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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
Warm-Up: October 19, 2012
  • Given the graph of f(x), graph f’(x)
no derivative
No Derivative
  • The derivative does not exist for any of the following:
  • Corner
  • Cusp
  • Vertical Tangent
  • Discontinuity
corner
Corner
  • When the one-sided derivatives differ.
  • Example: f(x) = |x| at x=0
slide5
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
Vertical Tangent
  • Both one-sided derivatives approach ∞ or both one-sided derivatives approach -∞
  • Example: at x=0
discontinuity
Discontinuity
  • Any point of discontinuity is not differentiable
local linearity
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
Derivatives on TI-83
  • [MATH] [8:nDeriv(]
  • nDeriv(f(x), x, a)
  • NOT ALWAYS CORRECT
example 1
Example 1
  • Find the following using TI-83:
symmetric difference quotient
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
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
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
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
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
Warm-Up: October 22, 2012
  • Calculate the following derivatives, or state that they do not exist. You may use your graphing calculator.
exploration activity
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?