On [a,b], ARC = On [a,b], ARC =

On [1, 16], find ARC forOn [1, 16], find ARC for

( ) ( )f b f a

b a

( )f x x

On [a,b], ARC = On [a,b], ARC =

On [1, 16], find ARC forOn [1, 16], find ARC for

ARC = = ARC = =

( ) ( )f b f a

b a

( )f x x

16 1

On [a,b], ARC = On [a,b], ARC =

On [1, 16], find ARC forOn [1, 16], find ARC for

ARC = = ARC = =

( ) ( )f b f a

b a

( )f x x

16 1

16 1

On [a,b], ARC = On [a,b], ARC =

On [1, 16], find ARC forOn [1, 16], find ARC for

ARC = = ARC = =

( ) ( )f b f a

b a

( )f x x

4 10.2

16 1

16 1

16 1

ContinuityContinuityThis function is continuous on [0,3]This function is continuous on [0,3]DefinitionDefinition

A function f is continuous at x = a A function f is continuous at x = a means three things.means three things.

1) a is in the domain of f,1) a is in the domain of f,2)2)3) 3) lim ( )exists ( )x a

f x as one real numberlim ( ) ( )x a

f x f a

ContinuityContinuityf is continuous on [0, 3] means that f is continuous on [0, 3] means that

f isf is

continuous at a for every a in [0, 3].continuous at a for every a in [0, 3].

ContinuityContinuityf is continuous on [-1, 3) U (3, 4] f is continuous on [-1, 3) U (3, 4]

means that f is continuous at x means that f is continuous at x = a = a

for every a in [-1, 3) U (3, 4] .for every a in [-1, 3) U (3, 4] .

ContinuityContinuity1) a is in the domain of f,1) a is in the domain of f,

2)2)

3) 3)

lim ( ) exists ( )x a

f x as one real numberlim ( ) ( )x a

f x f a

ContinuityContinuityWhy wasn’t f continuous when x = Why wasn’t f continuous when x =

3?3?

Must lift your pencilMust lift your pencil

Not a valid reasonNot a valid reason

ContinuityContinuity1) a is in the domain of f,1) a is in the domain of f,

2)2)

3)3)

Why not at x = 3? Why not at x = 3?

lim ( ) exists ( )x a

f x as one real numberlim ( ) ( )x a

f x f a

ContinuityContinuity1) a is in the domain of f,1) a is in the domain of f,

2)2)

3)3)

Yes on [-.2, 2) U (2, 3.1]Yes on [-.2, 2) U (2, 3.1]

Why not at x = 2? Why not at x = 2?

lim ( ) exists ( )x a

f x as one real numberlim ( ) ( )x a

f x f a

ContinuityContinuity1) a is in the domain of f,1) a is in the domain of f,

2)2)

3)3)

Yes on [-.2, 2) U (2, 3.1]Yes on [-.2, 2) U (2, 3.1]Why not at x = 2?Why not at x = 2?

Hole in the graphHole in the graphNot a valid reason Not a valid reason

lim ( ) exists ( )x a

f x as one real number

lim ( ) ( )x a

f x f a

Why is f not Why is f not continuous at x = 1?continuous at x = 1?

A.A. 1 is not in the domain of f1 is not in the domain of f

B.B. ..

C.C. ..

D.D. There is a hole in the graphThere is a hole in the graph

E.E. Must lift your pencilMust lift your pencil

1lim ( ) . .xf x d n e

1lim ( ) (1) 1xf x f

Why is f not Why is f not continuous at x = 1?continuous at x = 1?

A.A. 1 is not in the domain of f1 is not in the domain of f

B.B. ..

C.C. ..

D.D. There is a hole in the graphThere is a hole in the graph

E.E. Must lift your pencilMust lift your pencil

1lim ( ) . .xf x d n e

1lim ( ) (1) 1xf x f

f is continuous at x = f is continuous at x = 1.1.

A.A. TrueTrue

B.B. FalseFalse

f is continuous at x = f is continuous at x = 1.1.

A.A. TrueTrue

B.B. FalseFalse

f is continuous at x = 4.f is continuous at x = 4.

