Warmup – No calculator 1) is? 2) Sketch a function f(x) that has all of the following properties: could you write a function that would have this?

Warmup – No calculator

1) is?

2)

0)(

0)(

)(

)(

5)0(

lim

lim

lim

lim

4

4

xf

xf

xf

xf

f

x

x

x

x

Sketch a function f(x) that has all of the following properties:

could you write a functionthat would have this?

2.3: Continuity

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.

A function is continuous at a point if the limit is the same as the value of the function.

This function has discontinuities at x=1 and x=2.

It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

1 2 3 4

1

2

Continuity:You can draw a function without lifting your pencil off the paper

Ex.

exists )(lim xfax

f(a) )(lim

xfax

ALL 3 of these conditions need to be true for thefunction to be continuous at x = a .

Definition of continuity at a point:A function f(x) is said to be continuous at a point x = a if

1) f(a) exists

2)

3)

Free response type of question:

1for x x

1x2-for bax

-2for x 3

)(2

x

xf

Find a and b such that the function is continuous. Justify your answer using the definition of continuity

Find the values of the constants a and b such that the functionis continuous on the entire real line

ax

xxj

x

xbax

x

xg

xx

x

xxaxf

xax

xxxh

,8

a x,a-x

a-)()4

3 ,2

31- ,

1 ,2

)()3

0 ,sin4

0 ,2)()2

2 ,

2 ,1)()1

22

2

3

Removing a discontinuity:

3

2

1

1

xf x

x

has a discontinuity at .1x

Write an extended function that is continuous at .1x

3

21

1lim

1x

x

x

2

1

1 1lim 1 1x

x x xx x

1 1 1

2

3

2

3

2

1, 1

13

, 12

xx

xf x

x

Note: There is another discontinuity at that can not be removed.

1x

and -1

Removing a discontinuity:

Note: There is another discontinuity at that can not be removed. (VA)

1x

1 x,1

1

)(

2

3

x

x

xf1 x ,

2

3

Find the x-values (if any) at which f is not continuous.

2 x,14

2 ,2)()3

)(0,2 ,2csc)()2

cos3)( )1

2 xx

xxxf

xxf

xxxf

continuous on entire domain

2

3,,

2

Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

Also: Composites of continuous functions are continuous.

Intermediate Value Theorem

If a function is continuous between a and b, then it takes

on every value between and . f a f b

a b

f a

f b

Because the function is continuous, it must take on every y value between and .

f a f b

My son was born on February 10th, 2011. He weighed 6 lbs, 15 ounces and was 20.5 inches in length

At his 14 months he weighed 23 lbs, 12 ounces and was 33 inches in length

Using the intermediate value theorem,what kind of statements can I make that are known to be true?

Is any real number exactly one less than its cube?

(Note that this doesn’t ask what the number is, only if it exists.)

3 1x x

30 1x x

3 1f x x x

1 1f 2 5f

Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5.Therefore there must be at least one solution between 1 and 2.

Explain why the functions

have a zero in the domain indicated

No Calculator

][0, ,sec31 )h( 3)

][0, ,cos2 x g(x) 2)

[0,1] ,23)()1

2

3

x

xxxf

Recap:

Definition of continuity at a point:A function f(x) is said to be continuous at a point x = a if

1) f(a) exists

2)

3)

exists )(lim xfax

f(a) )(lim

xfax

ALL 3 of these conditions need to be true for thefunction to be continuous at x = a .

Free response type of question:

1for x ln x

1x2-for bax

-2for x 1x2

)( 2xf

Find a and b such that the function is continuous. Justify your answer using the definition of continuity

0 x, cos(kx)

3x0 , )ln()(

kxxf

What value of k makes the function continuous at x=0 ?

the end

Day 1:

p. 80 ( 1-26, 35, 36, 39, 41)

Day 2:

p. 80 ( 27-30, 31-34 state the domain only w/out calc 37,38,40)