1. Section 3.2b

2. The “Do Now” Find all values of x for which the given function is differentiable. This function is differentiable except possibly where Check for differentiability at x = 2: The function has a vertical tangent at x = 2. It is differentiable for all reals except x = 2.

3. The “Do Now” Find all values of x for which the given function is differentiable. What type of symmetry does the cosine function have???  Cosine is an even function (symmetric about the y-axis), so This is just a vertical stretch of the basic cosine function  It is differentiable for all reals.

4. The “Do Now” Find all values of x for which the given function is differentiable. Let’s rewrite C as a piecewise function… This function is differentiable for all x except possibly at x = 0: The function is differentiable for all reals. (support graphically?)

5. The difference quotient: For small values of h, this is a good approximation of To get an even better approximation, we can use the symmetric difference quotient: This is what our calculator uses to find the numerical derivative of a function, denoted We only need an h value of about 0.001 to get accurate values for derivatives  most calculators use

6. The Symmetric Difference Quotient Graphically Tangent line Which approximation is better???

7. Practice Problems Find the derivative of the cubing function: Look to your notes!!! What is the value of this derivative at x = 2? Compute, the numerical derivative of the cubing function at x = 2. With your calculator:

8. Practice Problems Compute the numerical derivative of the absolute value function at x = 0. Do you get the same answer with your calculator? Does this answer make sense?  Your calculator can be fooled!!! (It uses the symmetric difference quotient, which never detects the corner of this graph at x = 0…)

9. Use NDER to graph the derivative of the given function. Can you guess what function the derivative is by analyzing its graph? In your calculator: Use the window What function does the derivative look like???

10. Practice Problems Use NDER to graph the derivative of the given function. Can you guess what function the derivative is by analyzing its graph? In your calculator: Use the window What function does the derivative look like???

11. Practice Problems Use NDER to graph the derivative of the given function. Can you guess what function the derivative is by analyzing its graph? In your calculator: Use the window What function does the derivative look like???