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2010 calculusmvt3.2

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The document summarizes the Mean Value Theorem and Rolle's Theorem from calculus. The Mean Value Theorem states that for a differentiable function over a closed interval, there exists a point in the interval where the slope of the tangent line equals the average rate of change over the interval. Rolle's Theorem is a specific case of the Mean Value Theorem where the function value is equal at the endpoints of the interval. An example is provided to check if a function satisfies the hypotheses of Rolle's Theorem over an interval.

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If f (x) is a differentiable function over [a,b] and continuous over (a,b), then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − Mean Value Theorem for Derivatives
If f (x) is a differentiable function over [a,b] and continuous over (a,b), then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.
If f (x) is a differentiable function over [a,b] and continuous over (a,b, then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − 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] and continuous over (a,b), then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − 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. →
y x 0 A B a b Slope of chord: ( ) ( )f b f a b a − − Slope of tangent: ( )f c′ ( )y f x= Tangent parallel to chord. c →
If f (x) is a differentiable function over [a,b] and f(b) = f(a), then at some point between a and b: ( ) 0′ =f c Rolle‘s Theorem for Derivatives This is just a very specific version of the MVT where the endpoints are the same y-value. You are responsible for knowing both of these theorems by name.
Example: Show that satisfies all of the hypotheses (the “ifs”) of Rolle’s Theorem on [0,2]. 3 ( ) 0.5 2f x x x= − 1. Is f continuous on (0,2)? 2. Is f differentiable on [0,2]? 3. Does f(0) = f(2)?

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2010 calculusmvt3.2

  • 1. If f (x) is a differentiable function over [a,b] and continuous over (a,b), then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − Mean Value Theorem for Derivatives
  • 2. If f (x) is a differentiable function over [a,b] and continuous over (a,b), then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.
  • 3. If f (x) is a differentiable function over [a,b] and continuous over (a,b, then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous. The Mean Value Theorem only applies over a closed interval. →
  • 4. If f (x) is a differentiable function over [a,b] and continuous over (a,b), then at some point between a and b: ( ) ( ) ( ) f b f a f c b a − ′= − 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. →
  • 5. y x 0 A B a b Slope of chord: ( ) ( )f b f a b a − − Slope of tangent: ( )f c′ ( )y f x= Tangent parallel to chord. c →
  • 6. If f (x) is a differentiable function over [a,b] and f(b) = f(a), then at some point between a and b: ( ) 0′ =f c Rolle‘s Theorem for Derivatives This is just a very specific version of the MVT where the endpoints are the same y-value. You are responsible for knowing both of these theorems by name.
  • 7. Example: Show that satisfies all of the hypotheses (the “ifs”) of Rolle’s Theorem on [0,2]. 3 ( ) 0.5 2f x x x= − 1. Is f continuous on (0,2)? 2. Is f differentiable on [0,2]? 3. Does f(0) = f(2)?