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Chapter 30: Problem 20

Find the first three nonzero terms of the Maclaurin expansion of the given functions. $$f(x)=(2 x-1)^{2}$$

Short Answer

Expert verified
The first three nonzero terms are \( 1 - 4x + 4x^2 \).

Step by step solution

01

Identify the function and Maclaurin Series

The Maclaurin series is a specific case of the Taylor series expansion around 0. We need the first three nonzero terms of the series for the function \( f(x) = (2x-1)^2 \).
02

Express the function in terms of Maclaurin series

First, note that according to the binomial expansion, \( (2x - 1)^2 = 1 - 4x + 4x^2 \). Since this is a quadratic polynomial, it can also be considered as its own Maclaurin series.
03

Verify the first few terms

We expand \( (2x - 1)^2 = 1 - 4x + 4x^2 \) into the form of a power series about x=0 and identify the terms as such: the constant term is 1, the coefficient of the linear term \( x \) is \(-4\), and the coefficient of the quadratic term \( x^2 \) is \(4\).
04

Concatenate the nonzero terms

Since there are no higher-order terms needed for this function (due to its quadratic nature), the first three nonzero terms of the Maclaurin expansion are directly from the expression: \( 1 - 4x + 4x^2 \).

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Key Concepts

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

Understanding Binomial Expansion
The binomial expansion is a powerful tool in mathematics that allows us to expand expressions involving two terms raised to a power, such as \((a + b)^n\).This concept simplifies calculations and is widely applied in various mathematical problems, including calculus and algebra.For \((2x - 1)^2\), applying the binomial expansion simplifies the expression by following the formula: \((a - b)^2 = a^2 - 2ab + b^2\).
  • Identify the terms: here, \(a = 2x\) and \(b = 1\).
  • Calculate the expanded form: \((2x)^2 - 2(2x)(1) + 1^2\).
  • After simplification, we get: \((2x - 1)^2 = 4x^2 - 4x + 1\).
By using this approach, you can break down and simplify even complex expressions easily.
Exploring Power Series
A power series is an infinite series of the form \(a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots\).It allows representation of functions in a way that expresses them as sums of powers of \(x\).
In our specific function \(f(x) = (2x - 1)^2\), the first three terms of its power series expansion about \(x=0\) provide the approximation as \(1 - 4x + 4x^2\).This shows:
  • The constant term, \(a_0 = 1\).
  • The linear term, \(a_1 = -4\).
  • The quadratic term, \(a_2 = 4\).
Power series expansions are helpful because they can approximate functions by considering only finite terms, making them easier to work with for practical computations.
Quadratic Polynomials Demystified
Quadratic polynomials are algebraic expressions of the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a 鈮� 0\).This is essentially a polynomial of degree 2.For example, in the expression \((2x - 1)^2\), the function simplifies to \(4x^2 - 4x + 1\).
  • Here, \(a = 4\),
  • \(b = -4\),
  • and \(c = 1\).
Quadratic polynomials have various properties:
  • They graph as a parabola.
  • They have at most two roots.
  • The vertex form reveals maximum or minimum points.
Understanding these properties helps in many areas of math, including optimization problems, projectile motion, and more.
The Magic of Taylor Series
The Taylor series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. When the expansion is taken around \(x = 0\), it's called the Maclaurin series.
The general Taylor series formula around a point \(a\) is:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \]For our function \(f(x) = (2x-1)^2\), finding the Maclaurin series, which is simply the Taylor series about 0, gives us: \(1 - 4x + 4x^2\).
  • The constant term corresponds to \(f(a)\).
  • The linear term to \(f'(a)\).
  • The quadratic term to \(\frac{f''(a)}{2!}\).
The Taylor series can approximate functions to a very high degree of accuracy with sufficient terms, and serves as a fundamental tool in calculus and analysis.

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Most popular questions from this chapter

Determine the convergence or divergence of the given sequence. If \(a_{n}\) is the term of a sequence and \(f(x)\) exists for \(x \geq 1,\) then \(\lim _{x \rightarrow \infty} f(x)=L\) means \(a_{n} \rightarrow L\) as \(n \rightarrow \infty .\) This lets us analyze convergence or divergence by using the equivalent continuous function. Therefore, if applicable, L'Hospital's rule may be used. $$a_{n}=2+\frac{2}{n}$$

Test each of the given geometric series for convergence or divergence. Find the sum of each series that is convergent. $$4+1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots$$

Find the indicated series by the given operation. Find the first three terms of the expansion for \(\ln (1+\sin x)\) by using the expansions for \(\ln (1+x)\) and \(\sin x\)

Solve the given problems.A pulsating electric current \(i\) (in mA) with a period of 2 s can be described by \(i=e^{-t}\) for \(-1 \leq t<1\) s for one period (only positive values of \(t\) have physical significance). Find the Fourier seriec that represents this current.

Find the indicated series by the given operation. Find the first four nonzero terms of the expansion of the function \(f(x)=\frac{1}{2}\left(e^{x}-e^{-x}\right)\) by subtracting the terms of the appropriate series. The result is the series for \(\sinh x\). (See Exercise 55 of Section \(27.6 .)\)
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