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Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Photo by Vickie Kelly, 2002 Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

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Mean Value Theorem for Derivatives If f (x) is a continuous function over [a, b] and differentiable over ( a, b ), then at some point c between a and b : Mean Value Theorem for Derivatives The Mean Value Theorem only applies to a continuous function over a closed interval…

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If f (x) is a continuous function over [a, b] and differentiable over ( a, b ), then at some point c between a and b : Mean Value Theorem for Derivatives The Mean Value Theorem only applies to a well-behaved function that is also differentiable in the interior of the interval.

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If f (x) is a continuous function over [a, b] and differentiable over ( a, b ), then at some point c between a and b : Mean Value Theorem for DerivativesThe Mean Value Theorem says that the average slope across an interval equals the instantaneous slope at a point somewhere on the closed interval (where x=c).

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Average slope of chord from a to b: Slope of tangent AT c: Tangent parallel to chord.

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If the derivative of a function is always positive over an interval, then the function is increasing there. If the derivative of a function is always negative over an interval, then the function is decreasing there. A couple of definitions: Note the IF and the THEN…. they are NOT interchangeable!!! Trying to swap these is a typical misunderstanding!!!

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These two functions have the same slope at each value of x, but they are vertical translations of each other. Functions with the same derivative differ from each other by a constant.

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Example 6: Find the function whose derivative is sin(x), and whose graph passes through. could be, or could vary by a constant. Recognize that we need to have an initial value to determine the precise value of C !

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The process of finding the original function by working backward from the derivative is so pivotal that it has a name: Antiderivative A function is an antiderivative of a function if for all x in the domain of f. The process of finding an antiderivative is called antidifferentiation. You will hear much more about antiderivatives in the future; this section is just an introduction!

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Since acceleration is the derivative of velocity, velocity must be the antiderivative of acceleration. Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec 2 and an initial velocity of 1 m/sec downward; s(0) = -3. (Let down mean a negative value.) The power rule is now running in reverse: increase the exponent by one, then divide by the new exponent.

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Since velocity is the derivative of position, position must be the antiderivative of velocity. Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec 2 and an initial velocity of 1 m/sec downward; s(0) = -3. The power rule running in reverse: increase the exponent by one, then divide by the new exponent.

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Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec 2 and an initial velocity of 1 m/sec downward; s(0) = -3. The initial position is -3 at time zero.

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Exploration: Sketch a rectangular coordinate plane on a piece of paper. Label the points (1, 3) and (5, 3). Draw the graph of a differentiable.

Exploration: Sketch a rectangular coordinate plane on a piece of paper. Label the points (1, 3) and (5, 3). Draw the graph of a differentiable.

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