1. What is Calculus? The mathematics of tangent lines, slopes, areas, volumes, arc lengths, and...

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

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

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

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


Example 2: 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



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


Example 4: 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

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)

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

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

I. Finding limits Graphically


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



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


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

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

218 #7-12 all, 33, 35

Direction Change: #33, 35 Find the limit numerically and graphically (using your calculator)

1. What is Calculus? The mathematics of tangent lines, slopes, areas, volumes, arc lengths, and...

Documents

Transcript of 1. What is Calculus? The mathematics of tangent lines, slopes, areas, volumes, arc lengths, and...