AP Calculus Review First Semester Differentiation to the Edges of Integration Sections 1.2-3.7, 3.9, (7.7)

AP Calculus Review

First Semester

Differentiation to the Edges of Integration

Sections 1.2-3.7, 3.9, (7.7)

Limit Definition (1.2)

lim ( )x cf x L

• The number L is the limit of the function f(x) as x approaches c if, as the values of x get arbitrarily close (but not equal) to c, the values of f(x) approach (or equal) L:

•

When Limits Fail to Exist (1.2)

lim limx c x c

y = int(x)

y = 1/(xx)

Unbounded Behavior

y = sin(1/x)Oscillating Behavior

Evaluating Limits Analytically (1.3)

• Review the properties of limits, p. 57, if you don’t know:

2

4

2

2

lim3

lim

lim

x

x

x

x

x

3

-4

4

Strategies for Evaluating Limits (1.3)

• Ask yourself if the function can be evaluated by direct substitution (see previous slide)

• If it cannot, you can try – Cancelling and then using direct substitution

– Using L’Hopital’s Rule, provided you have 0/0

22

2lim

4x

x

x

2

( 2)lim

( 2)( 2)x

x

x x

2

1lim

2x x

1

4

22

2lim

4x

x

x

2

1lim

2x x

1

4

Strategies for Evaluating Limits, cont’d. (1.3)

• Or you can try– rationalizing and then using direct substitution

– Using L’Hopital’s Rule, provided you have 0/012

0 0 0

1( 1)1 1 1 12lim lim lim

1 22 1x x x

xx

x x

0 0

1 1 1 1 1 1 1lim lim

21 1x x

x x x

x x x

Evaluating Limits Analytically (1.3), cont’d.

– You can also try using one of the two special trig limits:

– For example:

– Note: You could also use l’Hopital’s Rule here.

0 0

sin 1 coslim 1 & lim 0x x

x x

x x

0 0

sin 3 3(1 cos )lim 3 & lim 3(0) 0x x

x x

x x

Questions on Limits

• Which limits exist?

20

1limx x

0limx

x

x

0

1lim

sinx x

2

0

2limx

x x

x

Only this one

Questions on Limits

• Find the limit

1

4

1

3 2lim

1x

x

x

Questions on Limits

2 2

2If 0, then lim

)0 )2 )2 )4 ) nonexistent

x k

x kk

x kx

a b c k d k e

Continuity (1.4)

• A function is continuous over an interval if we can draw its graph without lifting pencil from paper, i.e., the graph has no holes, breaks or jumps over the interval.

• More formally, the function y = f(x) is continuous at x = c if

lim ( ) ( )x cf x f c

Continuity, cont’d. (1.4)

• If

then all three of the following are true:

lim ( ) existsx cf x

lim ( ) ( )x cf x f c

lim ( ) = ( )x cf x f c

( ) exists, that is, is definedf c

Destroying Continuity (1)( ) does not exist, that is, is not definedf c

Continuity, cont’d. (1.4)

• If

then all three of the following are true:

lim ( ) existsx cf x

lim ( ) ( )x cf x f c

lim ( ) = ( )x cf x f c

( ) exists, that is, is definedf c

Destroying Continuity (2)lim ( ) does not existx cf x

y = 5-xx; -100.000000 <= x <= 2.000000y = x-2; 2.000000 <= x <= 100.000000

Continuity, cont’d. (1.4)

• If

then all three of the following are true:

lim ( ) existsx cf x

lim ( ) ( )x cf x f c

lim ( ) = ( )x cf x f c

( ) exists, that is, is definedf c

Destroying Continuity (3)lim ( ) ( )x cf x f c

y = 5-xx; -100.000000 <= x <= 2.000000y = .5x; 2.000000 <= x <= 100.000000

Continuity, cont’d. (1.4)

• If

then all three of the following are true:

lim ( ) existsx cf x

lim ( ) ( )x cf x f c

lim ( ) = ( )x cf x f c

( ) exists, that is, is definedf c

Types of Discontinuity

• Removeable: mere point discontinuity

• Non-removeable: infinite discontinuity (vertical asymptotes)

y = (x-3)/(xx-9)

2

3 3 1

9 ( 3)( 3) 3

x x

x x x x

2 2 1for 1

Let be defined by ( ) 1for 1

Determine the value of for which is continuous for all real .

x xx

f f x xk x

k f x

Questions on Continuity

0

Questions on Continuity

• The function f is continuous at the point (c, f(c)). Which of the following statements could be false?

) lim ( ) exists

) lim ( ) = ( )

) lim ( ) = lim ( )

) ( ) is defined

) ( ) exists

x c

x c

x c x c

a f x

b f x f c

c f x f x

d f c

e f c

This one

Intermediate Value Theorem

• If f is continuous on [a, b] and k is any number between f(a) and f(b), then there is a least one number c in [a, b] such that

f(c) = k

y = 5-xx; -100.000000 <= x <= 2.000000

Infinite Limits (1.5)• A limit in which f(x) increases or decreases

without bound as x approaches c is called an infinite limit.

• If f(x) approaches +/- infinity as x approaches c from the right or left, then the line x=c is a vertical asymptote of the graph of f.

y = (x-3)/(xx-9)

23

23

3lim

93

lim9

x

x

x

xx

x