NotebookTable of content

Page1

Learning Target 1

1) 1-1 A Preview of Calculus

2) 1-2 Finding limits graphically and numerically

3) 1-3 Evaluating limits analytically

4) 1-4 Continuity

Section HW Assignment Completed? Quiz Score

1.1 p.47: 4-6, 9

1.2 p. 551-27 odd, 57-60 all, 66-70 all

1.3 p. 67; 37,39,47-61,115,116,118,119

1.4 p. 68; 80-82, 89,90,p.79; 3-14,17-20, 35-47 odd,51, 61-66

1-4 Continuity

Continuous : Goes on forever with no breaks , no holes, jumps, asymptote

Discontinuity :1. Hole

2. Jump: Step Discontinuity (Piecewise function)

3. Asymptote (no bounds)

Removable or non-removable discontinuity

#

0

Definition of continuity at any point:

1. f(c) exist (y- coordinate exist)

Eliminates: functions w/ holes

2. lim ( ) existx cf x

The y-coordinate we are approaching

Eliminates: jumps & asymptote

3. lim ( ) = ( )x cf x f c

AP standard : Make sure we know the definition of continuous

Prove f(x) is continuous for all values?

2

1 0( )

1 0

x xf x

x x

1. f(0) = 0+1 = 1

2. One-side limits

0limx 0

limx

1 1

3. 0

(0) lim ( ) , therefore f(x) is continuous x

f f x

Doesn't work: therefore the function is not continuous

All polynomials are continuous

(0,1)

𝑥+12 1x

0lim ( ) 1xf x

You try: 2 1

( ) 3 1

1 1

x x

f x x

x x

1. f(c) exist

2. lim ( ) existx cf x

3. lim ( ) = ( )x cf x f c

( 1) 3f

1lim ( ) 2x

f x

1

lim ( ) 2x

f x

1( 1) lim ( ) , therefore f(x) is not continuous

xf f x

1lim ( ) 2xf x

AP TEST (Free response)

2

23 4( )

2

xxf x

xkx

Find the k value such that f(x) is continuous for all value

Guess

AP TEST (Free response)

2

23 4( )

2

xxf x

xkx

Find the k value such that f(x) is continuous for all value

1.

2. 2

lim x

(2) =5f

2lim x

3 4x 2kx5

22k 5

4k

What does this say ?

5

Guess

More examples find the value of the constant (a, b, or c) that makesthe function continuous.