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Aim: How can we use the Mean Value Theorem to solve problems? By: Nihir Shah

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The Mean Value Theorem The Mean Value Theorem is a very simple phenomenon. Here is a defination of the Mean Value Theorem from Wikipedia: In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the section. It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval. calculussmoothderivativecalculussmoothderivative You must be like what in the world is this? Allow me to explain in simple terms:

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My definition (the simple one) By this point in the so-called calculus era, many of are probably familiar with the first derivative. We have also used slopes before. So here is my defination of the MVT. Mean Value Theorem: Find the slope using the slope formula, then find the first derivative of the function. Then, connect the two. Easier than the abstruse definition of those Wikipidians!

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Example 1: Problem 1:Find a value of c such that the conclusion of the mean value theorem is satisfied for f(x) = -2x 3 + 6x - 2 on the interval [-2, 2] Problem 1:Find a value of c such that the conclusion of the mean value theorem is satisfied for f(x) = -2x 3 + 6x - 2 on the interval [-2, 2] (Borrowed from http://www.analyzemath.com) http://www.analyzemath.com Give it a shot, the answers on the next slide:

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Solution: What did I tell you? Slope time is first. Slope is simply the (f(b)-f(a))/ b-a. So lets find the y values first: f(-2)= -2(-2) 3 + 6(-2)-2= 2 f(2)= -2(2) 3 + 6(2)-2= -6 Plug that into the slope formula, and youll get the slope as -2 Now, find the first derivative: f '(x) = -6x 2 + 6 Get that… I knew you would! Now simply put them together. -2= -6x 2 + 6. Solve for that, and youll get C=+ or - 2radical2/3 element of [-2,2]

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Another one, Please!!! Okay then, your wish is granted: this is simply a change of numbers Just be careful: Borrowed from the Visual Calculus website. f(x)= x 3 -6x 2 +9x+2 [0,4] Now, a bell should ring in your head, it should tell you, SLOPEEEEEEEEE!!! So… f(0)=2 and f(2)= 6. You know the formula for the slope. You will simply use the formula and get, the slope as 1. Now, you find the first derivative: it is: f(x)= 3x 2 -12x+9. Now, comes the best part, the connection. So, it follows: 1=3x 2 -12x+9. Now just solve for x. You will find that x = 3.154700539 and x = 0.845299461. So the answer should be c= 3.154700539 and 0.845299461 over the interval of [0, 4] Simple enough right, now well get a bit more complex!!!

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The trigonometry Sometimes, intimidation arrives with the presence of the trig king. Well, dont worry, we use the same steps Sometimes, intimidation arrives with the presence of the trig king. Well, dont worry, we use the same steps So… Use the mean value theorem (MVT), to find c. f(x)= sin(2x)+cos(x) [0, 2pi] Take a deep breath relax), its not going to bite! Just try it as if it were a regular problem.

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The answer: Did you try the problem? Here it goes… find the slope: f(pi)-f(0)/ pi-0. You should wind up with a slope of -2/pi Now, remember, what did I tell you to do? FIND THE FIRST DERIVATIVE. f(x)=cos(2x)-sin(x). Now its time for the connection. Is it sticking to your head now??? Ok. So its -2/pi=cos(2x)- sin(x). After the equation is solved, you should get:x = 0.7704383989 and x = 2.371154255. So, your final answer would be c= 0.7704383989, 2.371154255 element of the interval [0, 2pi]

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