Rates of Change and Limits 2.1 Rates of Change and Limits

Quick Review In Exercises 1 – 4, find f (2).

Quick Review In Exercises 5 – 8, write the inequality in the form a < x < b.

Quick Review In Exercises 9 and 10, write the fraction in reduced form.

What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem Essential Question How can limits 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 the elapsed time. Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y = 16t 2 feet in the first t seconds. Wile E Coyote drops an anvil from the top of a cliff. What is its average rate of speed during the first 5 seconds of its fall?

Average and Instantaneous Speed A body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time. Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y = 16t 2 feet in the first t seconds. What is the speed of the anvil at the instant t = 5? Since we cannot calculate the speed right at 5 sec. Calculate the average between 5 sec and slightly after 5 sec. The smaller h becomes the closer the average will get to 160.

Definition of Limit Let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number e, there is a positive number d such that for all x, We write The sentence is read, “The limit of f of x as x approaches c equals L.” The notation means that the values of x approach (but does not equal) c. The next figures illustrate the fact that the existence of a limit as x → c never depend on how the function may not be defined at c.

Definition of Limit continued The function f has a limit 2 as x → 1 even though f is not defined at 1. The function g has a limit 2 as x → 1 even though g(1) ≠ 2. The function h is the only one whose limit as x → 1 equals its value at x = 1.

Properties of Limits

Properties of Limits continued Product Rule: Constant Multiple Rule:

Properties of Limits continued

Example Properties of Limits Use any properties of limits to find: Sum and difference rules Product and multiple rules Quotient rule Sum and difference rules Product and multiple rules

Polynomial and Rational Functions Example Limits Determine the limit by substitution. Support graphically.

Evaluating Limits Example 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. You can evaluate limits either graphically, numerically, or algebraically. Example Limits Determine the limit by substitution. Support graphically.

Example Limits Solve graphically: Confirm algebraically: Determine the limit graphically. Support algebraically. Solve graphically: Confirm algebraically:

Example Limits Limit does not exist. x y Solve graphically: Determine the limit graphically. Support algebraically. Solve graphically: Confirm algebraically: Confirm numerically: x y Limit does not exist.

Example Limits Solve graphically: Confirm algebraically: Determine the limit graphically. Support algebraically. Solve graphically: Confirm algebraically:

Example Limits Determine the limit graphically. Support algebraically. Solve graphically: Confirm algebraically: On page 60, the graph and table shows that this limit approaches 1.

One-Sided and Two-Sided Limits

Example One-Sided and Two-Sided Limits Solve graphically: Solve graphically:

Example One-Sided and Two-Sided Limits Given the following graph, compute the limits.

Example One-Sided and Two-Sided Limits Given the following graph, compute the limits.

Example One-Sided and Two-Sided Limits Given the following graph, compute the limits.

Sandwich Theorem

Pg. 66, 2.1 #1-47 odd