1. Sec 3.1: Tangents and the Derivative at a Point
Def:
Def:
The derivative of a function ƒ at a point x0
The difference quotient of ƒ at x0 with increment h.
Example:
Example:
Find the difference quotient of ƒ at x0=2 with increment h.
Find the derivative of ƒ at x0=2

2. Sec 3.1: Tangents and the Derivative at a Point
Example:
Find the derivative of ƒ at x0=2
Def:
The derivative of a function ƒ at a point x0
Example:
Find

3. Sec 3.1: Tangents and the Derivative at a Point
RATES OF CHANGE
Def:
The average rate of change of ƒ with respect to x over the interval [a, b]
Chane in x =
Chane in y =
Term 102

4. Sec 3.1: Tangents and the Derivative at a Point
Def:
Def:
The average rate of change of ƒ with respect to x over the interval [a, b]
The rate of change of ƒ with respect to x at x0
The instantaneous rate of change of ƒ with respect to x at x0
Term 102

5. Sec 3.1: Tangents and the Derivative at a Point
quotient
The slope of the tangent to the curve
The slope of the secant
The slope of the curve
instantaneous rate of change
Average rate of change

6. Sec 3.1: Tangents and the Derivative at a Point
quotient
The slope of the tangent to the curve
The slope of the secant
The slope of the curve
instantaneous rate of change
Average rate of change

7. Sec 3.1: Tangents and the Derivative at a Point

8. Sec 3.1: Tangents and the Derivative at a Point
Slopes :

9. Sec 3.1: Tangents and the Derivative at a Point

10. Sec 3.1: Tangents and the Derivative at a Point
Vertical Tangents
<-2
>2 2 -2 1 -1

11. Sec 3.1: Tangents and the Derivative at a Point
Example:
Example:
has a vertical tangent at x = 0.
has a vertical tangent at x = 0.

12. Sec 3.1: Tangents and the Derivative at a Point
Example:
has no vertical tangent at x = 0.

13. Sec 3.1: Tangents and the Derivative at a Point

14. Sec 3.1: Tangents and the Derivative at a Point

15. Sec 3.1: Tangents and the Derivative at a Point

16. Sec 3.1: Tangents and the Derivative at a Point