2.1 Rates of Change and Limits AP Calculus AB 2.1 Rates of Change and Limits

Average Speed – Example 0 Suppose you drive for 200 miles and it takes 4 hours. Find the average speed of your trip. Average speed = Distance Covered ÷ Time Elapsed Avg. Speed =

Instantaneous Speed Suppose, during your trip, you look at your speedometer and it reads 65 mi/hr. This is your instantaneous speed for that moment in time.

First, the equation for a free falling object on Earth is y = 16t2. Example 1 A rock breaks loose from the top of a tall cliff. What is its average speed during the first five seconds of fall? First, the equation for a free falling object on Earth is y = 16t2.

Average Speed – Example 1 (cont.) We need the distance covered in 5 seconds which is Δy and the change in time Δt which is 5 seconds. Since y = 16t2 = 80 feet/sec

Instantaneous Speed – Example 1 (cont.) Find the speed of the rock at the instant t = 5. What we need to look at is what happens to the formula when h (or Δt) gets very close to 0, but does not equal zero (or the limit as h approaches 0). h represents a slightly later time

Instantaneous Speed – Example 1 (cont.) Length of Time Interval h (sec) Avg. Speed for Interval Δy/Δt (ft/sec) 1 176 0.1 161.6 0.01 160.16 0.001 160.016 0.0001 160.0016 0.00001 160.00016 ∞ 160 First, let’s choose values for h that get closer and closer to 0. Then, using our calculators, find different values for the instantaneous speed. Therefore, we can see the instantaneous velocity approaches 160 ft/sec as h becomes very small.

Instantaneous Speed – Example 1 (cont.) Solve algebraically:

Limit Notation The limit of a function refers to the value that the function approaches, not the actual value (if any).

Example 2 Consider: What happens as x approaches zero? Let’s solve graphically: Y= WINDOW GRAPH

Example 2 (cont.) Looks like y = 1

Numerically: TblSet TABLE You can scroll up or down to see more values. It appears that the limit of as x approaches zero is 1

Limit notation: “The limit of f of x as x approaches c is L.” So:

The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

Example 3 – Using Properties of Limits

Example 4 – Determine the limit by substitution. Support graphically. Next, graph on the calculator and use either the table or trace functions to check the limit of the function at x = 1. Next, graph on the calculator and use either the table or trace functions to check the limit of the function at x = 1.

Example 5 – Determine the limit graphically. Confirm algebraically. π Day 1

Theorem 3 One-Sided and Two-Sided Limits For a limit to exist, the function must approach the same value from both sides. Or,

Example 6 Find Does the exist?

Example 7 Given c = 2, Draw the graph of f. Determine and Does exist? If so, what is it? If not, explain.

Example 7 (cont.)

Greatest Integer Function X Y 0.5 0.75 1 1.5 1.75 2 2.5 -0.5 On most calculators it is y = int(x). The Greatest Integer Function is also called a floor function and is denoted y = greatest integer ≤ x.

Greatest Integer Function (cont.) The Greatest Integer Function Is also called a step function. p

The Sandwich Theorem Use the Sandwich theorem to find

The Sandwich Theorem (cont.) If we graph , it appears that

The Sandwich Theorem (cont.) Next, let’s look at the graphs of and It appears is “sandwiched” between

The Sandwich Theorem (cont.) By taking the limits gets Since and , then by the Sandwich Theorem p