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The Mean Value Theorem and Rolle’s Theorem

The Mean Value Theorem and Rolle’s Theorem. 5-6, pg. 202-207 Lucas Guarino. Introduction. I chose this topic because I was not up to par in my knowledge of the MVT and Rolle’s Theorem A real world application of the section would be traveling through the New Jersey Turnpike.

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The Mean Value Theorem and Rolle’s Theorem

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  1. The Mean Value Theorem and Rolle’s Theorem 5-6, pg. 202-207 Lucas Guarino

  2. Introduction • I chose this topic because I was not up to par in my knowledge of the MVT and Rolle’s Theorem • A real world application of the section would be traveling through the New Jersey Turnpike

  3. Topics of Discussion • Learn MVT and Rolle’s Theorem and learn how to find the point in an interval at which the instantaneous rate of change equals the average rate of change.

  4. Relate to Earlier Lesson • Big Picture: Use the MVT to state why the velocity is true and use Rolle’s Theorem to prove MVT. • In 5-7 and 5-8 the MVT leads to an algerbraic method for finding exact definite integrals.

  5. Typical Problem F(x)=x1/3 on (0,8) F’(x)=1/3x-2/3 F’(0) would be 0-2/3= 1/02/3 =1/0 which is infinite F is differentiable at (0,8) and continuous at 0 and 8 Msec=81/3-01/3/8-0=1/4 F’(c)=1/4 1/3c-2/3 = ¼ then c-2/3 = ¾ C = +- (3/4) -3/2 = +-1.5396 only + val. In (0,8) so c=1.5396

  6. Step by Step Procedure • F(x) = xsinx • F(x) = 0 at x = 0 and at x = pi because sin x is 0 at those points, establish differentiability by, f’(x) = sinx + xcosx which exists for all x • It is differentiable on (0,pi). The function is continuous at this point. F’(c)= 0 if and only if sinc + ccosc = 0 c = 2.02875 • F’(0) is also 0 but c cannot = 0 because 0 is not in the open interval (0,pi)

  7. Summarize Investigation or Detail the Procedure • Mean Value Theorem • If f is differentiable for all values of x in the open interval (a,b) and • If f is continuous at x=a and at x=b, Then there is at least one number x=c in (a,b) such that f’(c) = f(b)-f(a)/b-a

  8. Rolle’s Theorem • If f is differentiable for all values of x in the open interval (a,b) and • If f is continuous at x = a and at x = b, and • If f (a) = f(b) = 0 • Then there is at least one number x = c in (a,b) such that f’(c) = 0

  9. Real Life Examples • When you enter the New Jersey Turnpike you receive a card that indicates your entrance point and the time at which you entered. When you exit, therefore, it can be determined how far you went and how long it took, and thus what your average speed was.

  10. Solution • Average velocity = f(b) – f(a)/b – a • Displacement a = 0m, displacement b = 1057m, time a = 0s, time b = 147s • 1057m – 0m/147s – 0s = 7.19m/s • Average velocity = 7.19m/s

  11. Conclusion • The Mean Value Theorem and Rolle’s Theorem • When you enter the New Jersey Turnpike you receive a card that indicates your entrance point and the time at which you entered. When you exit, therefore, it can be determined how far you went and how long it took, and thus what your average speed was. • The audience learned what the MVT and Rolle’s Theorems are and how to relate them to the real world.

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