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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.1.74
Chapter 1, Problem 1.1.74

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.
ƒ(x) = (1/x) - x²

Verified step by step guidance
1
First, identify the function ƒ(x) = (1/x) - x² and the point a. We need to find ƒ(a) by substituting a into the function: ƒ(a) = (1/a) - a².
Next, substitute ƒ(x) and ƒ(a) into the difference quotient formula: (ƒ(x) - ƒ(a)) / (x - a). This gives us: (((1/x) - x²) - ((1/a) - a²)) / (x - a).
Simplify the numerator by combining like terms: ((1/x) - (1/a)) - (x² - a²). Notice that (x² - a²) is a difference of squares, which can be factored as (x - a)(x + a).
Factor the expression (x² - a²) in the numerator: ((1/x) - (1/a)) - (x - a)(x + a). This allows us to cancel the (x - a) term in the numerator and denominator.
After canceling the (x - a) term, simplify the remaining expression. You should be left with a simplified form of the difference quotient.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Difference Quotient

The difference quotient is a fundamental concept in calculus that represents the average rate of change of a function over an interval. It is defined as (ƒ(x) - ƒ(a)) / (x - a), where ƒ(x) is the function and a is a specific point. This expression is crucial for understanding the derivative, as it approaches the instantaneous rate of change as x approaches a.
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The Quotient Rule

Function Composition

Function composition involves combining two functions to create a new function. In the context of the given function ƒ(x) = (1/x) - x², it is important to recognize how the components interact. Understanding how to manipulate and simplify functions is essential for effectively working with the difference quotient and finding limits.
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Evaluate Composite Functions - Special Cases

Limits

Limits are a core concept in calculus that describe the behavior of a function as its input approaches a certain value. They are essential for evaluating the difference quotient, especially when determining the derivative. In this case, simplifying the difference quotient may involve taking the limit as x approaches a, which provides insight into the function's behavior at that point.
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Related Practice
Textbook Question

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = - (3/√x)

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Textbook Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

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Textbook Question

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.


cos⁻¹ √3/2

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Textbook Question

Solve each equation.

48=6e4k48=6e^{4k}

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Textbook Question

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function  h equals ƒ o g .


h(x) = √ (x⁴ + 2 )

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Textbook Question

Finding all inverses Find all the inverses associated with the following functions, and state their domains.


ƒ(x) = x² -2x + 6

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