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Chapter 3: Problem 7

Estimate the indicated value without using a calculator. \(e^{0.0013}\)

Short Answer

Expert verified
The estimated value of \(e^{0.0013}\) without using a calculator is approximately 1.0013.

Step by step solution

01

Approximate the value of e

The number e is an important mathematical constant, approximately equal to 2.718281828459045. Since we cannot use a calculator, we will use an approximation for e. A common approximation is: \(e \approx 2.72\). We will use this approximation in the following steps.
02

Use the power series for exponential function

To estimate the value of \(e^{0.0013}\), we can use the Taylor series of \(e^x\), which is: \[e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...\] Since our exponent is quite small (0.0013), we can truncate the series after the second or third term and still get a reasonably accurate estimation.
03

Estimate the value with the truncated series

Using the truncated Taylor series and approximating e, let's estimate the given value: \[e^{0.0013} \approx 1 + (0.0013) + \frac{(0.0013)^2}{2!}\] By doing the arithmetic: \[e^{0.0013} \approx 1 + 0.0013 + \frac{0.00000169}{2}\] \[e^{0.0013} \approx 1 + 0.0013 + 0.000000845\] \[e^{0.0013} \approx 1.001300845\] So, the estimated value of \(e^{0.0013}\) without using a calculator is approximately 1.0013.

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Most popular questions from this chapter

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=5 e^{9 x}\)

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