1. 3.2 Differentiability

2. Yes No

3. All Reals 3.2 5

4. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

5. Most of the functions we study in calculus will be differentiable.

6. Derivatives on the TI-89: You must be able to calculate derivatives with the calculator and without. Today you will be using your calculator, but be sure to do them by hand when called for. Remember that half the test is no calculator.

7. Example: Find at x = 2. d ( x ^ 3, x ) ENTER returns This is the derivative symbol, which is. 8 2nd It is not a lower case letter “d”. Use the up arrow key to highlight and press. ENTER returns or use: ENTER

8. Warning: The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable. Examples: returns

9. Graphing Derivatives Graph:What does the graph look like? This looks like: Use your calculator to evaluate: The derivative of is only defined for, even though the calculator graphs negative values of x.

10. Two theorems: If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

11. Intermediate Value Theorem for Derivatives Between a and b, must take on every value between and. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and. 

12. 3.3 Rules for Differentiation

13. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.

14. If we find derivatives with the difference quotient: (Pascal’s Triangle) We observe a pattern: …

15. examples: power rule We observe a pattern:…

16. examples: constant multiple rule: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

17. (Each term is treated separately) constant multiple rule: sum and difference rules:

18. Example: Find the horizontal tangents of: Horizontal tangents occur when slope = zero. Plugging the x values into the original equation, we get: (The function is even, so we only get two horizontal tangents.)

24. First derivative (slope) is zero at:

25. product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:

26. quotient rule: or

27. Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative.is the fourth derivative. We will learn later what these higher order derivatives are used for. 