Explanation Given: The function f is any quadratic polynomial. Concept used: The mean value theorem: The mean value theorem states that if the function f is continuous on the interval [ a , b ] and differentiable on the interval ( a , b ) then a number c exists in the interval such that the slope at c can be expressed as follows: f ′ ( c ) = f ( b ) − f ( a ) b − a ...... (1) Proof: Consider f ( x ) as a quadratic polynomial expressed as follows: f ( x ) = A x 2 + B x + D ...... (2) Differentiate the above equation with respect to x as shown below: d d x f ( x ) = d d x ( A x 2 + B x + D ) f ′ ( x ) = 2 A x + B Substitute c for x in the above equation as follows: f ′ ( c ) = 2 A c + B Here, c is the mid-point. Substitute ( a + b 2 ) for c in the above equation as follows: f ′ ( a + b 2 ) = 2 A ( a + b 2 ) + B = A ( a + b ) + B Label the above equation as equation (3). f ′ ( a + b 2 ) = A ( a + b ) + B ...... (3) Now, apply the mean value theorem on the interval [ a , b ] . Substitute a for x in the equation (2) as follows: f ( a ) = A a 2 + B a + D Substitute b for x in the equation (2) as follows: f ( b ) = A b 2 + B b + D To obtain the value of f ′ ( a + b 2 ) , substitute A a 2 + B a + D for f ( a ) , A b 2 + B b + D for f ( b ) and a + b 2 for c in the equation (1)

CALCULUS EARLY TRANSCENDENTALS W/ACHIEVE

4th Edition

ISBN: 9781319057664

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.3, Problem 68E

To determine

To show:

That the midpoint c=a+b2 satisfies the conclusion of the MVT on [a,b] for any a and b if the function f is any quadratic polynomial.

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Chapter 4.3, Problem 67E

Chapter 4.3, Problem 69E

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Chapter 4 Solutions

CALCULUS EARLY TRANSCENDENTALS W/ACHIEVE

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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That the midpoint c = a + b 2 satisfies the conclusion of the MVT on [ a , b ] for any a and b if the function f is any quadratic polynomial.

Solution Summary: The author explains that the midpoint c=a+b2 satisfies the conclusion of the MVT.

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To show:

That the midpoint c=a+b2 satisfies the conclusion of the MVT on [a,b] for any a and b if the function f is any quadratic polynomial.

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Chapter 4 Solutions

That the midpoint c = a + b 2 satisfies the conclusion of the MVT on [ a , b ] for any a and b if the function f is any quadratic polynomial.

Solution Summary: The author explains that the midpoint c=a+b2 satisfies the conclusion of the MVT.

Videos

To show: That the midpoint c=a+b2 satisfies the conclusion of the MVT on [a,b] for any a and b if the function f is any quadratic polynomial.

Want to see the full answer?

Chapter 4 Solutions

To show:

That the midpoint c=a+b2 satisfies the conclusion of the MVT on [a,b] for any a and b if the function f is any quadratic polynomial.