Problem 1 - Calculus: One and Several Variables

Question

Find the Taylor polynomial \(P_{4}\) for the function \(f\) $$f(x)=x-\cos x$$

Step-by-Step Solution

Answer

The Taylor polynomial of degree 4 for the function \(f(x) = x - \cos x\) is \(P_{4}(x) = -1 + x + \frac{1}{2}x^2 - \frac{1}{24}x^4\).

Step 1: Find the derivatives of the function

Find the first four derivatives of the function \(f(x) = x - \cos x\). - \(f(x) = x - \cos x\) - \(f'(x) = 1 + \sin x\) - \(f''(x) = \cos x\) - \(f'''(x) = -\sin x\) - \(f^{(4)}(x) = -\cos x\)

Step 2: Evaluate the derivatives at a = 0

Evaluate the derivatives at the point a = 0. - \(f(0) = 0 - \cos(0) = -1\) - \(f'(0) = 1 + \sin(0) = 1\) - \(f''(0) = \cos(0) = 1\) - \(f'''(0) = -\sin(0) = 0\) - \(f^{(4)}(0) = -\cos(0) = -1\)

Step 3: Construct the Taylor polynomial using the Taylor series formula

Using the Taylor series formula and plugging in the values obtained from steps 1 and 2, we obtain the Taylor polynomial of degree 4: \[P_{4}(x) = \sum_{k=0}^4 \frac{f^{(k)}(0)}{k!}(x-0)^k\] \[P_{4}(x) = \frac{-1}{0!}x^{0} + \frac{1}{1!}x^{1} +\frac{1}{2!}x^{2} + \frac{0}{3!}x^{3} - \frac{1}{4!}x^{4}\] \[P_{4}(x) = -1 + x + \frac{1}{2}x^2 - \frac{1}{24}x^4\] So the Taylor polynomial of degree 4 for the function \(f(x) = x - \cos x\) is: \[P_{4}(x) = -1 + x + \frac{1}{2}x^2 - \frac{1}{24}x^4\]

Key Concepts

Derivatives

To build a Taylor polynomial, the first step is to find the derivatives of the function. A derivative indicates how a function changes as its input changes, revealing the function's rate of change. In our exercise, the function given is \(f(x) = x - \cos x\).

Here's how the process unfolds:

• The first derivative, \(f'(x) = 1 + \sin x\), shows the change of the original function.
• The second derivative, \(f''(x) = \cos x\), describes the rate of change of the first derivative.
• The third derivative, \(f'''(x) = -\sin x\), further examines the change in the second derivative.
• Finally, the fourth derivative, \(f^{(4)}(x) = -\cos x\), provides a deeper understanding of the behavior of the function.

Each derivative gives us valuable information needed to construct the Taylor series. The derivatives don't just provide insight; they are essential in calculating each term of the series.

Taylor Series

A Taylor series helps approximate complex functions using simple polynomial expressions. It's a fundamental concept for understanding how functions behave around specific points, and involves an infinite sum of terms calculated from the function's derivatives.

This series relies heavily on derivatives to create an approximation that represents the function. Each derivative is evaluated at a specific point, typically \(a = 0\). The formula for the Taylor series is: \[ \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k \] In our case:

• The polynomial is of degree 4, meaning we compute up to the fourth derivative.
• We're evaluating the function around \(a = 0\).

These elements are essential in communicating how close our polynomial gets to the function's actual output, providing an effective approximation over a range of values.

Function Evaluation

Function evaluation is about substituting a particular value into our function or its derivatives to calculate their specific results. This is vital in forming the Taylor polynomial, as it dictates the coefficients used in your polynomial expression.

For the given function \(f(x) = x - \cos x\), you evaluate its derivatives at \(x = 0\):

• \(f(0) = -1\)
• \(f'(0) = 1\)
• \(f''(0) = 1\)
• \(f'''(0) = 0\)
• \(f^{(4)}(0) = -1\)

Each derivative at \(x = 0\) gives rise to different coefficients for the Taylor series' terms. Remember, these coefficients determine the weight of each term in the polynomial, playing a crucial role in creating an accurate approximation.

Degree of Polynomial

The degree of a polynomial tells you the highest power of \(x\) featured in your polynomial expression. It's a critical aspect when constructing a Taylor polynomial, as it denotes how many derivatives you've computed and included in your series.

For our function \(f(x) = x - \cos x\), we developed a polynomial of degree 4. This involves calculating up to the fourth derivative and ensuring up to the \(x^4\) term is in the polynomial.

This degree is crucial because:

• A higher degree typically means a better approximation to the function over a wider range.
• However, it involves more calculations and can make polynomial solving more complex.

Choosing the degree balances precision with complexity, determining how close you want your polynomial to approximate the original function.