Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=(1+x)^{5}, \quad b=0 $$

Step-by-Step Solution

Answer

The Taylor series for \(f(x) = (1+x)^5\) centered at \(0\) is \(f(x) = 1 + 5x + 10x^2 + 10x^3 + ...\).

Step 1: Understanding the Taylor Series Formula

The formula for a Taylor series centered at \(b\) is given by \(f(b) + f'(b)(x-b) + \frac{f''(b)(x-b)^2}{2!} + \frac{f'''(b)(x-b)^3}{3!} + ...\). Since \(b=0\), this simplifies to \(f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + ...\)

Step 2: Calculating Derivatives

We need to calculate the function and its derivatives at \(0\). \n - The function at \(0\) is \(f(0)= (1+0)^5=1\). - The first derivative \(f'(x) = 5(1+x)^4\), so \(f'(0) = 5\). - The second derivative \(f''(x) = 5*4(1+x)^3 = 20(1+x)^3\), so \(f''(0) = 20\). - The third derivative \(f'''(x) = 20*3(1+x)^2 = 60(1+x)^2\), so \(f'''(0) = 60\). - We observe a pattern: the \(n\)th derivative at \(0\) is \(n!\), where \(n!\) denotes the factorial of \(n\).

Step 3: Formulating the Taylor Series

Plugging in these values into the Taylor series formula, we have: \(f(x) = 1 + 5x + \frac{20x^2}{2!} + \frac{60x^3}{3!} + .... Normalize all fractions and remember the pattern we found in step 2: \(f(x) = 1 + 5x + 10x^2 + 10x^3 + ... \).

Key Concepts

Calculus

Calculus is an advanced branch of mathematics that deals with change and motion. It is divided mainly into two areas: differential calculus, concerned with the rate of change of quantities, and integral calculus, which focuses on the accumulation of quantities and the areas under and between curves.

In the context of our exercise, differential calculus plays a central role since we deal with derivatives when finding a Taylor series expansion. The ability to break down functions into their derivatives is a fundamental skill in calculus and provides the foundation for understanding series expansions.

Series Expansion

Series expansion is a concept in mathematics whereby a function is represented as a sum of terms, called a series. For more complex functions, a series can be an approximation method that allows us to calculate values and behaviors of functions near certain points.

The Taylor series is a type of series expansion that represents a function as an infinite sum of terms calculated from its derivatives at a single point. It provides a powerful tool for approximating functions with polynomials that can be as accurate as needed by including more terms in the series.

Derivatives

Derivatives represent the rate at which a function is changing at any given point. In simple terms, it's the slope of the function's graph at a specific point. Derivatives are crucial when working with series expansions because they determine the coefficients for each term of the series.

In our exercise example, we calculated successive derivatives of the function \( f(x) = (1+x)^5 \) at \( x = 0 \). This process is vital to establish the pattern and coefficients that will be used in the Taylor series expansion.

Factorial

The factorial, denoted by an exclamation mark, is a function that multiplies a number by all positive integers less than itself. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It appears frequently in mathematics, particularly in combinatorics and series expansions.

In the step-by-step solution to the Taylor series, we observed that the pattern of derivatives leads us to use the factorial in the denominator. This not only simplifies the representation of the series but also plays a significant role in determining the value of each term in the Taylor series expansion.