

4

2

-5

g x = ex

ex is never 0 x + 1 = 0

x = -1

4

2

-2

-5

h x = xex

A


Course: Calculus

Model Problem

2 /2

Show that the normal probability density function

1( ) has points of inflection when 1.

2xf x e x

When 2nd derivative equals zero.

2 /21'( )

2xf x x e

u ud due e

dx dx

u = -x2/2; u’ = -x

2 2/2 /21''( ) 1

2x xf x x x e e

2 22 /2 /211 0

2x xx e e

2 /2 211 0

2xe x

x = ±1

0.5

0.4

0.3

0.2

0.1

-0.1

-2 2


Course: Calculus

Model Problem

For 1980 through 1993, the number y of medical doctors in the U.S. can be modeled by y = 476,260e0.026663t where t = 0 represents 1980. At what rate was the number of M.D.’s changing in 1988?

When t in 1st derivative equals 8.

0.026663'( ) 0.026663 476,260 tf x e

0.026663 80.026663 476,260 e

15,718 doctors per year


Course: Calculus

Integrals of Exponential FunctionsLet be a differentiable function of .

1. 2. x x u u

u x

e dx e C e du e C 3 1Evaluate xe dx

3 1 3 113

3x xe dx e dx

u = 3x + 1du/dx = 3;du = 3dx

multiple and divide by 3

1

3ue du

1

3ue C u ue du e C 3 11

3xe C Back-substitute


Course: Calculus

Model Problem2

Evaluate 5 xxe dxu = -x2

du/dx = -2x du = -2xdx xdx = du/2

u ue du e C

2 2

5 5x xxe dx e xdx regroup integrand

52

u due

substitute

5

2ue du factor out -5/2

u ue du e C 5

2

ue C

25

2xe C Back-substitute


Course: Calculus

Model Problem1

2

xedx

x u = 1/x 2 2

1 1dudu dx

dx x x

1

2

xedx

x

12

1u

u

x

de

e dxx

1/ xe C

cossin xx e dx u = cosx sin sindu

x du xdxdx

cos sin

u

x

due

e xdx

cos xe C

cossin xx e dx


Course: Calculus

Model Problem

Find the areas1

0

xe dx 1

01 xe

1 1e

11 0.632

e

1.2

1

0.8

0.6

0.4

0.2

-0.2

0.5 1

t x = e-x

1

0 1

x

x

edx

e 1

0ln 1 xe

ln 1 ln2e

0.620

1.2

1

0.8

0.6

0.4

0.2

-0.2

0.5 1

u x = ex

1+ex


Course: Calculus

Do Now:

Aim: How do we differentiate and integrate the exponential function?

x x

x x

e edx

e e

2

Find the indefinite integral

ln xdx

x

3 2

Find using log differentiation

1

1

dy

dx

x xy

x

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