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Magazine Chapter 9: Problem 23
Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). $$ 1+x^{2}+x^{3}, a=1 $$
Short Answer
Expert verified
The Taylor series is \(3 + 5(x-1) + 4(x-1)^2 + 1(x-1)^3\).
Step by step solution
01
Understand the Problem
We are tasked with finding the Taylor series expansion of the given function \(f(x) = 1 + x^2 + x^3\) around the point \(a = 1\) up to \((x - a)^3\). The Taylor series formula is \(f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots\).
02
Compute the Function Value at a
First, evaluate the function at \(a = 1\):\[f(1) = 1 + 1^2 + 1^3 = 3.\]This gives the first term of the series.
03
Find the First Derivative and Evaluate at a
The first derivative of the function \(f(x) = 1 + x^2 + x^3\) is:\[f'(x) = 2x + 3x^2.\]Evaluate this at \(x = 1\):\[f'(1) = 2 \times 1 + 3 \times 1^2 = 5.\]The second term of the series is \(f'(1)(x-1) = 5(x-1)\).
04
Find the Second Derivative and Evaluate at a
Differentiate \(f'(x) = 2x + 3x^2\) to find the second derivative:\[f''(x) = 2 + 6x.\]Evaluate at \(x = 1\):\[f''(1) = 2 + 6 \times 1 = 8.\]The third term of the series is \(\frac{8}{2!}(x-1)^2 = 4(x-1)^2\).
05
Calculate the Third Derivative and Evaluate at a
Differentiate \(f''(x) = 2 + 6x\) for the third derivative:\[f'''(x) = 6.\]Evaluate at \(x = 1\):\[f'''(1) = 6.\]The fourth term of the series is \(\frac{6}{3!}(x-1)^3 = 1(x-1)^3\).
06
Write the Taylor Series
Combine the terms calculated in the previous steps to write the Taylor series expansion:\[f(x) = 3 + 5(x-1) + 4(x-1)^2 + 1(x-1)^3.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivatives
Derivatives are a cornerstone in calculus, representing the rate of change of a function with respect to a variable. In layman's terms, the derivative tells you how a function's output changes as its input changes. Learning to compute derivatives is key for understanding how functions behave. In the original exercise, the function given is a simple polynomial:
- The first derivative, denoted as \( f'(x) \), reflects how the function is changing at any point \( x \). For the function \( f(x) = 1 + x^2 + x^3 \), the first derivative is \( 2x + 3x^2 \).
- The second derivative, \( f''(x) \), shows how the rate of change itself is changing, calculated as \( 2 + 6x \).
- The third derivative, \( f'''(x) = 6 \), at any point gives the changes in the rate of acceleration of the function.
Polynomial Functions
Polynomial functions like \( f(x) = 1 + x^2 + x^3 \) are algebraic expressions consisting of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is crucial for creating Taylor series, as the series itself becomes a new polynomial. In the original problem:
- The polynomial \( 1 + x^2 + x^3 \) is expanded around \( a = 1 \), translating it into a simpler form.
- The aim is to express this polynomial in terms of \( (x-a) \) up to the third degree, resulting in terms like \( (x-1) \) and \( (x-1)^2 \).
Mathematical Analysis
Mathematical analysis involves rigorous reasoning to study change and motion, predominantly using concepts like limits, continuity, and series expansions. The Taylor series is a fundamental tool in this analysis as it transforms complex functions into polynomial approximations, simplifying calculus problems. In mathematical analysis:
- The Taylor series representation \( f(x) = 3 + 5(x-1) + 4(x-1)^2 + 1(x-1)^3 \) serves to approximate \( f(x) = 1 + x^2 + x^3 \) near \( a = 1 \).
- This approximated polynomial makes it easier to study the behavior and properties of functions.
- With higher order derivatives, refinements in the approximation are possible up to the desired degree of accuracy, which helps in practical computation.
