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 = 16t 2 …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 xcxc Lets investigate: y = sin(x)/x

Definition of Limit continued xcxc xcxc

x1x1 x1x1

Properties of Limits xcxc xcxc xcxc xcxc

Properties of Limits continued Product Rule: Constant Multiple Rule: (f(x)  g(x)) = L  M xcxc (k  f(x)) = k  L xcxc xcxc

Properties of Limits continued xcxc xcxc xcxc provided that L r/s is a real number. (f(x)) r/s = L r/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 (x 3 + 4x 2 - 3) –lim Properties of Limits continued xcxcxcxc xcxc xcxc x 4 + x x Using the two observations above, we can immediately work our way to the next theorems…

Polynomial and Rational Functions xcxc xcxc

Example 4: Using the Properties of Limits –Use the theorem of Polynomials and Rational Functions: lim (4x 2 - 2x + 6) Example 5: Using the Properties of Limits –Use the Product Rule (hint: remember lim x→0 sinx/x = 1) lim Polynomial and Rational Functions x5x5 x0x0 tan x x

Example 6: Exploring a Nonexistent limit –Use a graph to show that the following limit does not exist. lim Polynomial and Rational Functions x2x2 x 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 x0x0 Graphically: Analytically:

More Example of Limits Graphically: Analytically: x0x0

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 #3-30 multiples of 3