(a) Answer Both the functions are same as x = a + h . Explanation Calculation: As x = a + h , h = x − a . Substitute x − a for h in the difference quotient formula. f ( a + h ) − f ( a ) h = f ( a + ( x − a ) ) − f ( a ) x − a = f ( x ) − f ( a ) x − a Therefore, both the functions are same. (b) To determine To check: The reason for the two forms of difference quotient. Answer The first form of difference quotient is used for a big point as the limit h tends to 0 and the second function is used for a small value of a . Explanation The second formula for difference quotient is used for small value of a as the limit x will tends to a which is possible only when a is small. The first formula for difference quotient is used for large values of a as a new variable h is used which tends to 0 . Therefore, the first form of difference quotient is used for a big point as the limit h tends to 0 and the second function is used for a small value of a .

Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 2.4, Problem 55E

(a)

To determine

To check: The reason that the difference quotient of f at a , f(a+h)−f(a)h is equivalent to an alternate form f(x)−f(a)x−a.

(a)

Expert Solution

Answer to Problem 55E

Both the functions are same as x=a+h.

Explanation of Solution

Calculation:

As x=a+h , h=x−a.

Substitute x−a for h in the difference quotient formula.

f(a+h)−f(a)h=f(a+(x−a))−f(a)x−a=f(x)−f(a)x−a

Therefore, both the functions are same.

(b)

To determine

To check: The reason for the two forms of difference quotient.

(b)

Expert Solution

Answer to Problem 55E

The first form of difference quotient is used for a big point as the limit h tends to 0 and the second function is used for a small value of a.

Explanation of Solution

The second formula for difference quotient is used for small value of a as the limit x will tends to a which is possible only when a is small.

The first formula for difference quotient is used for large values of a as a new variable h is used which tends to 0.

Therefore, the first form of difference quotient is used for a big point as the limit h tends to 0 and the second function is used for a small value of a.

Chapter 2.4, Problem 54E

Chapter 2.4, Problem 1QQ

Chapter 2 Solutions

Calculus: Graphical, Numerical, Algebraic

Ch. 2.1 - Prob. 1QR Ch. 2.1 - Prob. 2QR Ch. 2.1 - Prob. 3QR Ch. 2.1 - Prob. 4QR Ch. 2.1 - Prob. 5QR Ch. 2.1 - Prob. 6QR Ch. 2.1 - Prob. 7QR Ch. 2.1 - Prob. 8QR Ch. 2.1 - Prob. 9QR Ch. 2.1 - Prob. 10QR

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prob. 38E Ch. 2.1 - Prob. 39E Ch. 2.1 - Prob. 40E Ch. 2.1 - Prob. 41E Ch. 2.1 - Prob. 42E Ch. 2.1 - Prob. 43E Ch. 2.1 - Prob. 44E Ch. 2.1 - Prob. 45E Ch. 2.1 - Prob. 46E Ch. 2.1 - Prob. 47E Ch. 2.1 - Prob. 48E Ch. 2.1 - Prob. 49E Ch. 2.1 - Prob. 50E Ch. 2.1 - 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Prob. 50RE Ch. 2 - Prob. 51RE Ch. 2 - Prob. 52RE Ch. 2 - Prob. 53RE Ch. 2 - Prob. 54RE Ch. 2 - Prob. 55RE

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To check: The reason that the difference quotient of f at a , f ( a + h ) − f ( a ) h is equivalent to an alternate form f ( x ) − f ( a ) x − a .

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To check: The reason that the difference quotient of f at a , f(a+h)−f(a)h is equivalent to an alternate form f(x)−f(a)x−a.

Answer to Problem 55E

Explanation of Solution

To check: The reason for the two forms of difference quotient.

Answer to Problem 55E

Explanation of Solution

Chapter 2 Solutions

Additional Math Textbook Solutions