Mean Value Theorem for Derivatives 4.10 Mean Value Theorem for Derivatives Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives

If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.

If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous. The Mean Value Theorem only applies over a closed interval.

If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

Tangent parallel to chord. Slope of tangent: Slope of chord:

Example: Verify that the Mean-Value Theorem holds true for the following function over the given interval.

If f (x) is a differentiable function over [a,b]. If , then there is at least one point c in [a, b] where Rolle’s Theorem for Derivatives The Rolle’s Theorem says that at some point in the closed interval, the slope equals 0.

Example: Verify that Rolle’s Theorem holds true for the following function over the given interval.