/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 Evaluating a Difference Quotient... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 81

Evaluating a Difference Quotient In Exerciseses \(77-84\) , find the difference quotient and simplify your answer. $$g(x)=\frac{1}{x^{2}}, \quad \frac{g(x)-g(3)}{x-3}, \quad x \neq 3$$

Short Answer

Expert verified
The difference quotient of the given function is \(-\frac{(x-3)(x+3)}{18x^{3}}\)

Step by step solution

01

Identify the function and substitute

Given function is \(g(x) = \frac{1}{x^{2}}\) and we need to find \(\frac{g(x)-g(3)}{x-3}\). First, substitute \(g(x)\) and \(g(3)\) in the expression. \(g(3) = \frac{1}{3^{2}} = \frac{1}{9}\). So, \(\frac{g(x)-g(3)}{x-3}=\frac{\frac{1}{x^{2}}-\frac{1}{9}}{x-3}\)
02

Simplify the expression

The expression on the numerator is complex fraction. We need to simplify it by finding a common denominator, which is \(\frac{9-x^{2}}{9x^{2}}\). So, \((\frac{g(x)-g(3)}{x-3})= \frac{\frac{9-x^{2}}{9x^{2}}}{x-3}\)
03

Further Simplification

Multiply numerator and denominator by \(9x^{2}\) to eliminate complex fraction. So, \(\frac{9-x^{2}}{9x^{2}\times(x-3)}= \frac{9-x^{2}}{9x^{2}x-27x^{2}} = \frac{9-x^{2}}{-18x^{3}}\)
04

Final Simplification

Simplify the expression and get the final form of this difference quotient. The final form is \(\frac{x^{2}-9}{18x^{3}} = -\frac{x^{2}-9}{18x^{3}} = -\frac{(x-3)(x+3)}{18x^{3}}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
Function evaluation is a fundamental concept in algebra and calculus. It involves determining the output of a function for a given input. In this exercise, we focus on evaluating the function \(g(x) = \frac{1}{x^2}\) at specific values of \(x\). To understand this better, let's break it down:
  • Given a function \(g(x) = \frac{1}{x^2}\), we aim to find the output when \(x = 3\).
  • This requires substituting \(x = 3\) into the function: \(g(3) = \frac{1}{3^2} = \frac{1}{9}\).
Conceptually, function evaluation helps us understand how changes in the input \(x\) affect the function. This is a critical step before applying the difference quotient, as you'll have to know function values at different points to analyze the rate of change.
Simplifying Fractions
Simplifying fractions is a key skill in both algebra and calculus, especially when working with expressions like difference quotients. The problem involves simplifying the expression \(\frac{\frac{1}{x^2} - \frac{1}{9}}{x-3}\). Let's simplify it step-by-step:
  • First, handle the numerator: \(\frac{1}{x^2} - \frac{1}{9}\).
  • To simplify this, find a common denominator, which in this case is \(9x^2\).
  • This makes the expression: \(\frac{9}{9x^2} - \frac{x^2}{9x^2} = \frac{9 - x^2}{9x^2}\).
Simplifying fractions involves manipulating them into a form that's easier to work with in calculus operations. Doing this properly ensures accuracy in further steps and simplifies calculations.
Complex Fractions
Complex fractions can appear intimidating, but with practice, they become manageable. In this exercise, the goal is to simplify the expression \(\frac{\frac{9 - x^2}{9x^2}}{x-3}\). Here鈥檚 how it's done:
  • The expression \(\frac{9 - x^2}{9x^2}\) is in the numerator.
  • To eliminate the complex fraction, multiply the entire expression by \(9x^2\), which is the common factor for the numerator's denominator.
  • This leads to: \(\frac{(9 - x^2)}{9x^2 \times (x-3)} = \frac{(9 - x^2)}{-18x^3}\).
  • Finally, factor the numerator as \(-(x-3)(x+3)\) and simplify the expression.
Mastering complex fractions is necessary for working with more advanced calculus concepts. Taking them step-by-step allows us to simplify these structures for further analysis or application.

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Most popular questions from this chapter

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. Predicting Graphical Relationships Use a graphing utility to graph \(f, g\) , and \(h\) in the same viewing window. Before looking at the graphs, try to predict how the graphs of \(g\) and \(h\) relate to the graph of \(f .\) (a) $$f(x)=x^{2}, \quad g(x)=(x-4)^{2}h(x)=(x-4)^{2}+3 h(x)=(x-4)^{2}+3$$ (b) $$f(x)=x^{2}, g(x)=(x+1)^{2} h(x)=(x+1)^{2}-2 $$ (c) $$f(x)=x^{2}, \quad g(x)=(x+4)^{2} h(x)=(x+4)^{2}+2$$

Restricting the Domain In Exercises \(73-82,\) restrict the domain of the function \(f\) so that the function is one-to-one and has an inverse function. Then find the inverse function \(f^{-1} .\) State the domains and ranges of \(f\) and \(f^{-1} .\) Explain your results. (There are many correct answers.) $$f(x)=\frac{1}{2} x^{2}-1$$

From the top of a mountain road, a surveyor takes several horizontal measurements \(x\) and several vertical measurements \(y\) as shown in the table \((x\) and \(y\) are measured in feet). $$\begin{array}{|c|c|c|c|c|}\hline x & {300} & {600} & {900} & {1200} \\\ \hline y & {-25} & {-50} & {-75} & {-100} \\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|}\hline x & {1500} & {1800} & {2100} \\ \hline y & {-125} & {-150} & {-175} \\ \hline\end{array}$$ $$\begin{array}{l}{\text { (a) Sketch a scatter plot of the data. }} \\\ {\text { (b) Use a straightedge to sketch the line that you }} \\ {\text { think best fits the data. }} \\ {\text { (c) Find an equation for the line you sketched in }} \\ {\text { part (b). }}\end{array}$$ $$ \begin{array}{l}{\text { (d) Interpret the meaning of the slope of the line in }} \\ {\text { part (c) in the context of the problem. }} \\ {\text { (e) The surveyor needs to put up a road sign that }} \\ {\text { indicates the steepness of the road. For instance, }} \\ {\text { a surveyor would put up a sign that states "8% }}\end{array}$$ $$ \begin{array}{l}{\text { grade" on a road with a downhill grade that has }} \\\ {\text { a slope of }-\frac{8 .}{100 .} \text { . What should the sign state for }} \\ {\text { the road in this problem? }}\end{array}$$

Think About lt Restrict the domain of \(f(x)=x^{2}+1\) to \(x \geq 0 .\) Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

Finding an Equation of a Line ,find an equation of the line passing through the points. Sketch the line. $$(-8,0.6),(2,-2.4)$$
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