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Mean Value Theorem for Derivatives4.2
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If you drive 100 miles north …in 2 hours… What was your average velocity for the trip? 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. 100 miles
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Remember Mr. Murphy’s plunge from the diving platform? = 56 feet/sec …is an equation that you know finds… …the slope of the line through the initial and final points. from time 0 to time 3.5…Mr Murphy’s average velocity from 0 to 3.5 seconds s(t) = Height off of the ground(in feet) t = Time in seconds
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Is there ever a time during Mr Murphy’s fall that his instantaneous velocity is also –56 feet/sec ? = 1.75 seconds Absolutely. We just need to find out where s´(t) = –56 feet/sec This means that the slope of the secant line through the initial and final points… …is parallel to the slope of the tangent line through the point t = 1.75 seconds s(t) = Height off of the ground(in feet) t = Time in seconds It is also the point at which Mr. Murphy’s instantaneous velocity is equal to his average velocity
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If the function f (x) is continuous over [ a, b ] and differentiable over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives Average Velocity Instantaneous Velocity
![If the function f (x) is continuous over [ a, b ] and differentiable over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives Average Velocity Instantaneous Velocity](http://images.slideplayer.com/32/9821503/slides/slide_5.jpg "If the function f (x) is continuous over [ a, b ] and differentiable over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives Average Velocity Instantaneous Velocity")
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Slope of chord: Slope of tangent: Tangent parallel to chord.
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If the function f (x) is continuous over [ a, b ] and differentiable 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 the function f (x) is continuous over [ a, b ] and differentiable over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.](http://images.slideplayer.com/32/9821503/slides/slide_7.jpg "If the function f (x) is continuous over [ a, b ] and differentiable over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.")
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If the function f (x) is continuous over [ a, b ] and differentiable 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 the function f (x) is continuous over [ a, b ] and differentiable over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.](http://images.slideplayer.com/32/9821503/slides/slide_8.jpg "The Mean Value Theorem only applies over a closed interval..")
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If the function f (x) is continuous over [ a, b ] and differentiable 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.
![If the function f (x) is continuous over [ a, b ] and differentiable 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.](http://images.slideplayer.com/32/9821503/slides/slide_9.jpg ".")
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