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4.2 Mean Value Thm for Derivatives

4.2 Mean Value Thm for Derivatives. Rolle’s Theorem. If y = f(x) is continuous on [a,b] and differentiable on (a, b) and f(a) = f(b) = 0 then there exists some c in (a, b) such that f ` (c)=0. Examples. Determine if Rolle’s theorem applies. If so find c that satisfies the theorem.

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4.2 Mean Value Thm for Derivatives

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  1. 4.2 Mean Value Thm for Derivatives

  2. Rolle’s Theorem • If y = f(x) is continuous on [a,b] and differentiable on (a, b) and f(a) = f(b) = 0 then there exists some c in (a, b) such that f`(c)=0.

  3. Examples • Determine if Rolle’s theorem applies. If so find c that satisfies the theorem.

  4. If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives

  5. If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.

  6. If f (x) is a differentiable function 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.

  7. If f (x) is a differentiable function over [a,b], then at some point between a and b: The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope. Mean Value Theorem for Derivatives

  8. Tangent parallel to chord. Slope of tangent: Slope of chord:

  9. Examples • Determine if MVT applies. If so find c that satisfies the theorem.

  10. 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:

  11. Examples: Determine where the function is increasing and decreasing

  12. HW • A: p. 192/15-18, 21-24 • B: p.192/1-14

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

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

  15. 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.

  16. 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 antidifferentiation. The process of finding the original function from the derivative is so important that it has a name: You will hear much more about antiderivatives in the future. This section is just an introduction.

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