Problem 37 - Calculus

Question

(a) Find the Taylor polynomial approximation of degree 4 about \(x=0\) for the function \(f(x)=e^{x^{2}}\) (b) Compare this result to the Taylor polynomial approximation of degree 2 for the function \(f(x)=e^{x}\) about \(x=0 .\) What do you notice? (c) Use your observation in part (b) to write out the Taylor polynomial approximation of degree 20 for the function in part (a). (d) What is the Taylor polynomial approximation of degree 5 for the function \(f(x)=e^{-2 x} ?\)

Step-by-Step Solution

Answer

(a) Polynomial: \(1 + x^2 + \frac{1}{2}x^4\); (b) Pattern of even powers noted; (c) Degree 20: Follow even \(x\) terms; (d) Polynomial: \(1 - 2x + 2x^2 - \frac{4}{3}x^3 + \frac{2}{3}x^4 - \frac{1}{15}x^5\).

Step 1: Derivatives for Taylor Polynomial of Degree 4 for \(e^{x^2}\)

To find the Taylor polynomial of degree 4 for \(f(x) = e^{x^2}\) about \(x=0\), we need to calculate the derivatives of \(f(x)\) evaluated at \(x=0\). The general formula for the Taylor series is:\[P(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4\]First, calculate derivatives and evaluate at zero:- \(f(0) = e^{0} = 1\).- \(f'(x) = 2xe^{x^2}, \; f'(0) = 2 \cdot 0 = 0\).- \(f''(x) = (4x^2 + 2)e^{x^2}, \; f''(0) = 2\).- \(f'''(x) = (8x^3 + 12x)e^{x^2}, \; f'''(0) = 0\).- \(f^{(4)}(x) = (16x^4 + 48x^2 + 12)e^{x^2}, \; f^{(4)}(0) = 12\).

Step 2: Constructing the Taylor Polynomial for \(e^{x^2}\)

Substitute the evaluated derivatives back into the Taylor polynomial formula:\[P_4(x) = 1 + \frac{0}{1!}x + \frac{2}{2!}x^2 + \frac{0}{3!}x^3 + \frac{12}{4!}x^4\]Simplify:\[P_4(x) = 1 + x^2 + \frac{1}{2}x^4\]

Step 3: Taylor Polynomial for \(e^x\) of Degree 2

Now, find the Taylor polynomial approximation of degree 2 for \(g(x) = e^x\) about \(x=0\):The Taylor series for \(e^x\) is:\[g(x) = 1 + \frac{x}{1!} + \frac{x^2}{2!}\]So the degree 2 polynomial is:\[P_2(x) = 1 + x + \frac{x^2}{2}\]

Step 4: Comparing Results

Compare the two:- For \(f(x) = e^{x^2}\), the degree 4 polynomial is: \(1 + x^2 + \frac{1}{2}x^4\).- For \(e^x\), the degree 2 polynomial is: \(1 + x + \frac{x^2}{2}\).Notice the pattern: Terms of the form \(\frac{x^n}{n!}\) appear based on powers of \(x\).

Step 5: Taylor Polynomial for \(e^{x^2}\) of Degree 20

Using the pattern observed, construct the degree 20 polynomial for \(e^{x^2}\):Following the even powers:\[P_{20}(x) = 1 + x^2 + \frac{x^4}{2} + \frac{x^6}{3!} + \frac{x^8}{4!} + \cdots + \frac{x^{20}}{10!}\]

Step 6: Derivatives for Taylor Polynomial of Degree 5 for \(e^{-2x}\)

Find the Taylor polynomial of degree 5 for \(h(x) = e^{-2x}\) about \(x=0\):First, calculate derivatives:- \(h'(x) = -2e^{-2x}, \; h'(0) = -2\).- \(h''(x) = 4e^{-2x}, \; h''(0) = 4\).- \(h'''(x) = -8e^{-2x}, \; h'''(0) = -8\).- Continuing similarly, \(h^{(4)}(0) = 16\), \(h^{(5)}(0) = -32\).Form the series:\[P_5(x) = 1 - 2x + \frac{4}{2!}x^2 - \frac{8}{3!}x^3 + \frac{16}{4!}x^4 - \frac{32}{5!}x^5\]Simplify:\[P_5(x) = 1 - 2x + 2x^2 - \frac{4}{3}x^3 + \frac{2}{3}x^4 - \frac{1}{15}x^5\]

Key Concepts

Polynomial Approximation

Polynomial approximation is a mathematical technique used to approximate functions with polynomials. This is particularly useful for simplifying complex functions and making calculations easier. One popular method for polynomial approximation is the Taylor series expansion. The Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

In the original exercise, we dealt with polynomial approximation using Taylor polynomials of different degrees, which are finite sums of derivatives. For the function \( f(x) = e^{x^2} \), the Taylor polynomial of degree 4 is determined by taking several derivatives of the function, evaluating them at the center point (usually \( x=0 \)), and plugging them into the Taylor formula. This resulted in a polynomial, \( P_4(x) = 1 + x^2 + \frac{1}{2}x^4 \), which approximates \( f(x) \) near \( x=0 \).

• The degree of the polynomial controls how closely the approximation matches the function.
• Higher degree polynomials typically provide a more accurate approximation over a larger range.
• Each additional degree adds another term to the polynomial, refining its accuracy.

Understanding polynomial approximation helps in simplifying functions, predicting function behavior, and is a crucial tool in numerical methods and engineering.

Exponential Functions

Exponential functions are a class of functions characterized by a constant base raised to a variable exponent. The standard form is \( y = a^x \), with e (approximately 2.71828) being the most common base in mathematics due to its natural occurrence in growth and decay processes. The function \( f(x) = e^x \) is the quintessential exponential function, showcasing unique properties such as having the same derivative as itself.

In comparing the Taylor polynomial of \( f(x) = e^{x^2} \) to that of \( g(x) = e^x \), we noticed structural similarities. For \( e^x \), each term in its Taylor series expansion about \( x=0 \) has the form \( \frac{x^n}{n!} \), contributing to its precise approximation ability.

• Exponential functions model a wide variety of natural processes, including population growth and radioactive decay.
• They have unique properties like continuous growth at every point and a constant rate of proportional change.
• The natural exponential function, \( e^x \), is uniquely defined as the limit of \( (1 + \frac{1}{n})^n \) as \( n \to \infty \).

Grasping exponential function behavior is essential for calculations involving growth, interest rates, or any phenomenon exhibiting exponential changes.

Derivative Calculations

Derivatives measure how a function's output changes as its input changes, and they are fundamental in obtaining the terms of a Taylor polynomial. Calculating derivatives of a function at a specific point gives you insight into the function's behavior near that point.

In our exercise, we derived functions like \( f(x) = e^{x^2} \) and \( g(x) = e^{-2x} \) to construct their Taylor polynomials. Each derivative calculated contributes a term to the polynomial, enhancing the approximation.

For instance, to find the Taylor polynomial of \( e^{x^2} \) to degree 4, we calculated derivatives up to the fourth order:

• \( f'(x) = 2xe^{x^2} \), evaluated as \( f'(0) = 0 \).
• \( f''(x) = (4x^2 + 2)e^{x^2} \), evaluated as \( f''(0) = 2 \).

To form Taylor polynomials, each subsequent derivative provides more information for increasingly accurate approximations. Understanding derivatives and their calculations underpin much of calculus and applied mathematics. They allow for evaluating changes and optimizing functions in a proto-real-world setting.