Question

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

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

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.

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.

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.

Key concepts

Taylor Series

The Taylor series is a fundamental concept in calculus, used to approximate functions with polynomials. It essentially allows you to rewrite a function as an infinite sum of terms calculated from the derivatives of the function at a single point. The general form of a Taylor series expansion for a function \( f(x) \) about a point \( a \) is given by:

• \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \)

This expansion helps simplify the complexity of evaluating functions by converting them into polynomials, which are easier to manipulate.
When a Taylor series is centered at zero, it is often referred to as a Maclaurin series. Taylor series are incredibly powerful in both theoretical and practical contexts, particularly for solving complex differential equations or analyzing systems where only local information is available. By leveraging a few terms from the Taylor series, we can precisely approximate the value of a function near the point \( a \). This makes Taylor series a versatile tool in mathematical and engineering applications.

Approximation Methods

Approximation methods in mathematics are techniques used to find values that are close to the exact solution, which may be difficult or impossible to obtain directly. These methods are particularly useful when dealing with functions that do not have closed-form solutions or when computational simplicity is desired. One common example is using series expansions, such as the Taylor series, for approximation.

• By expanding a function into a series, we truncate it to a finite number of terms to get an approximate value.
• This process simplifies the calculation and makes it feasible to analyze complex mathematical expressions.

In practical applications, the number of terms retained in the expansion depends on the desired precision. The more terms we include, the more accurate the approximation but also more computational resources are required. Approximations find wide applications in engineering, physics, and economics, where they provide a manageable way to work with complex models or systems.

Mathematical Expressions Evaluation

Evaluating mathematical expressions involves computing the exact or approximate value of the expression at specific points. This process is crucial in various fields like statistics, physics, and engineering, where the behavior of a function at certain values provides valuable insights. In practice, students use calculators or computer software to perform this evaluation, especially when the expressions involve complicated operations or transcendental functions.

• The first step in this process is substituting the variable values into the expression.
• For more precision, numerical methods or series expansions are often utilized.

By using series expansions, we convert complex mathematical expressions into polynomials. This conversion simplifies the evaluation process, providing a close approximation to the true value with minimal computational effort. This method is particularly effective for small values near the expansion point, where the approximated series closely follows the actual function.