2.1- rates of change and limits warm-up: “quick review” page 65 #1- 4 homework: page 66 #3-30...
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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