Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Concept explainers

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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Question

Chapter 2.4, Problem 21E

(a)

To determine

To find: The slope of the curve y=1x−1 at x=a.

(a)

Expert Solution

Answer to Problem 21E

The slope of the curve y=1x−1 at x=a is −1(a−1)2.

Explanation of Solution

Given information:

The curve is y=1x−1 and point is x=a.

Calculation:

The formula for the slope of the curve f(x) at (a,f(a)) is limh→0f(a+h)−f(a)h.

The function is given as f(x)=1x−1.

Substitute a+h for x in the function.

f(a+h)=1a+h−1

Substitute a for x in the function.

f(a)=1a−1

Substitute 1a+h−1 for f(a+h) and 1a−1 for f(a) in the formula for slope.

limh→0f(a+h)−f(a)h=limh→01a+h−1−1a−1h=limh→0a−1−(a+h−1)(a−1)(a+h−1)h=limh→0−hh(a−1)(a+h−1)=limh→0(−1(a−1)(a+h−1))

Further simplify.

limh→0f(a+h)−f(a)h=−1(a−1)(a+0−1)=−1(a−1)2

Therefore, the slope of the curve y=1x−1 at x=a is −1(a−1)2.

(b)

To determine

To find: The change in the tangent at x=a as the value of a changes.

(b)

Expert Solution

Answer to Problem 21E

The slope of the tangent increases as the value of a increases and the tangent is steeper near the point a=1.

Explanation of Solution

Given information:

The curve is y=1x−1 and point is x=a.

Calculation:

As calculated in part (a), the slope of the curve y=1x−1 at x=a is equal to −1(a−1)2.

So, the slope of the tangent of the curve is always negative and the value of slope of tangent increases as the value of a increases. The tangent is steeper near the point a=1.

Therefore, the slope of the tangent decreases as the value of a increases.

Chapter 2.4, Problem 20E

Chapter 2.4, Problem 22E

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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The slope of the curve y = 1 x − 1 at x = a .

Concept explainers

Videos

To find: The slope of the curve y=1x−1 at x=a.

Answer to Problem 21E

Explanation of Solution

To find: The change in the tangent at x=a as the value of a changes.

Answer to Problem 21E

Explanation of Solution

Chapter 2 Solutions

Additional Math Textbook Solutions