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Drill

Drill. Tell whether the limit could be used to define f’(a). no. yes. yes. yes. no. Differentiability. Lesson 3.2 HW: page 114: 2, 4, 10-22 (even), 31-35. Objectives. Students will be able to

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Drill

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  1. Drill Tell whether the limit could be used to define f’(a). no yes yes yes no

  2. Differentiability Lesson 3.2 HW: page 114: 2, 4, 10-22 (even), 31-35

  3. Objectives • Students will be able to • find where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents. • approximate derivatives numerically and graphically.

  4. Where Might a Derivative Not Exist? • At a corner • Where the one-sided derivatives differ • Example: y = |x|; there is a corner at x = 0 • At a cusp • Where the slopes of the secant lines approach ∞ from one side and -∞ from the other • Example: y = x2/3 , there is a cusp at x = 0

  5. Where Might a Derivative Not Exist? • At a vertical tangent • Where the slopes of the secant lines approach either ±∞ from both sides • Example: f(x) = x1/3 • At a discontinuity • Which will cause one or both of the one-sided derivatives be non-existent • Example: a step function

  6. Compare the right-hand and left-hand derivatives to show that the function is NOT differentiable. y = x2 y = x P=(0,0)

  7. The graph of a function over a closed interval D is given. At what points does the function appear to be: Differentiable? Continuous but not differentiable? Neither continuous or differentiable? [-3.2] none none (-3. 2) (2. -2)

  8. The graph of a function over a closed interval D is given. At what points does the function appear to be: Differentiable? Continuous but not differentiable? Neither continuous or differentiable? All points [-3,3], but x = 0 none x = 0 (3. 0) (-3. 0)

  9. Derivatives on a Calculator For small values of h, the difference quotient is often a good numerical approximation of f’(a). However, using the same value of h will actually yield a much better approximating if we use the symmetric difference quotient: Numerical Derivative at f at some point a: NDER

  10. Example: Computing a Numerical Derivative Compute NDER (x2, 3). Using h = 0.001 and

  11. Numerical Derivatives on the TIs MATH nDeriv(function, variable, number)

  12. Example 2 More Numerical Derivatives Compute each numerical derivative.

  13. Theorems • Theorem 1: If f has a derivative at x = a, then f is continuous at x = a • Theorem 2: 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) • Example: Does any function have the Unit Step Function as its derivative? • No, choose some a = -.5 and some b =.5, Then U(a) = -1 and U(b) = 1, but U does NOT take on any value between -1 and 1.

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