Mean Value Theorem for Derivatives

1. 4.2 Mean Value Theorem for Derivatives

2. If you drive 100 miles north …in 2 hours… What was your average velocity for the trip? 100 miles 50 miles/hour Does this mean that you were going 50 miles/hour the whole time? No. Were you at any time during the trip going 50 mi/hr? Absolutely. There is no way that you couldn’t have been.

3.  Remember Mr. Murphy’s plunge from the diving platform? s(t) = Height off of the ground(in feet) …is an equation that you know finds…  t = Time in seconds …the slope of the line through the initial and final points. = 56 feet/sec …Mr Murphy’s average velocity from 0 to 3.5 seconds from time 0 to time 3.5

4.  Is there ever a time during Mr Murphy’s fall that his instantaneous velocity is also –56 feet/sec?  Absolutely. We just need to find out where s´(t) =–56 feet/sec s(t) = Height off of the ground(in feet)  t = Time in seconds This means that the slope of the secant line through the initial and final points… = 1.75 seconds …is parallel to the slope of the tangent line through the point t = 1.75 seconds It is also the point at which Mr. Murphy’s instantaneous velocity is equal to his average velocity

5. Mean Value Theorem for Derivatives If the function f (x)is continuous over [a,b] and differentiable over (a,b), then at some point between a and b: Instantaneous Velocity Average Velocity

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

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

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

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