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Chapter 5: Problem 63

Find the average rate of change, by the \(\Delta\) process, for: \(\mathrm{y}=1 / \mathrm{x}\)

Short Answer

Expert verified
The average rate of change for the function \(y = \frac{1}{x}\) using the \(\Delta\) process is \(\frac{-1}{x(x + \Delta x)}\).

Step by step solution

01

Find \(f(x + \Delta x)\)

To find \(f(x + \Delta x)\), substitute \(x + \Delta x\) into the given function: \(f(x + \Delta x) = \frac{1}{x + \Delta x}\)
02

Write down the expression for the difference quotient

We will use the formula of the average rate of change: Average rate of change = \(\frac{f(x + \Delta x) - f(x)}{\Delta x}\) Plug in \(f(x + \Delta x) = \frac{1}{x + \Delta x}\) and \(f(x) = \frac{1}{x}\): Average rate of change = \(\frac{\frac{1}{x + \Delta x} - \frac{1}{x}}{\Delta x}\)
03

Simplify the expression

To simplify, we need to get rid of the fraction within the fraction. We will accomplish this by finding a common denominator for \(\frac{1}{x + \Delta x}\) and \(\frac{1}{x}\), then combine both terms into a single fraction: Common denominator: \((x + \Delta x)x = x(x + \Delta x)\) Combine both terms: \(\frac{x - (x + \Delta x)}{x(x + \Delta x)}\) Simplify the expression: \(\frac{-\Delta x}{x(x + \Delta x)}\) Now, insert this result back into the difference quotient: Average rate of change = \(\frac{\frac{-\Delta x}{x(x + \Delta x)}}{\Delta x}\)
04

Simplify further and find the average rate of change

To simplify further, we can cancel out the \(\Delta x\) term from both the numerator and the denominator: Average rate of change = \(\frac{-1}{x(x + \Delta x)}\) This is the average rate of change of the function \(y = \frac{1}{x}\) using the \(\Delta\) process.

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Most popular questions from this chapter

Find the rate of change of \(\mathrm{y}\) with respect to \(\mathrm{x}\) at the point \(\mathrm{x}=5\), if \(2 y=x^{2}+3 x-1\)

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