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If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives
![If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives](http://images.slideplayer.com./32/9955924/slides/slide_1.jpg "If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives")
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If f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.
![If f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.](http://images.slideplayer.com./32/9955924/slides/slide_2.jpg "If f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.")
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If f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, 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 continuous over [a,b] and differentiable in ( a, b ), then at some point, c, between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.](http://images.slideplayer.com./32/9955924/slides/slide_3.jpg "The Mean Value Theorem only applies over a closed interval..")
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If f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, 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 f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, 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./32/9955924/slides/slide_4.jpg "If f (x) is continuous over [a,b] and differentiable in ( a, b ), then at some point, c, 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.")
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Slope of secant: Slope of tangent: Tangent parallel to secant.
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Let f (x) be a differentiable function over [ a, b ]. If f(a) = f(b), then there is at least one number, c, in (a,b) such that f ’(c) = 0 Rolle’s Theorem
![Let f (x) be a differentiable function over [ a, b ].](http://images.slideplayer.com./32/9955924/slides/slide_6.jpg "If f(a) = f(b), then there is at least one number, c, in (a,b) such that f ’(c) = 0 Rolle’s Theorem.")
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