Bell Work for Quarter I … listed in reverse order.

Bell Work for Quarter I

… listed in reverse order

Essential Question(s)September 25, 2013

How do we use vertical asymptotes, horizontal asymptotes, and holes in the graph to sketch the graph of a function?

Vol. I No. 14BSeptember 25, 2013

3 2

5

5 41) ( )

x x xf x

x x

Essential Question(s)September 24, 2013

How do we use vertical asymptotes, horizontal asymptotes, and holes in the graph to sketch the graph of a function?


31) ( )

4f x

x

1

2) ( )2

f xx

2

23) ( )

1f x

x

2

34) ( )

2f x

x

Make a sketch of each function without a calculator


31 ) lim

4x

ax

4

31 ) lim

4x

bx

4

31 ) lim

4x

cx


12 ) lim

2x

ax

2

12 ) lim

2x

bx

2

12 ) lim

2x

cx


2

23 ) lim

1x

ax

21

23 ) lim

1x

bx

21

23 ) lim

1x

cx


2

34 ) lim

2x

ax

22

34 ) lim

2x

bx

22

34 ) lim

2x

cx


3 25 4

5) ( )4

x x xf x

x x

Make a sketch of each function with a calculator

3 2

3

5 46) ( )

16

x x xf x

x x

Identify: a) VA b) HA c) hole(s)


3 25 4

5) ( )4

x x xf x

x x


3 25 4

5) ( )4

x x xf x

x x

VA:

HA: hole(s):


3 2

3

5 46) ( )

16

x x xf x

x x


VA:

HA: hole(s):

3 2

3

5 46) ( )

16

x x xf x

x x

Vol. I No. 14H

Section 1.5 (Infinite Limits)Page 88: 1, 3, 7, 15, 19,

28, 33, 37, 39, 41, 43, 45, 47, 49, 51, 53,

61, 64, 68

15

Vol. I No. 15H

Section 3.5 (Limits at Infinity)Page 205: 9, 13, 15, 17, 19,

21, 23, 25, 27, 29, 31, 33,

35, 37, 39, 43, 57, 62, 63,

64, 7116

Essential Question(s)

How do we find vertical asymptotes?

How do we find horizontal asymptotes?


2

2 1781) lim

3 13x

x

x

2

12 92) lim

8 3x

x

x

3

2

2 33) lim

8 5x

x

x

As x approaches infinityLimits at Infinity

September 23, 20132

2

2 1781) lim

3 13x

x

x

September 23, 2013

2

12 92) lim

8 3x

x

x

September 23, 2013

3

2

2 33) lim

8 5x

x

x

As x approaches c

September 23, 2013

2

31) lim

2x x

2

32) lim

2x x

2

33) lim

2x x

September 23, 2013

2

31) lim

2x x

September 23, 2013

2

32) lim

2x x

September 23, 2013

2

33) lim

2x x

September 23, 20132

22

2 84) lim

4x

x x

x

2

1

35) lim

1x

x x

x

2

1

36) lim

1x

x x

x

Below is the graph of j(x) No. 1. 2B List as many facts about j(x) as you can.

September 19, 20132

2

2 1781) lim

3 13x

x

x

2

12 92) lim

8 3x

x

x

3

2

2 33) lim

8 5x

x

x

As x approaches infinity

September 19, 20132

2

2 1781) lim

3 13x

x

x

September 19, 2013

2

12 92) lim

8 3x

x

x

September 19, 2013

3

2

2 33) lim

8 5x

x

x

As x approaches c

September 19, 2013

2

31) lim

2x x

2

32) lim

2x x

2

33) lim

2x x

September 19, 2013

2

31) lim

2x x

September 19, 2013

2

32) lim

2x x

September 19, 2013

2

33) lim

2x x

September 19, 20132

22

2 84) lim

4x

x x

x

2

1

35) lim

1x

x x

x

2

1

36) lim

1x

x x

x

September 19, 20132

22

2 84) lim

4x

x x

x

September 19, 2013

2

1

35) lim

1x

x x

x

September 19, 20132

1

36) lim

1x

x x

x

Need to Know for Test• Find limit as x approaches a value• Find left limit• Find right limit• Find points of discontinuity• Find Vertical Asymptotes• Find Horizontal Asymptotes• Find when a function is continuous• Function Analysis

• Sketch the graph of a function• Discuss a function without a graph• Discuss a function with a graph• Squeeze Theorem• Special Limits• Identify types of discontinuities–From graph–From equation

• Do calculations from graph

• Difference betweenDNE and DNE and

Need to Know for Test

Work

Vol. I No. 12HPage 88: 37 – 47 (odd)


2 2

01) lim

x

x x x

x

0

sin( ) sin2) lim

x

x x x

x


0

sin( ) sin2) lim

x

x x x

x


2 2

01) lim

h

x h x

h

0

sin( ) sin2) lim

h

x h x

h

The Squeeze Theorem

This theorem concerns the limit of a function that is squeezed between two other functions, each of which has the same limit at a given x-value, as shown in Figure 1.21

The Squeeze Theorem

The Squeeze Theorem

Figure 1.21

Squeeze Theorem is also called the Sandwich Theorem or the Pinching Theorem.

