2.1- Rates of Change and Limits Warm-up: “Quick Review” Page 65 #1- 4 Homework: Page 66 #3-30 multiples of 3,

2.1- Rates of Change and Limits

•Warm-up:• “Quick Review”

Page 65 #1- 4

•Homework:• Page 66 #3-30 multiples of 3,

2.1- Rates of Change and Limits

• “Quick Review” Solutions

Chapter 2: Limits and Continuity

•The concept of limits is one of the ideas that distinguish calculus from algebra and trigonometry.•In this chapter you will learn how to define and calculate limits of function values.•One of the uses of limits is to test functions for continuity•Continuous functions arise frequently in scientific work because they model a wide range of natural behaviors.

Chapter 2: Limits and Continuity

•L2.1 Rates of Change and Limits•L2.2 Limits Involving Infinity•L2.3 Continuity•L2.4 Rates of Change and Tangent Lines

2.1- Rates of Change and Limits

•What you’ll learn about: Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem

…and why

Limits can be used to describe continuity, the derivative

and the integral: the ideas giving the foundation of

calculus.

Average and Instantaneous Speed

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

•Example 1: Finding an Average Speed– A rock breaks loose from the top of a tall cliff. What is the

average speed during the first 2 seconds of fall? • Hint #1: y = 16t2 …why?

• Hint #2: Δy/ Δt

Average and Instantaneous Speed

• Example 2: Finding an Instantaneous Speed– Find the speed of the rock in Example 1 at the instant t = 2.

• Numerically (pick value really close to t=2, i.e. t=2+h, and look at values where h is approaching the value of 0)

• Algebraically (expand the numerator)

Definition of Limit

• Limits give us a language for describing how the outputs of a function behave as the inputs approach some particular value.• Sometimes we use direct substitution or factoring to calculate a limit.• We this can’t be done, we will need to use the definition of limits to confirm its value.

Definition of Limit

xc

Lets investigate: y = sin(x)/x

Definition of Limit continued

xc

xc

Definition of Limit continued

x1

x1

Properties of Limits

xc

xc

xc

xc

Properties of Limits continued

Product Rule:

Constant Multiple Rule:

(f(x) g(x)) = L Mxc

(k f(x)) = k Lxc

xc

Properties of Limits continued

xc

xc

xc

provided that Lr/s is a real number.

(f(x))r/s = Lr/s

• Example 3: Using Properties of Limits

– Use the observations lim k = k and lim x = c, and the properties

of limits to find the following limits.

– lim (x3 + 4x2 - 3)

– lim

Properties of Limits continued

xcxc

xc

xc

x4 + x2 - 1

x2 + 5

Using the two observations above, we can immediately work our way to the next theorems…

Polynomial and Rational Functions

xc

xc

• Example 4: Using the Properties of Limits– Use the theorem of Polynomials and Rational Functions:

lim (4x2 - 2x + 6)

•Example 5: Using the Properties of Limits– Use the Product Rule (hint: remember limx→0 sinx/x = 1)

lim

Polynomial and Rational Functions

x5

x0

tan xx

• Example 6: Exploring a Nonexistent limit– Use a graph to show that the following limit does not exist.

lim

Polynomial and Rational Functions

x2

x3 - 1

x - 2

Evaluating Limits

• As with polynomials, limits of many familiar

functions can be found by substitution at points

where they are defined. • This includes trigonometric functions,

exponential and logarithmic functions, and composites of these functions.

More Example of Limits

x0

Graphically:

Analytically:

More Example of Limits

Graphically:

Analytically:

x0

2.1- Rates of Change and Limits

• Summary of Today’s Topics: Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem

• Homework:• Page 66-68 #3-30 multiples of 3