Example Ex. Find

Sol.

So

lim( ln ).x

x x

ln lnxe

x xx

lim lim1

x x

x x

e e

x

lim( ln ) .x

x x

Example Ex. Find (1) (2) (3)

Sol. (1)

(2)

(3)

2 20 0

1 ln cos sin 1lim lim2 cos 2

0lim(cos ) .

x x

x x

x x x x

xx e e e

2

1

0lim(cos )

x

xx

1sin

lim

x

xx sin

0lim .

x

xx

1 1 ln 1sin lim sin ln lim lim

lim 1.

x x x

xx

x x x x

xx e e e

0 0lim sin ln lim lnsin 0

0lim 1.

x x

x x x xx

xx e e e

Question Find

Sol. Use L’Hospital’s rule.

30

sin (1 )lim .

x

x

e x x x

x

3 20 0

0 0 0

sin (1 ) (sin cos ) 2 1lim lim

3

2 cos 2 1 cos 1 1 (cos sin ) 1lim lim lim .

6 3 3 1 3

x x

x x

x x x

x x x

e x x x e x x x

x x

e x e x e x x

x x

Question Find

Sol.

3lim ( 2 2 1 ).x

x x x x

1 12 2 1

2 1 1x x x

x x x x

2

( 2 1)( 1 )

x x

x x x x

2

( 2 1)( 1 )( 2)x x x x x x

1

4

Optimization problems Optimization: minimize costs and/or maximize profits

Steps in solving optimization problems: first understand the problem and formulate the cost function, then find the global minimum/maximum using the closed interval method.

Example Ex. Find the area of the largest rectangle that can be inscribed

in a semicircle of radius r.

Sol. Set up the coordinate system. The semicircle has the

equation Let (x,y) be the vertex lying in the first

quadrant. Then the rectangle has length 2x and width y, so its

area is A=2xy. Since we can eliminate y:

thus

The domain is [0,r]. Use the closed interval method, A(x) has

maximum value

2 2 2.x y r

2 2 2 ,x y r 2 2 ,y r x 2 22 .A x r x

2( / 2) .A r r

Newton’s method Find a root of f(x)=0 Idea: successively replace f(x) by its linear approximation.

Given an initial guess of a root, say, approximate f(x) by the linear approximation of f(x) at Use the root of as the approximate root of f(x)=0, and denote it as That is, so

Repeat this process, we obtain a sequence with recurrence relationship:

0 ,x0.x

0 ( )L x0 ( ) 0L x

1.x

1

( ), 0.

( )n

n nn

f xx x n

f x

0 1( ) 0,L x 01 0

0

( ).

( )

f x

x xf x

Newton’s method Under appropriate conditions, the sequence generated from

Newton’s method is convergent to the root of f(x)=0. Ex. Without using the operation of taking roots, find

correct to four decimal places. Sol. is a root of Since Newton’s method gives

Choosing an arbitrary say we have

Since and agree to 4 decimal places, we conclude that correct to four decimal places.

2

1

2 1, 0.

2 2

n n

n nn n

x xx x n

x x

22( ) 2 0. f x x2 ( ) 2 , f x x

0 ,x0 1,x

1 2 3 41.5, 1.4167, 1.4142, 1.4142. x x x x

3x 4x2 1.4142

Antiderivatives Definition A function F is called an antiderivative of f on

an interval I if for all x in I. For example, is an antiderivative of Question: given f(x), is the antiderivative of f unique?

Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is

F(x)+C

where C is an arbitrary constant.

( ) ( ) F x f x2( ) F x x ( ) 2 .f x x

Example Ex. Find the most general antiderivative of :

(1) (2)

(3) Sol. (1)

(2)

(3)

( ) 1/f x x ( ) ( 1) f x x ( ) ( 0, 1) xf x a a a

( ) ln | | F x x C1

( )1

xF x C

( )ln

xa

F x Ca

Example Ex. Find all functions g such that

Sol. By the sum rule of derivative, we can find the

antiderivative for each term and add together.

2

4 1( ) 3sin .

1

g x x x

x x

3

22

( ) 3cos 8 arcsin .3

g x x x x x C

Example Ex. Find all functions g such that

Sol. Write the function into the sum of the functions, for which we can find antiderivative.

cos 2( ) .

sin cos

x

g xx x

2 2cos 2 cos sinsin cos .

sin cos sin cos

x x x

x xx x x x

( ) cos sin . g x x x C

Example Ex. Find g if g(1)=0 and

Sol.

2 2

1( ) .

( 1)

g x

x x

2 2 2 2

1 1 1.

( 1) 1

x x x x

1( ) arctan . g x x C

x

(1) 1 0 1.4 4

g C C

1( ) arctan 1.

4 g x x

x

Direction fields The geometry of antiderivatives can be described by a dire

ction field: given f, to draw the graph of F, at an arbitrary point x, the tangent line has slope f(x)

Ex. If sketch the graph of the

antiderivative F that satisfies the initial condition F(-1)=0.

3( ) 1 ,f x x x

Homework 11 Section 4.7: 17, 22

Section 4.10: 27, 29, 32

Review exercises (P362): 13, 14, 51

Question for midterm reviewSuppose that f(x) is defined for all and that

for any real number x,y, where

Suppose also that

Find

Sol.

( ) ( ) ( ) ( ) ( )f x y f x g y f y g x

(0) 0f

( , )x

sin( ) cos .xg x e x x

0

( )lim 1.x

f x

x

(0), (0), ( ).f f f x

(0) 1f ( ) ( )f x g x