Problem 1 - Basic Analysis: Introduction to Real Analysis

Question

Compute the nth Taylor Polynomial at 0 for the exponential function.

Step-by-Step Solution

Answer

The nth Taylor Polynomial for \( e^x \) at 0 is \( T_n(x) = \sum_{k=0}^{n} \frac{1}{k!} x^k \).

Step 1: Understand the Function

We want to find the Taylor polynomial of the exponential function, which is given by \( f(x) = e^x \). This function is special because the exponential function is equal to its own derivative.

Step 2: Taylor Series Formula

The Taylor series of a function \( f(x) \) at \( a = 0 \) is given by the formula:\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]where \( f^{(n)}(0) \) represents the \( n \)-th derivative of \( f(x) \) evaluated at \( x=0 \).

Step 3: Derivative of the Exponential Function

For \( f(x) = e^x \), each derivative is the same as the original function. This means:\[ f^{(n)}(x) = e^x \]Evaluating this at \( x = 0 \) gives \( f^{(n)}(0) = e^0 = 1 \) for all \( n \).

Step 4: Construct the nth Taylor Polynomial

Substitute the derivatives into the Taylor series formula:\[ T_n(x) = \sum_{k=0}^{n} \frac{1}{k!} x^k \]This gives us the nth Taylor polynomial of the exponential function centered at 0.

Key Concepts

Exponential Function

The exponential function, commonly represented as \( f(x) = e^x \), is a unique and fundamental function in mathematics. Its key property is that its rate of growth (derivative) is always proportional to its current value. This means

• The exponential function is equal to its own derivative.
• Therefore, the function itself is unchanged by differentiation.

This powerful property allows it to model a wide range of phenomena such as growth processes in nature, financial calculations, and even complex systems like population dynamics. The base of the exponential function \( e \) is an irrational number approximately equal to 2.71828, and it is known as Euler's number. This special function always yields positive values, and it increases rapidly as the input increases, demonstrating a steady and constant growth rate.

Taylor Series

A Taylor series is an infinite sum of terms that are derived from the values of a function's derivatives at a single point. It's used to approximate functions with polynomials, making complex functions easier to work with. For any function \( f(x) \), the Taylor series centered at 0 is represented as

• The formula: \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
• Each term of the series is determined by the nth derivative of \( f(x) \) evaluated at 0, divided by \( n! \), with \( x^n \) as a multiplying factor.

Using the Taylor series, functions can be expressed as sums of an infinite number of simple polynomial terms, which is extremely useful in approximation problems.

Derivative

The derivative of a function is a core concept in calculus that describes the rate at which a function is changing at any given point. For the exponential function \( f(x) = e^x \), the derivative holds the incredible property of being the same as the original function. This can be stated mathematically as

• \( f'(x) = e^x \)
• Higher order derivatives also yield the same expression, \( f^{(n)}(x) = e^x \), for all orders \( n \).

When these derivatives are evaluated at \( x = 0 \), each value is 1. This means

• Each coefficient in the Taylor polynomial \( T_n(x) = \sum_{k=0}^{n} \frac{1}{k!} x^k \) for the exponential function is simply \( \frac{1}{k!} \).

Understanding derivatives and their behavior for different functions is crucial in applications that involve varying rates, such as physics, engineering, and economics.