2.1 Rates of Change and Limits• Average and Instantaneous Speed

– A moving body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time.

• The unit of measure is length per unit time – ex. Miles per hour, etc.

Finding an Average Speed• A rock breaks loose from the top of a tall cliff.

What is its average speed during the first 2 seconds of fall?

• Experiments show that objects dropped from rest to free fall will fall y = 16t² feet in the first t seconds.

• For the first 2 seconds of, we change t = 0 to t = 2.

2 216(2) 16(0) .32

2 0 sec

y ft

t

Finding an Instantaneous Speed• Find the speed of the rock in example 1 at the

instant t = 2.

• Since we cannot use h = 0 because it will give us an undefined answer, evaluate the formula at values close to 0.

• See the table 2.1 on p. 60 in your textbook.

• Notice, the average speed approaches the limiting value of 64 ft/sec.

2 216(2 ) 16(2)y h

t h

Average Speeds over Short Time Intervals Starting at t = 2

Finding an Instantaneous Speed• Confirm algebraically:

• So, we can see why the average speed has the limiting value of 64 + 16(0) = 64 ft/sec as h approaches 0.

2 216(2 ) 16(2)y h

t h

216(4 4 ) 64h h

h

264 16h h

h

64 16h

Limits• Most limits of interest in the real world can be viewed

as numerical limits of values of functions.• A calculator can suggest the limits, and calculus can

give the mathematics for confirming the limits analytically.

Properties of Limits

Properties of Limits

Using Properties of Limits• Use the observations

and and the properties of limits to find the following limits.

a. b.3 2lim( 4 3)

x cx x

4 2

2

1lim

5x c

x x

x

limx c k k limx c x c

3 2lim lim4 lim3x c x c x cx x

3 24 3c c

4 2

2

lim( 1)

lim( 5)x c

x c

x x

x

4 2

2

lim lim lim1

lim lim5x c x c x c

x c x c

x x

x

4 2

2

1

5

c c

c

Polynomial and Rational Functions

Using Theorem 2a.

b.

2

3lim[ (2 )]x

x x

2(3) (2 3)

92

2

2 4lim

2x

x x

x

2(2) 2(2) 4

2 2

12

34

Using the Product Rule

• Determine

0

tanlimx

x

x

0

tanlim .x

x

x

0

sin 1lim

cosx

x

x x

0 0

sin 1lim lim

cosx x

x

x x

11cos0

1

1 11

Exploring a Nonexistent Limit

• Use a graph to show that does not exist.

• Notice that the denominatoris 0 when x is replaced by2, so we cannot usesubstitution.

• The graph suggests that asx approaches 2 from eitherside, the absolute values getvery large. This suggests thatthe limit does not exist.

3

2

1lim

2x

x

x

One-Sided and Two-Sided Limits• Limits can approach a function from opposite

sides.

• Right-hand limit – limit approaches from the right side.

• Left-hand limit – limit approaches from the left side.

lim ( )x c

f x

lim ( )x c

f x

One-sided and Two-sided Limits

Exploring Right- and Left-Hand Limits

Sandwich Theorem

Using the Sandwich Theorem• Show that 2

0lim[ sin(1 )] 0.x

x x

Homework!!!!!

• Textbook p. 66 – 67 #1, 2, 5, 6, 7 – 14, 20 – 28 even, 37, 40 – 44 even.