1. what is calculus? the mathematics of tangent lines, slopes, areas, volumes, arc lengths, and...
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1. What is Calculus?
The mathematics of tangent lines, slopes, areas, volumes, arc lengths, and curvatures
2. Pre-Calculus vs Calculus
The mathematics of change (Velocities and Acceleration)
Look at page 206 of your text for the 4 examples given
Pre-Calculus Limit Process Calculus
What we will be studying this unit:
Section 3.2: Finding Limits Numerically and Graphically
Section 3.3: Finding Limits Algebraically
Section 3.4: Continuity and One-Sided Limits
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Objectives:1. Be able to state the limit notation.
2. Be able to describe where limits are used.3. Be able to find the limit of a function numerically.
Critical Vocabulary:Limit
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I. Limit Notation
Lxfcx
)(lim
“The limit of f(x) as x approaches c is L”
II. Where are Limits Used
1. Define the tangent line to a curve
2. Define the velocity of an object that moves along a straight line
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III. Finding limits Numerically
Example 1: Evaluate _____)(lim1
xfx
As x approaches 1 from the left
x
f(x)
-2
-4
-1
-2
0
0
.9
1.80
.99
1.98
.999
1.998
What is y approaching from the left
As x approaches 1 from the right
What is y approaching from the right
4
8
3
6
2
4
1.1
2.2
1.01
2.02
1.001
2.002
1
?
2)(lim1
xfx
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III. Finding limits Numerically
Example 2: Evaluate _____1
1lim
2
1
x
xx
As x approaches 1 from the left
x
f(x)
-2
-1
-1
0
0
1
.9
1.9
.99
1.99
.999
1.999
What is y approaching from the left
As x approaches 1 from the right
What is y approaching from the right
4
5
3
4
2
3
1.1
2.1
1.01
2.01
1.001
2.001
1
?
21
1lim
2
1
x
xx
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III. Finding limits Numerically
Example 3: Evaluate _____1
1lim
1
x
xx
As x approaches 1 from the left
x
f(x)
-2
-1
-1
-1
0
-1
.9
-1
.99
-1
.999
-1
What is y approaching from the left
As x approaches 1 from the right
What is y approaching from the right
4
1
3
1
2
1
1.1
1
1.01
1
1.001
1
1
?
DNEx
xx
1
1lim
1
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III. Finding limits Numerically
Example 4: Evaluate _____1,0
1,)(lim
1
x
xxxf
x
As x approaches 1 from the left
x
f(x)
-2
-2
-1
-1
0
0
.9
.9
.99
.99
.999
.999
What is y approaching from the left
As x approaches 1 from the right
What is y approaching from the right
4
4
3
3
2
2
1.1
1.1
1.01
1.01
1.001
1.001
1
?
11,0
1,)(lim
1
x
xxxf
x
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Page 217 – 218 #1-6 all, 21-31 odd
Direction Change: #21-31 Find the limit numerically (you only need to find three values on each side)
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Objectives:1. Be able to find the limit of a function graphically.
2. Be able to summarize the big ideas of Limits.
Critical Vocabulary:Limit
Warm Up: Find the limit Numerically
2
23lim
2
2
x
xxx
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WARM UP:Evaluate
x
f(x)
-1
-2
0
-1
1
0
1.9
.9
1.99
.99
1.999
.999
5
4
4
3
3
2
2.1
1.1
2.01
1.01
2.001
1.001
2
?
_________2
23lim
2
2
x
xxx
12
23lim
2
2
x
xxx
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I. Finding limits Graphically
Example 1: Evaluate _____1
1lim
2
1
x
xx
x
f(x)
-2
-1
-1
0
0
1
.9
1.9
.99
1.99
.999
1.999
4
5
3
4
2
3
1.1
2.1
1.01
2.01
1.001
2.001
1
?
21
1lim
2
1
x
xx
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I. Finding limits Graphically
Example 2: Evaluate _____1
1lim
1
x
xx
x
f(x)
-2
-1
-1
-1
0
-1
.9
-1
.99
-1
.999
-1
4
1
3
1
2
1
1.1
1
1.01
1
1.001
1
1
?
DNEx
xx
1
1lim
1
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I. Finding limits Graphically
Example 3: Evaluate _____1,0
1,)(lim
1
x
xxxf
x
x
f(x)
-2
-2
-1
-1
0
0
.9
-9
.99
.99
.999
.999
4
4
3
3
2
2
1.1
1.1
1.01
1.01
1.001
1.001
1
?
11,0
1,)(lim
1
x
xxxf
x
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II. Summarize the Big Ideas!!!!
1. A limit is a y-value if it exists
2. When you say lim f(x) it means we choose “x’s” very close to “c” and
look at the behavior of the function.
xc
3. For a limit to exist, you must allow “x’s” to approach “c” from both sides of “c”. If f(x) approaches a different number from the left and right, the limit does not exist.
4. We don’t care how the function is defined at “c” We do care about the behavior surrounding where x = c. (Journey)
*even if x = c is undefined1
1lim
2
1
x
xx
*even if x = c doesn’t equal the limit
1,0
1,)(lim
1 x
xxxf
x
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Page 217 – 218 #7-12 all, 33, 35
Direction Change: #33, 35 Find the limit numerically and graphically (using your calculator)