Warm-Up: October 19, 2012

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Warm-Up: October 19, 2012. Given the graph of f(x), graph f’(x). Differentiability. Section 3.2. No Derivative. The derivative does not exist for any of the following: Corner Cusp Vertical Tangent Discontinuity. Corner. When the one-sided derivatives differ.

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Warm-Up: October 19, 2012
• Given the graph of f(x), graph f’(x)

### Differentiability

Section 3.2

No Derivative
• The derivative does not exist for any of the following:
• Corner
• Cusp
• Vertical Tangent
• Discontinuity
Corner
• When the one-sided derivatives differ.
• Example: f(x) = |x| at x=0
Cusp
• Extreme case of a corner where one sided derivatives approach ±∞ (one side positive, one side negative)
• Example: f(x) = x2/3 at x=0
Vertical Tangent
• Both one-sided derivatives approach ∞ or both one-sided derivatives approach -∞
• Example: at x=0
Discontinuity
• Any point of discontinuity is not differentiable
Local Linearity
• Local linearity means that in a small interval, the graph is close to a straight line.
• On a graphing calculator, zoom in repeatedly to check local linearity.
• If a graph is locally linear near a point, then it is differentiable at that point.
Derivatives on TI-83
• [MATH] [8:nDeriv(]
• nDeriv(f(x), x, a)
• NOT ALWAYS CORRECT
Example 1
• Find the following using TI-83:
Symmetric Difference Quotient
• TI-83 calculates numerical derivatives using a symmetric difference quotient:
• This is the same as f’(a) when the derivative exists.
• TI-83 uses h=0.001
Graphing Derivatives
• Y1=nDeriv(_______,X,X)
• Graph the derivatives of the following functions and try to identify the equation of the derivative:
Differentiability Implies Continuity
• Theorem 1:
• If f has a derivative at x=a, then f is continuous at x=a.
• Proof on page 110.
Intermediate Value Theorem
• Theorem 2 – Intermediate Value Theorem for Derivatives:
• If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).
Assignment
• Read Section 3.2 (pages 105-110)
• Page 111 Exercises #1-22 all
• Read Section 3.3 (pages 112-119)
Warm-Up: October 22, 2012
• Calculate the following derivatives, or state that they do not exist. You may use your graphing calculator.
Exploration Activity
• Is either of these functions differentiable at x=0?
• Graph f and g together in a standard viewing window. How do they compare?
• Turn off the graph of g. Graph f and zoom in on (0,1) several times. Does the graph show signs of straightening out?
• Turn off the graph of f and turn on the graph of g. Return to a standard viewing window, and then zoom in on (0.1).
• How many zooms until glooks horizontal?