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Understanding Derivatives and Differentiability in Calculus

Learn about derivatives, higher-order derivatives, and differentiability in calculus. Understand how to identify local maximums or minimums and concavity of curves. Explore the concept of continuous and smooth functions.

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Understanding Derivatives and Differentiability in Calculus

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  1. Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Photo by Vickie Kelly, 2003 The Second Derivative Arches National Park

  2. is the first derivative of y with respect to x. is the second derivative. is the third derivative. is the fourth derivative. Higher Order Derivatives: (y double prime) We will learn later what these higher order derivatives are used for. p

  3. is positive is negative is zero is positive is negative is zero First derivative: Curve is rising. Curve is falling. Possible local maximum or minimum. Second derivative: Curve is concave up. Curve is concave down. Possible inflection point (where concavity changes).

  4. Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Differentiability

  5. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp discontinuity vertical tangent

  6. Most of the functions we study in calculus will be differentiable.

  7. If f has a derivative at x = a, then f is continuous at x = a. Two theorems: Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

  8. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and . Between a and b, must take on every value between and . Intermediate Value Theorem for Derivatives p

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