1. Section 2.1 Limits, Rates of Change, and Tangent Lines
MAT 1234 Calculus I
Section 2.1
Limits, Rates of Change, and Tangent Lines

2. WebAssign
Homework 2.1

3. Two Worlds and Two Problems?

4. What do we care? How fast “things” are going
The velocity of a particle
The “speed” of formation of chemicals
The rate of change of a population

5. What is “Rate of Change”?
Context
60 miles/hour at t=40s
30 ml/s at t=5s
-30ml/s at t=5s
-$5/min at t=8:05am

6. What is “Rate of Change”?
We are going to look at how to understand and how to find the “rate of change” in terms of functions.

7. The Problems
The Tangent Problem
The Velocity Problem

8. Example 1 The Tangent Problem
Slope=?

9. Example 1 The Tangent Problem
Slope=?
We are going to use an “limiting” process to “find” the slope of the tangent line at x=1.

10. Example 1 The Tangent Problem
Slope=?
First we compute the slope of the secant line between x=1 and x=3.

11. Example 1 The Tangent Problem
Slope=?
Then we compute the slope of the secant line between x=1 and x=2.

12. Example 1 The Tangent Problem
Slope=?
As the point on the right hand side of x=1 getting closer and closer to x=1, the slope of the secant line is getting closer and closer to the slope of the tangent line at x=1.

13. Example 1 The Tangent Problem
Slope=?
First we compute the slope of the secant line between x=1 and x=3.

14. Example 1 The Tangent Problem
Slope=?
First we compute the slope of the secant line between x=1 and x=3.

15. Example 1 The Tangent Problem
Let us record the results in a table. h
slope 2 1
0.1
0.01

16. Example 1 The Tangent Problem
We see from the table that the slope of the tangent line at x=1 should be _________.

17. Limit Notations When h is approaching 0, is approaching ___.
We say as h0,
Or,

18. Definition For the graph of , the slope of the tangent line at is
if it exists.

19. Two Worlds and Two Problems

21. Example 2 The Velocity Problem
y = distance dropped (ft)
t = time (s)
Displacement Function
(Positive Downward)
Find the velocity of the ball at t=2.

22. Example 2 The Velocity Problem
Again, we are going to use the same “limiting” process.
Find the average velocity of the ball from t=2 to t=2+h by the formula

23. Example 2 The Velocity Problem
Average Velocity (ft/s)
2 to 3 1
2 to 2.1
0.1
2 to 2.01
0.01
2 to 2.001
0.001

24. Example 2 The Velocity Problem
We see from the table that velocity of the ball at t=2 should be _________ft/s.

25. Limit Notations When h is approaching 0, is approaching____.
We say as h0,
Or,

26. Definition
For the displacement function , the instantaneous velocity at is
if it exists.

27. Two Worlds and Two Problems

28. Review and Preview
Example 1 and 2 show that in order to solve the tangent and velocity problems we must be able to find limits.
In the next few sections, we will study the methods of computing limits without guessing from tables.