A.A. TrueTrue

B.B. FalseFalse

f is continuous at x = 4.f is continuous at x = 4.

A.A. TrueTrue

B.B. FalseFalse

Why is s continuous at Why is s continuous at 20?20?

A.A. 20 is in the domain20 is in the domain

B.B. ..

C.C. ..

D.D. a, b, and c abovea, b, and c above

E.E. none of the abovenone of the above

20lim ( )existsx

s x

20lim ( ) (20)x

s x s

Why is s continuous at Why is s continuous at 20?20?

A.A. 20 is in the domain20 is in the domain

B.B. ..

C.C. ..

D.D. a, b, and c abovea, b, and c above

E.E. none of the abovenone of the above

20lim ( )existsx

s x

20lim ( ) (20)x

s x s

f is continuous on [-2, 3].f is continuous on [-2, 3].

A.A. YesYes

B.B. NoNo

f is continuous on [-2, 3].f is continuous on [-2, 3].

A.A. YesYes

B.B. NoNo

Why is f not continuous at Why is f not continuous at x=1?x=1?

A.A. 1 is not in the domain1 is not in the domain

B.B. ..

C.C. ..

D.D. All of the aboveAll of the above

1lim ( ) . . .

xf x d n e

1lim ( ) (1) 2xf x f

Why is f not continuous at Why is f not continuous at x=1?x=1?

A.A. 1 is not in the domain1 is not in the domain

B.B. ..

C.C. ..

D.D. All of the aboveAll of the above

1lim ( ) . . .

xf x d n e

1lim ( ) (1) 2xf x f

Why is f not continuous at Why is f not continuous at x=2?x=2?

A.A. 2 is not in the domain2 is not in the domain

B.B. ..

C.C. ..

D.D. All of the aboveAll of the above

2lim ( ) . .x

f x d n e

24 lim ( ) (2)

xf x f

Why is f not continuous at Why is f not continuous at x=2?x=2?

A.A. 2 is not in the domain2 is not in the domain

B.B. ..

C.C. ..

D.D. All of the aboveAll of the above

2lim ( ) . .x

f x d n e

24 lim ( ) (2)

xf x f

Having thrown a pebble from the roof of Having thrown a pebble from the roof of MPP, will it pass the floor of the 3rd MPP, will it pass the floor of the 3rd floor?floor?

The domain of this function is [a, b] The domain of this function is [a, b] representing the time it was thrown until representing the time it was thrown until the time the pebble struck the ground. the time the pebble struck the ground. The rule of the function, f, is "height of The rule of the function, f, is "height of the pebble at time x".the pebble at time x".

Intermediate Value Intermediate Value TheoremTheorem

If f is continuous on [a, b] and If f is continuous on [a, b] and dd is is inbetween inbetween f(f(aa) ) and and f(f(bb)),,

then c (a,b) such that then c (a,b) such that f (f (cc) ) = = dd..

f(x) = xf(x) = x22 + x – 1.7 + x – 1.7

Prove f has a zero on [-1, 1]Prove f has a zero on [-1, 1] f(-1) = -1.7, f(1) = 0.3f(-1) = -1.7, f(1) = 0.3 f is continuous on [-1, 1]f is continuous on [-1, 1] 0 (-1.7, 0.3) so by the IVT0 (-1.7, 0.3) so by the IVT a c (-1, 1) so that f(c)= 0.a c (-1, 1) so that f(c)= 0.

Find c to make f Find c to make f continuouscontinuous c/x must equalc/x must equal 4x + c4x + c When x = 2When x = 2 c/2 = 8 + cc/2 = 8 + c -8 = c/2-8 = c/2 -16 = c-16 = c f(2) = -8f(2) = -8

ContinuityContinuityThis function is continuous on [0,3]This function is continuous on [0,3]

d.n.e.d.n.e.

Similarly, Similarly, even thougheven though d.n.e.d.n.e.

0lim ( ) (0)x

f x f

0lim ( )x

f x

0

0lim ( ) (0) 0xf x f

3lim ( ) (3) 210xf x f

3lim ( )x

f x