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Chapter 4: Problem 26

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$\sqrt{5 / 29}$$

Short Answer

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Question: Use linear approximations to estimate the value of \(\sqrt{5 / 29}\). Answer: The linear approximation of \(\sqrt{5 / 29}\) is approximately 0.385.

Step by step solution

01

Choose a suitable value for \(a\)

Since we are looking for a square root of a fraction, we will choose a value of \(a\) such that its square root is easy to compute and that it is close to the given value. We consider the following: $$a = \frac{4}{25}$$ Since \(\sqrt{a} = \sqrt{\frac{4}{25}} = \frac{2}{5}\), which is easy to compute. Now, we will use the function \(f(x) = \sqrt{x}\) to compute the linear approximation.
02

Calculate the derivative of the function

Compute the derivative of \(f(x) = \sqrt{x}\): $$f'(x) = \frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}}$$
03

Find the tangent line equation for the chosen value of \(a\)

Using \(f(a) = \sqrt{\frac{4}{25}} = \frac{2}{5}\) and the derivative at \(a\), we have: $$f'(\frac{4}{25}) = \frac{1}{2\sqrt{\frac{4}{25}}} = \frac{1}{2\frac{2}{5}} = \frac{1}{\frac{4}{5}} = \frac{5}{4}$$ Now, we can write the tangent line equation: $$L(x) = f(a) + f'(a)(x - a) = \frac{2}{5} + \frac{5}{4}(x - \frac{4}{25})$$
04

Use the tangent line equation to approximate the desired value

Now, use the tangent line equation to estimate \(\sqrt{\frac{5}{29}}\). Let \(x = \frac{5}{29}\): $$L(\frac{5}{29}) = \frac{2}{5} + \frac{5}{4}(\frac{5}{29} - \frac{4}{25}) = \frac{2}{5} + \frac{5}{4}(\frac{1}{29} + \frac{1}{25})$$ Compute the approximation: $$L(\frac{5}{29}) \approx \frac{2}{5} + \frac{5}{4}(\frac{1}{29} + \frac{1}{25}) \approx 0.385$$ So, the linear approximation of \(\sqrt{\frac{5}{29}}\) using the chosen value of \(a\) is approximately 0.385.

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Most popular questions from this chapter

Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function \(D(p)=500-10 p\) says that at a price of \(p=10,\) a quantity of \(D(10)=400\) units of the commodity can be sold. The elasticity \(E=\frac{d D}{d p} \frac{p}{D}\) of the demand gives the approximate percent change in the demand for every \(1 \%\) change in the price. a. Compute the elasticity of the demand function \(D(p)=500-10 p\) b. If the price is \(\$ 12\) and increases by \(4.5 \%,\) what is the approximate percent change in the demand? c. Show that for the linear demand function \(D(p)=a-b p\) where \(a\) and \(b\) are positive real numbers, the elasticity is a decreasing function, for \(p \geq 0\) and \(p \neq a / b\) d. Show that the demand function \(D(p)=a / p^{b}\), where \(a\) and \(b\) are positive real numbers, has a constant elasticity for all positive prices.

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