3.2 Differentiability Arches National Park - Park Avenue

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Presentation transcript:

3.2 Differentiability Arches National Park - Park Avenue Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003

North Window Arch Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003

Balanced Rock Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2010

Delicate Arch Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2010

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

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

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

Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and . Because the function is continuous, it must take on every y value between and .

p Intermediate Value Theorem for Derivatives 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 . p