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Calculus Vocabulary 3.2

Calculus Vocabulary 3.2. f ' might fail to exist: A function will not have a derivative at a point P ( a, f ( a )) where the slopes of the secant lines fail to approach a limit as x approaches a . There are four different instances where this occurs: A corner:. A cusp:.

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Calculus Vocabulary 3.2

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  1. Calculus Vocabulary 3.2 • f'might fail to exist: A function will not have a derivative at a point P (a, f (a)) where the slopes of the secant lines fail to approach a limit as x approaches a. • There are four different instances where this occurs: • A corner:

  2. A cusp: A vertical tangent line: A discontinuity:

  3. Example 1 • Find all points in the domain of f (x) where f is not differentiable. • Do Exploration 1 on page 110 in your text.

  4. Derivatives on a Calculator • For small values of h, the difference quotient is often a good numerical approximation of f ´, but an even better approximation of f ´ is a symmetric difference quotient. • Our calculator uses this formula to calculate NDER f (a), the numerical derivative of f at point a.

  5. It can be proven that the limit as h approaches 0 of NDER f (a) = f ´ (a), wherever f ´ exists or • Most calculators, including the TI-89 use h = 0.001 to approximate NDER f (a). • Unless otherwise specified, use h = 0.001. • Example: Compute NDER • Example: Compute NDER

  6. Differentiability implies Continuity • If fhas a derivative at x = a, then f is continuous at x = a. • Intermediate Value Theorem for Derivatives:

  7. Example • Example: Does any function have the unit step function as its derivative?

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