1. Calculus and analytic geometry
Lecture 04
CS001
Calculus and analytic geometry

2. Average Rates of Change and Secant Lines

3. Rates of Change: Derivative at a Point

4. Example
we studied the speed of a rock falling freely from rest near the surface of the earth. We knew that the rock fell y = 16t2 feet during the first t sec, and we used a sequence of average rates over increasingly short intervals to estimate the rock’s speed at the instant t = 1. What was the rock’s exact speed at this time?

6. Derivatives
The process of calculating a derivative is called differentiation. To emphasize the idea that differentiation is an operation performed on a function y = ƒ(x), we use the notation

7. Differentiation Rules
X ≠ 0

8. Differentiation Rules
Example

9. Example
Differentiate the following powers of x.

10. Example

11. Differentiation Rules

12. Example

15. Differentiation Rules

18. Differentiation Rules

21. Second- and Higher-Order Derivatives

22. Second- and Higher-Order Derivatives

24. Try this