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Chapter 4: Problem 23

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$\sqrt{146}$$

Short Answer

Expert verified
Question: Using linear approximation, estimate the value of 鈭�146. Answer: With linear approximation, the estimated value of 鈭�146 is approximately 12.08333.

Step by step solution

01

Choose a value for a

An appropriate value for \(a\) should be close to \(146\) and easy to work with. The easiest way to proceed is to find the square root of a perfect square close to \(146\). The closest perfect square to \(146\) is \(144\), since \(12^2 = 144\). Therefore, we'll choose \(a=144\).
02

Calculate the derivative of the function

We are working with the function \(f(x) = \sqrt{x}\) and using linear approximations, which involve the tangent line to the curve. The slope of the tangent line is given by the derivative of the function. We need to find \(f'(x)\): $$f'(x) = \frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}$$
03

Find the value of the derivative at a

Now, we need to find the value of the derivative at our chosen value for \(a=144\): $$f'(144) = \frac{1}{2\sqrt{144}} = \frac{1}{2\cdot 12} = \frac{1}{24}$$
04

Find the equation of the tangent line

Using the slope \(f'(144) = \frac{1}{24}\) and the point \((144,\sqrt{144}) = (144,12)\), we can find the equation of the tangent line: $$y = f(144) + f'(144)(x-144)$$ $$y = 12 + \frac{1}{24}(x-144)$$
05

Estimate the value of the function

Finally, we can use this tangent line equation to estimate the value of \(\sqrt{146}\): $$\sqrt{146} \approx 12 + \frac{1}{24}(146-144) = 12 + \frac{1}{24}(2) = 12 + \frac{1}{12} = 12+\frac{1}{12} = 12.08333$$ The linear approximation for \(\sqrt{146}\) is \(\approx 12.08333\).

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Most popular questions from this chapter

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