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We Calculus!!! 3.2 Rolle s Theorem and the Mean Value Theorem

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Rolle s Theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If then there is at least one number c in (a, b). ab c

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Ex.Find the two x-intercepts of and show that at some point between the two intercepts. x-int. are 1 and 2 Rolles Theorem is satisfied as there is a point at x = 3/2 where.

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Let. Find all c in the interval (-2, 2) such that. Since, we can use Rolle s Theorem. Thus, in the interval (-2, 2), the derivative is zero at each of these three x-values. 8

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If f (x) is continuous 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 over a closed interval. 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 chord: Slope of tangent: Tangent parallel to chord.

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A function is increasing over an interval if the derivative is always positive. A function is decreasing over an interval if the derivative is always negative. A couple of somewhat obvious definitions:

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These two functions have the same slope at any value of x. Functions with the same derivative differ by a constant.

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Example 6: Find the function whose derivative is and whose graph passes through. so: could beor could vary by some constant.

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Example 6: Find the function whose derivative is and whose graph passes through. so: Notice that we had to have initial values to determine the value of C. HW Pg. 176 1-7 odd, 11-27 odd, 33-45 odd

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The Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then a number c in (a,b) a b secant line c represents slope of the secant line.

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Given, find all c in the interval (1,4) such that the slope of the secant line = the slope of the tangent line. ? But in the interval of (1,4), only 2 works, so c = 2.

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Rolle’s Theorem/Mean-Value Theorem Objective: Use and interpret the Mean-Value Theorem.

Rolle’s Theorem/Mean-Value Theorem Objective: Use and interpret the Mean-Value Theorem.

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