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Chapter 10: Problem 21

using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$e^{t} \cos t$$

Short Answer

Expert verified
The first four nonzero terms are: 1, \(t\), \(-\frac{t^2}{2}\), and \(\frac{t^3}{6}\).

Step by step solution

01

Understand the Known Taylor Series

The Taylor series for both \(e^t\) and \(\cos t\) about 0 are known. The Taylor series for \(e^t\) is \(1 + t + \frac{t^2}{2!} + \frac{t^3}{3!} + \cdots \) and the Taylor series for \(\cos t\) is \(1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \cdots\).
02

Multiply the Series

To find the Taylor series for \(e^t \cos t\), multiply the two known series by aligning the series terms according to increasing powers of \(t\), just like you would multiply polynomials.
03

Combine Like Terms

Compute the product term-by-term, gathering terms for each power of \(t\). Here are the first few terms calculated:- Constant term: \(1 \cdot 1 = 1\)- Term with \(t^1\): \(t \cdot 1 = t\)- Term with \(t^2\): \(\frac{t^2}{2!} \cdot 1 - t \cdot \frac{t^2}{2!} = -\frac{t^2}{2}\)- Term with \(t^3\): \(\frac{t^3}{3!} \cdot 1 = \frac{t^3}{6}\)
04

Verify Four Nonzero Terms

Verify the above-calculated terms are indeed correctly computed and check if there are more non-zero terms beyond the \(t^3\) term to gather the first four nonzero terms. The terms are simply copied and calculated correctly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Multiplication
When you multiply two series, such as the Taylor series of two functions, the process is similar to multiplying polynomials. Each term in one series is multiplied by every term in the other series.
This involves pairing the terms from both series and summing the results.

Here's a simple approach to visualize it:
  • Align the series with increasing powers of the variable (in this case, "t").
  • Start by multiplying the constant terms first, and gradually move to higher powers.
  • Remember to combine like terms after multiplying, which means summing up the coefficients for each power of the variable.
This technique allows you to compute a new series which represents the product of the original functions in their Taylor-expanded form.
Polynomial Approximation
Polynomial approximation is a powerful tool in mathematics that allows complex functions to be represented as simple polynomials.
Using Taylor series, we can approximate a function using the sum of its derivatives at a specific point.

This approximation can be very useful:
  • It provides a way to approximate difficult functions with polynomials that are easier to handle.
  • The approximation becomes more precise as more terms are added to the series.
  • In cases where functions are too complex for standard calculations, polynomial approximation offers a practical alternative.
For example, the Taylor series for the exponential function or trigonometric functions can be used to estimate values with high accuracy just by using a few terms.
Exponential Function
The exponential function, often denoted as \(e^t\), is a fundamental mathematical function representing exponential growth or decay.
It has a continuous growth rate and is defined by the infinite series expansion \(e^t = 1 + t + \frac{t^2}{2!} + \frac{t^3}{3!} + \ldots\).

The exponential function has several critical properties:
  • It's the only function whose rate of change is proportional to its value.
  • It's used extensively in calculus, physics, and financial mathematics.
  • The elegance of its Taylor series allows it to be used in conjunction with other functions, such as trigonometric functions, for complex analyses.
The convergence of its Taylor series is an excellent way to approximate it over the whole real number line.
Trigonometric Function
Trigonometric functions like cosine and sine are periodic functions commonly used to model oscillatory phenomena.
The Taylor series expansion for cosine \(\cos t\) is given by \(1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \cdots\), allowing us to approximate it using a few terms.

These series expansions are essential for numerous applications:
  • They help approximate solutions to differential equations by simplifying complex trigonometric expressions.
  • Engineers and scientists use them to model waves and harmonic motions.
  • These polynomial approximations can simplify computational tasks, making algorithms faster and more efficient.
The simplicity of Taylor series for trigonometric functions makes them a practical choice for many theoretical and applied mathematics fields.

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Most popular questions from this chapter

The theory of relativity predicts that when an object moves at speeds close to the speed of light, the object appears heavier. The apparent, or relativistic, mass, \(m\) of the object when it is moving at speed \(v\) is given by the formula $$ m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}} $$ where \(c\) is the speed of light and \(m_{0}\) is the mass of the object when it is at rest. (a) Use the formula for \(m\) to decide what values of \(v\) are possible. (b) Sketch a rough graph of \(m\) against \(v,\) labeling intercepts and asymptotes. (c) Write the first three nonzero terms of the Taylor series for \(m\) in terms of \(v\) (d) For what values of \(v\) do you expect the series to converge?

If \(f(2)=g(2)=h(2)=0,\) and \(f^{\prime}(2)=h^{\prime}(2)=0\) \(g^{\prime}(2)=22,\) and \(f^{\prime \prime}(2)=3, g^{\prime \prime}(2)=5, h^{\prime \prime}(2)=7\) calculate the following limits. Explain your reasoning. (a) \(\lim _{x \rightarrow 2} \frac{f(x)}{h(x)}\) (b) \(\lim _{x \rightarrow 2} \frac{f(x)}{g(x)}\)

Find the \(n^{\text {th }}\). Fourier polynomial for the given functions, assuming them to be periodic with period \(2 \pi\) Graph the first three approximations with the original function. $$g(x)=x, \quad-\pi

Pad茅 approximants are rational functions used to approximate more complicated functions. In this problem, you will derive the Pad茅 approximant to the exponential function. (a) Let \(f(x)=(1+a x) /(1+b x),\) where \(a\) and \(b\) are constants. Write down the first three terms of the Taylor series for \(f(x)\) about \(x=0\) (b) By equating the first three terms of the Taylor series about \(x=0\) for \(f(x)\) and for \(e^{x},\) find \(a\) and \(b\) so that \(f(x)\) approximates \(e^{x}\) as closely as possible near \(x=0\)

Decide if the statements in Problems are true or false. Give an explanation for your answer. The linear approximation to \(f(x)\) near \(x=-1\) shows that if \(f(-1)=g(-1)\) and \(f^{\prime}(-1) < g^{\prime}(-1),\) then \(f(x) < g(x)\) for all \(x\) sufficiently close to -1 (but not equal to -1 ).
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