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Chapter 4: Problem 21

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$1 / 203$$

Short Answer

Expert verified
Question: Use linear approximation to estimate the value of \(\frac{1}{203}\). Answer: Approximately \(0.00495\).

Step by step solution

01

Calculate the derivative of the function f(x)

First, we need to find the derivative of the function \(f(x) = \frac{1}{x}\). Using the power rule: $$f'(x) = -\frac{1}{x^2}$$
02

Choose an appropriate value for a

We want to choose an appropriate value for \(a\) close to 203, for which the function's value is easy to calculate. The closest and simplest value is \(a = 200\). Thus, we will use the tangent line at the point \((200, \frac{1}{200})\) to approximate the value of the function at \(x = 203\).
03

Calculate the equation of the tangent line

We can use the slope-point form of a line to find the equation of the tangent line to the graph of f(x) at the point \((200, \frac{1}{200})\): $$y - f(a) = f'(a)(x - a)$$ Plugging in the values for the function and its derivative as well as \(a = 200\): $$y - \frac{1}{200} = -\frac{1}{200^2}(x - 200)$$
04

Estimate by plugging in x = 203

Now we can estimate the value of \(\frac{1}{203}\) by plugging in \(x = 203\) into the equation of the tangent line: $$y = \frac{1}{200} + \frac{1}{200^2}(200 - 203)$$ $$y = \frac{1}{200} - \frac{3}{200^2} \approx 0.00495$$ So, using linear approximation, the estimated value for \(\frac{1}{203}\) is approximately \(0.00495\).

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