Problem 25 - Thomas Calculus

Question

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=x^{3}-2 x+4, \quad a=2\)

Step-by-Step Solution

Answer

The Taylor series is \(T(x) = 4 + 10(x-2) + 6(x-2)^2 + (x-2)^3\).

Step 1: Understand the Taylor Series

The Taylor series of a function \(f(x)\) at \(x = a\) is given by the formula: \[T(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots\]It represents the function as an infinite sum of terms calculated from the values of its derivatives at \(a\).

Step 2: Calculate the Derivatives of \(f(x)\)

Given the function \(f(x) = x^3 - 2x + 4\):- First derivative: \(f'(x) = 3x^2 - 2\)- Second derivative: \(f''(x) = 6x\)- Third derivative: \(f'''(x) = 6\)- Fourth derivative and higher: \(f^{(n)}(x) = 0\) where \(n\geq 4\)

Step 3: Evaluate the Derivatives at \(a = 2\)

Now, evaluate each derivative at \(x = 2\):- \(f(2) = 2^3 - 2\times2 + 4 = 4\)- \(f'(2) = 3(2)^2 - 2 = 10\)- \(f''(2) = 6\times2 = 12\)- \(f'''(2) = 6\)These values will be used to construct the Taylor series.

Step 4: Construct the Taylor Series

Substitute the values into the Taylor series formula:\[T(x) = 4 + 10(x - 2) + \frac{12}{2!}(x - 2)^2 + \frac{6}{3!}(x - 2)^3\]Simplify the terms:\[T(x) = 4 + 10(x - 2) + 6(x - 2)^2 + 1(x - 2)^3\]

Key Concepts

Derivatives

Derivatives are fundamental tools in calculus that measure how a function changes as its input changes. Simply put, the derivative tells us the slope of the tangent line to the function at any given point. For the function \(f(x) = x^3 - 2x + 4\), the derivatives give us insights into its local behavior.

• The first derivative \(f'(x) = 3x^2 - 2\) signifies the rate of change of the function itself. It helps determine where the function is increasing or decreasing.
• The second derivative \(f''(x) = 6x\) tells us about the curvature or concavity of the function. Positive values indicate that the function is concave up, while negative values indicate a concave down shape.
• The third derivative \(f'''(x) = 6\) is constant, suggesting uniform changes in curvature and providing insights into the "jerk" or the rate at which acceleration is changing.

In the context of a Taylor series, each derivative evaluated at a specific point \(a\) is used to form the series' terms. This allows a function to be expressed as a polynomial around the point, offering a polynomial approximation.

Infinite Series

An infinite series refers to the sum of infinitely many terms. In calculus, infinite series are used to represent complex functions as sums of simpler terms. The Taylor series is a specific type of infinite series that represents a function in terms of its derivatives at a single point.

The general formula for the Taylor series is \(f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots\). Each term in this series adds more precision to the representation of the original function.

• The first term \(f(a)\) captures the value of the function at \(a\).
• Subsequent terms involve derivatives and powers of \((x-a)\), providing a way to capture behavior around \(a\).

Instead of using all infinitely many terms, we often use a finite number of them for practical approximations. This creates a polynomial expression that approximates the function around \(x = a\). Such approximations are particularly useful in calculations involving complex functions.

Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents. In mathematics, polynomials are easy to differentiate, integrate, and evaluate, making them useful in various applications, including approximations with Taylor series.

When we approximate a function using a Taylor series, we express it as a polynomial. The order of this polynomial (i.e., linear, quadratic, cubic, etc.) depends on how many derivative terms are included. The function \(f(x) = x^3 - 2x + 4\) is itself a polynomial of degree three.

• Linear polynomials have the form \(ax + b\).
• Quadratic polynomials take the shape \(ax^2 + bx + c\).
• Cubic polynomials are given by \(ax^3 + bx^2 + cx + d\), containing terms up to degree three.

The Taylor series derived in the solution gives us a polynomial that approximates \(f(x)\) around a specific point \(a\). Here, by including up to the third derivative, we ensure a cubic approximation that closely mirrors the changes in \(f(x)\) near \(x = 2\). This is useful in simplifying calculations and understanding the function's local behavior.