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Differentiability 3.2

Differentiability 3.2. Goal. I will determine when a function is differentiable. . Theorem. Differentiability implies continuity. So if f has a derivative at x=a, then f is continuous at x=a. Smooth Functions.

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Differentiability 3.2

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  1. Differentiability3.2

  2. Goal • I will determine when a function is differentiable.

  3. Theorem • Differentiability implies continuity. So if f has a derivative at x=a, then f is continuous at x=a.

  4. Smooth Functions • We want functions with no Cusps, Corners, Vertical Asymptotes, or Discontinuities.

  5. Local Derivative • It is possible for a function to NOT be differentiable at a point x=a, but it may be differentiable everywhere else.

  6. Example • Think about the graph: Where does the corner occur? • So, f is not differentiable at x=2, but it is everywhere else.

  7. Right and left hand Derivatives • We can calculate the derivative from both sides (like we did with limits). • If the derivatives are different, then the derivative does not exist at that point.

  8. Example • List all values where f is differentiable. Graph it! Do you see a corner, cusp, V.A, discontinuity? Answer: All real numbers except x=-3

  9. Homework • REMOVE 27 and 28 from calendar.

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