Table of Contents 21. Section 4.3 Mean Value Theorem.

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Presentation transcript:

Table of Contents 21. Section 4.3 Mean Value Theorem

Mean Value Theorem Essential Question – What 2 concepts that we have learned about does the mean value theorem connect?

Mean Value Theorem Connects concepts of average rate of change over an interval (secant line) with the instantaneous rate of change at a point within that interval (tangent line) Somewhere between points a and b on a differentiable curve, there is at least one tangent line parallel to secant AB.

Mean Value Theorem If f(x) is continuous at every point on [a,b] and differentiable on (a,b), then there is at least one point c in (a,b) where Rolle’s Theorem is the special case of the MVT where f(a) = f(b) Slope of tangent line Slope of secant line

Example Show that f(x) = x2 satisfies the Mean Value Theorem on [0,2]. Continuous on [0,2] and differentiable on [0,2]

Example Why does the mean value theorem not work on [-1,1] for the following functions? Not differentiable at x=0 Not continuous at x=1

Example Find a tangent that is parallel to secant AB

Uses of Mean Value Theorem Speed Mean Value Theorem says that the instantaneous rate of change (velocity) at some point must equal the average rate of change over the whole interval Example A car accelerating from zero takes 8 sec to go 352 feet. The average velocity is 352/8 = 44 ft/sec or 30 mph. At some point during the acceleration, the car’s speed must be exactly 30 mph.

Uses of Mean Value Theorem Increasing and Decreasing When the derivative is positive, function is increasing When the derivative is negative, function is decreasing If function only increases or decreases on an interval, then it is called monotonic Example Where is y = x2 increasing and decreasing?

Example Where is f(x)=x3-4x increasing and decreasing? Increasing

Testing Critical points A function’s first derivative tells how the graph looks At points where f has a minimum value, f’<0 to the left and f’>0 to the right Similarly, where f has a maximum value, f’>0 to the left and f’<0 to the right

1st derivative test for local extrema At a critical point c If f’ changes sign from positive to negative, then f has a local max If f’ changes sign from negative to positive, then f has a local min If f’ doesn’t change sign, then no local extrema At left endpoint if f’<0 on interval to the right, f has a local max At left endpoint if f’>0 on interval to the right, f has a local min At right endpoint if f’<0 on interval to the left, f has a local min At right endpoint if f’>0 on interval to the right, f has a local max

Example -2 2 Plug in values in each interval to f’ Local max Local min

Example -3 1 Plug in values in each interval to f’ 3 -5 Min max min max

Assignment Pg. 236: #1-9 odd, 13-29 odd, 41, 45, 57