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Mean Value Theorem

The document discusses applying the Mean Value Theorem to the function f(x) = x+2/(x+5) on the interval [-3,7]. It is shown that the function satisfies the conditions of the theorem by being continuous on the interval. The average rate of change is calculated as 1/8. The values c=3 and c=5 are identified as points where the tangent line agrees with the average rate of change based on the Mean Value Theorem. An example of how police can infer speeding using the theorem is also provided.

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55 views3 pages

Mean Value Theorem

The document discusses applying the Mean Value Theorem to the function f(x) = x+2/(x+5) on the interval [-3,7]. It is shown that the function satisfies the conditions of the theorem by being continuous on the interval. The average rate of change is calculated as 1/8. The values c=3 and c=5 are identified as points where the tangent line agrees with the average rate of change based on the Mean Value Theorem. An example of how police can infer speeding using the theorem is also provided.

Uploaded by

api-376448000
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Math 1210 Signature Assignment

The Mean Value Theorem

Begin by making a complete statement of the mean value theorem.


Consider the function f (x) = x+2
x+5
on the interval [-3, 7]. Explain how this meets the conditions
of the Mean Value Theorem.
{x|x 5} The value function f (x) = x+2 x+5
is continuous on the interval [-3,7] because it is never
undefined at any point from [-3,7]. Continuity is the condition for the mean value theorem to be
true. Therefore there is a number c between [-3,7] where f (c) = k .

Find the average rate of change of f (x) on [-3, 7], and find the equation of the line segment that
connects the points (-3, f (-3)) and (7, f (7)).
Let y 2 = f (7) and y 1 = f ( 3), also let x2 = 7 and x1 = 3.
( 3 + 2)/( 3 + 5) = 1/2 and (7 + 2)/(7 + 5) = 9/12 = 3/4.

Then the average rate of change is ((3/4) ( 1/2))/(7 ( 3) = 1/8.

Find any numbers c, which are guaranteed by the Mean Value Theorem, and find the equation of
the tangent line to the function f (x) at each value of c.

f (c) = (x2 + 5x (x2 + 2x))/(x + 5)2 3x/(x + 5)2

3x/(x + 5)2 = 1/8 3x/(x + 5)2 * (x + 5)2 = 1/8 * (x + 5)2 3x = 1/8(x + 5)2
3x = 1/8(x2 + 10x + 15) 3x * 8 = 8 * 1/8(x2 + 10x + 15) 18x = x2 + 10x + 15

18x 18x = x2 + 10x + 15 18x = x2 8x 15 = 0

By factoring the equation x2 8x 15 = 0 we find that (x 5)(x 3) which means that


x = 3 or x = 5 . Since both 3 and 5 are on the interval [-3,7], both are numbers c .

Using technology, graph the function f (x) = x+2


x+5
and the above lines on the same axes. Choose
an appropriate viewing window. Include the graph in your report.
Explain what the Mean Value Theorem tells us about rates of change referring to your graph as
an example. Give one real world example of an application of the Mean Value Theorem. Cite
any sources used.
The Mean Value Theorem is an extension of the Intermediate Value Theorem, stating that
between the continuous interval [a,b], there must exist a point c where the tangent at f(c) is equal
to the slope of the interval.
An example of real world application of the mean value theorem would be the ability of police to
infer whether or not you are speeding at a certain point during your commute.
Source:
https://www.khanacademy.org/math/ap-calculus-ab/ab-existence-theorems/ab-mvt/v/getting-a-tic
ket-because-of-the-mean-value-theorem

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