Explanation Given information: The Mean value theorem states that “if the function f is continuous on [ a , b ] and differential on ( a , b ) , then there is some value c ∈ ( a , b ) with f ' ( c ) = f ( b ) − f ( a ) b − a ”. Concept used: Roll’s theorem : if the function f is continuous on [ a , b ] and differential on ( a , b ) , such That, f ( a ) = f ( b ) , the f ' ( x ) = 0 for some x with a ≤ x ≤ b . Proof: Let us Consider the function f is shown below. The equation of the secant line passing through A and B is y = f ( a ) + f ( b ) − f ( a ) b − a . ( x − a ) Consider the function g ( x ) is the difference between f ( x ) and the secant line equation. Then g ( x ) = f ( x ) − ( f ( a ) + f ( b ) − f ( a ) b − a

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ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3.1, Problem 66P

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To prove: The mean value theorem.

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Chapter 3.1, Problem 65P

Chapter 3.1, Problem 67P

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Chapter 3 Solutions

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To prove: The mean value theorem.

Solution Summary: The author proves the mean value theorem by comparing the equation of the secant line passing through AandB and g(x).

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To prove: The mean value theorem.

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Chapter 3 Solutions