The Second Derivative Arches National Park Photo by

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

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

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

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

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

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

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