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Exploring Differentiability and Continuity in Honors Precalculus

Understand the concepts of differentiability and continuity in Honors Precalculus. Learn how to identify discontinuities, holes, sharp turns, and vertical tangents in functions. Practice finding x-values where functions are not differentiable and determine if functions are continuous.

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Exploring Differentiability and Continuity in Honors Precalculus

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  1. Honors Precalculus Differentiability Vs. Continuity!

  2. Able to take the derivative You should expect functions to be differentiable everywhere until you find otherwise. Be sure to check for problems with differentiability in the order listed. a. A discontinuity A hole, V.A. or any other time the limit may not exist at x = c. b. A sharp turn c. A vertical tangent A function is not differentiable at x = c if any of the following occur at x = c: Any time the derivative is undefined at x = c. Vertical tangent at x = 0.

  3. Find any x-values where f(x) is not differentiable and tell why. f(x) is not differentiable at x = 4, because the function has a sharp turn. Is f(x) continuous? The individual functions are parabolas, but will they come together at x = 0? f(x) is not differentiable at x = 0, because the function is discontinuous.

  4. Find any x-values where f(x) is not differentiable and tell why. Is f(x) continuous? The individual functions are, but will they come together at x = 0? f(x) is continuous Does f(x) have a sharp turn? Yes, at x = 0. f(x) is not differentiable at x = 0 because of a sharp turn.

  5. Honors Precalculus Differentiability!

  6. a. A discontinuity b. A sharp turn c. A vertical tangent A function is not differentiable at x = c if any of the following occur at x = c:

  7. Find any x-values where f(x) is not differentiable and tell why.

  8. Find any x-values where f(x) is not differentiable and tell why.

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