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Chapter 4: Problem 27

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$e^{0.06}$$

Short Answer

Expert verified
Question: Estimate the value of $$e^{0.06}$$ using linear approximations. Answer: Approximately 1.06

Step by step solution

01

Identify the function and its derivative

We are working with the exponential function: $$f(x) = e^x$$. The derivative of the exponential function is: $$f'(x) = e^x$$.
02

Choose an appropriate value for 'a'

Since $$x = 0.06$$ and it's a small value, let's choose $$a = 0$$ as it is close to $$0.06$$ and easy to work with.
03

Find the value of the function and its derivative at a

Now we need to find the values of the function and its derivative at $$a = 0$$: $$f(a)=e^0=1$$ $$f'(a)=e^0=1$$
04

Apply the linear approximation formula

Using the linear approximation formula, we have: $$f(x) \approx f(a) + f'(a)(x-a)$$ So, $$e^{0.06} \approx 1 + 1(0.06 - 0)$$
05

Calculate the approximation

Now, calculate the approximation: $$e^{0.06} \approx 1+ 0.06$$ $$e^{0.06} \approx 1.06$$ Hence, using linear approximation, we go on to estimate that $$e^{0.06}$$ is approximately $$1.06$$.

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