The Squeeze Theorem

The Squeeze Theorem

Find

Vol. I No. 11H

Page 67: 27-35 (odd); 49 – 63 (odd); 65 – 75 (odd)

2 3 , 0( )

1, 0

xx xf x

x x

At what point(s) is NOT continuous?( )f x


2 3 , 0( )

1, 0

xx xf x

x x

Which condition fails?

( ) lim ( ) existsx c

ii f x

( ) ( ) is definedi f c

( ) lim ( ) ( )x c

iii f x f c



ii f x

( ) lim ( ) ( )x c

iii f x f c




ii f x

( ) lim ( ) ( )x c

iii f x f c


Continuity (AB)1

2

3

3, 1

( ) 1, 1 2

4, 2

x x

g x x x

x x

At what point(s) is g(x) NOT continuous?

Continuity (AB)2

2 1, 1

( ) 3 , 1 1

2 1, 1

x

x x

h x x

x x

At what point(s) is NOT continuous?( )h x

Continuous at x = 1

Not Continuous at x = 1

Not Continuous at x = 1

Continuous at x = 1

Continuous at x = 1

Vol. I No. 9H

Page 78: 3, 5, 7, 9, 15, 17, 18, 19, 20, 33, 39, 43, 47, 49, 51, 53, 63, 65, 67, 75, 98

Vol. I No. 9BFind the limit

2 2 2 2cos sin cos sin

2

2lim 2 2

3x x x x

x

EQSeptember 16, 2013

How do you show that a function is continuous at a point?

Vol. I No. 9 (Notes)

September 16, 2013

What is Continuity at a Point?

2(1) ( )f x x

This function iscontinuous for all values of x

Continuous or Not?2 4

(2) ( )2

xf x

x

This function iscontinuous for all values of x except at x=2

Continuous or Not?

2(3) ( )

2

xf x

x

This function iscontinuous for all values of x except for x = -2

Continuous or NOT?1

(4) ( )1

f xx

This function iscontinuous for all values of x except for x = 1

Definition of Continuity

( ) lim ( ) ( )x c

iii f x f c

A function is continuous atif all of the following conditions are true:

f x c


ii f x


2 3 , 0( )

1, 0

xx xf x

x x

At what point(s) is NOT continuous?( )f x

Vol. I No. 10B

2 3 , 0( )

1, 0

xx xf x

x x



ii f x


( ) lim ( ) ( )x c

iii f x f c



ii f x

( ) lim ( ) ( )x c

iii f x f c




ii f x

( ) lim ( ) ( )x c

iii f x f c


Continuity (AB)1

2

3

3, 1

( ) 1, 1 2

4, 2

x x

g x x x

x x

At what point(s) is g(x) NOT continuous?

Continuity (AB)2

2 1, 1

( ) 3 , 1 1

2 1, 1

x

x x

h x x

x x

At what point(s) is NOT continuous?( )h x

Vol. I No. 9H

Page 78: 3, 5, 7, 9, 15, 17, 18, 19, 20, 33, 39, 43, 47, 49, 51, 53, 63, 65, 67, 75, 98

EQ:How do we score an AP-Style Problem?

September 13, 2013Vol. I No. 8( )

(a) +1(b) +4(c) +4

9

Vol. I No. 8 ( )

Page AP1 (after p. 94): 1 – 10 Work as a team of 2, 3, or 4

EQSeptember 9, 2013

How do you find the limit …… Graphically?… Numerically?… Analytically?… Verbally?

Vol. I No. 7B

2

lim tanx

x

Evaluate

Graphically, Numerically, Analytically,

Verbally

EQSeptember 5-6, 2013

How do you find the limit at a given point …

… Graphically?… Numerically?… Analytically?… Verbally?

Evaluate Graphically

23

0(1) lim

xx

20

1(2) lim

x x

0(4) lim cot

xx

0(3) lim

2 1xx

x

Evaluate Numerically

23

0(1) lim

xx

20

1(2) lim

x x

0(4) lim cot

xx

0(3) lim

2 1xx

x

Evaluate Analytically

23

0(1) lim

xx

20

1(2) lim

x x

0(4) lim cot

xx

0(3) lim

2 1xx

x

EQSeptember 4, 2013

What is a limit and how do we find it?

Evaluate

3

(1) lim 3x

x

3

(2) lim 3xx

3

3(4) lim

3x

x

x

2

3

9(3) lim

3x

x

x

EQSeptember 3, 2013

How do we describe the behavior of functions?

Vol. I No. 4G (AB)August 29, 2013

Complete discussion criteria 1 – 13 and 20 for the function.

Note: Bring Calculus Book Tomorrow … and every day this week

y x

Vol. I No. 3G (AB)(August 28, 2013)

Make a careful graph of the graph of the following function on your paper.



2

2

4( )

1

xy f x

x

Vol. I No. 4G (BC)(August 28, 2013)




3( )y f x x x

Vol. I No. 2G(August 27, 2013)


(1) y x x2

(2)1

xy

x

Complete the discussion criteria for each function.


Vol. I No. 1G(August 26, 2013)

Make a careful graph of each of the following functions on the paper provided.

(1) y x (2) cosy x

1(3) y

x

, 0(4)

0, 0

xx

xy

x

Bell Work for Quarter I … listed in reverse order.

Documents

b slide

limit slide

odd slide

calculator slide

test slide

values of x slide

squeeze theorem slide

reverse order slide