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Chapter 1: Problem 20

Evaluate the following expressions at the given point. Use your calculator or your computer (such as Maple). Then use series expansions to find an approximation of the value of the expression to as many places as you trust. a. \(\frac{1}{\sqrt{1+x^{3}}}-\cos x^{2}\) at \(x=0.015\). b. \(\ln \sqrt{\frac{1+x}{1-x}}-\tan x\) at \(x=0.0015\). c. \(f(x)=\frac{1}{\sqrt{1+2 x^{2}}}-1+x^{2}\) at \(x=5.00 \times 10^{-3}\). d. \(f(R, h)=R-\sqrt{R^{2}+h^{2}}\) for \(R=1.374 \times 10^{3} \mathrm{~km}\) and \(h=1.00 \mathrm{~m}\). e. \(f(x)=1-\frac{1}{\sqrt{1-x}}\) for \(x=2.5 \times 10^{-13}\).

Short Answer

Expert verified
This is a calculative task. Approximations can be computed using series expansions, where the terms need to be computed according to the required precision, and then in each expression, the given variables need to be substituted with the provided numbers. The short answer is the set of approximated values obtained from each expression.

Step by step solution

01

Computing the exact values.

The first step involves evaluating the expressions at the specified points using a calculator or an appropriate software. This is done by substituting the given values of the variables in the expressions. This process is repeated for every single expression given.
02

Computing the series expansions.

The next step is computing the series expansions. Using the principles of calculus, calculate the Taylor or Maclaurin series for the expressions around the point at which the approximation is to be made. The expansion should be made up to a sufficient number of terms to achieve the precision specified in the question. If no precision is given, a common practice is to expand up to the term with the power of 5.
03

Using the series expansions for approximation.

Now, use the series expansion obtained in Step 2 to approximate the expressions. Substitute the given values of the variables into the series expansion. The result will give a close approximation of the original expression at the point in consideration.

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Most popular questions from this chapter

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In the event that a series converges uniformly, one can consider the derivative of the series to arrive at the summation of other infinite series. a. Differentiate the series representation for \(f(x)=\frac{1}{1-x}\) to sum the series \(\sum_{n=1}^{\infty} n x^{n},|x|<1\) b. Use the result from part a to sum the series \(\sum_{n=1}^{\infty} \frac{n}{5^{n}}\) c. Sum the series \(\sum_{n=2}^{\infty} n(n-1) x^{n},|x|<1\) d. Use the result from part \(c\) to sum the series \(\sum_{n=2}^{\infty} \frac{n^{2}-n}{5^{n}}\) e. Use the results from this problem to sum the series \(\sum_{n=4}^{\infty} \frac{n^{2}}{5^{n}}\)
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