1. 2.1 Rates of Change and Limits
AP Calculus AB
2.1 Rates of Change and Limits

2. 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 =

3. 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.

4. 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.

5. 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

6. 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

7. 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
0.0001 ∞
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.

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

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

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

11. Example 2 (cont.)
Looks like y = 1

12. 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

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

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

15. 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.

16. Example 3 – Using Properties of Limits

17. 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.

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

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

20. Example 6
Find
Does the
exist?

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

22. Example 7 (cont.)

23. Greatest Integer FunctionX 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.

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

25. The Sandwich Theorem
Use the Sandwich theorem to find

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

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

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