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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?
Vol. I No. 14BSeptember 24, 2013
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
Vol. I No. 14BSeptember 24, 2013
31 ) lim
4x
ax
4
31 ) lim
4x
bx
4
31 ) lim
4x
cx
Vol. I No. 14BSeptember 24, 2013
12 ) lim
2x
ax
2
12 ) lim
2x
bx
2
12 ) lim
2x
cx
Vol. I No. 14BSeptember 24, 2013
2
23 ) lim
1x
ax
21
23 ) lim
1x
bx
21
23 ) lim
1x
cx
Vol. I No. 14BSeptember 24, 2013
2
34 ) lim
2x
ax
22
34 ) lim
2x
bx
22
34 ) lim
2x
cx
Vol. I No. 14BSeptember 24, 2013
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)
Vol. I No. 14BSeptember 24, 2013
3 25 4
5) ( )4
x x xf x
x x
Vol. I No. 14BSeptember 24, 2013
3 25 4
5) ( )4
x x xf x
x x
VA:
HA: hole(s):
Vol. I No. 14BSeptember 24, 2013
3 2
3
5 46) ( )
16
x x xf x
x x
Vol. I No. 14BSeptember 24, 2013
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?
Vol. I No. 13BSeptember 23, 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 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)
Vol. I No. 11BSeptember 18, 2013
2 2
01) lim
x
x x x
x
0
sin( ) sin2) lim
x
x x x
x
Vol. I No. 11BSeptember 18, 2013
0
sin( ) sin2) lim
x
x x x
x
Vol. I No. 11BSeptember 18, 2013
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
Vol. I No. 10BSeptember 17, 2013
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
( ) ( ) is definedi f c
( ) lim ( ) existsx c
ii f x
( ) lim ( ) ( )x c
iii f x f c
Which condition fails?
( ) ( ) is definedi f c
( ) lim ( ) existsx c
ii f x
( ) lim ( ) ( )x c
iii f x f c
Which condition fails?
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
( ) lim ( ) existsx c
ii f x
( ) ( ) is definedi f c
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
Which condition fails?
( ) lim ( ) existsx c
ii f x
( ) ( ) is definedi f c
( ) lim ( ) ( )x c
iii f x f c
( ) ( ) is definedi f c
( ) lim ( ) existsx c
ii f x
( ) lim ( ) ( )x c
iii f x f c
Which condition fails?
( ) ( ) is definedi f c
( ) lim ( ) existsx c
ii f x
( ) lim ( ) ( )x c
iii f x f c
Which condition fails?
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.
Complete discussion criteria 1 – 13 and 20 for the function.
Note: Bring Calculus Book Tomorrow … and every day this week
2
2
4( )
1
xy f x
x
Vol. I No. 4G (BC)(August 28, 2013)
Make a careful graph of the graph of the following function on your paper.
Complete discussion criteria 1 – 13 and 20 for the function.
Note: Bring Calculus Book Tomorrow … and every day this week
3( )y f x x x
Vol. I No. 2G(August 27, 2013)
Make a careful graph of the graph of the following function on your paper.
(1) y x x2
(2)1
xy
x
Complete the discussion criteria for each function.
Note: Bring Calculus Book Tomorrow … and every day this week
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
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