Problem 15 - Calculus for Biology and Medicine

Question

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\tan x, n=2, x=0.1 $$

Step-by-Step Solution

Answer

The Taylor polynomial \(P_2(x)\) is \(x\), and at \(x=0.1\), it approximates \(\tan(0.1)\) well since both values are close: 0.1 and 0.10033.

Step 1: Recall the Taylor Polynomial Formula

The Taylor polynomial of degree \(n\) for a function \(f(x)\) about \(x=0\) is given by:\[P_n(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \cdots + \frac{f^{(n)}(0)x^n}{n!}\]Since \(n=2\), we need to calculate up to the second derivative of \(f(x)\).

Step 2: Calculate the Derivatives at Zero

Let's compute the derivatives of \(f(x) = \tan(x)\) and evaluate them at \(x = 0\).- \(f(x) = \tan(x)\), so \(f(0) = \tan(0) = 0\).- \(f'(x) = \sec^2(x)\), so \(f'(0) = \sec^2(0) = 1\).- \(f''(x) = 2\sec^2(x)\tan(x)\), so \(f''(0) = 2\cdot 1 \cdot 0 = 0\).

Step 3: Form the Second-Degree Taylor Polynomial

Substitute the derivatives evaluated at \(x=0\) into the Taylor polynomial:\[P_2(x) = 0 + 1\cdot x + \frac{0\cdot x^2}{2} = x\]Thus, the Taylor polynomial of degree 2 for \(\tan(x)\) about \(x=0\) is \(P_2(x) = x\).

Step 4: Evaluate the Function and Polynomial at the Given Point

Now we compare the value of \(f(x)\) and \(P_2(x)\) at \(x=0.1\):- \(f(0.1) = \tan(0.1) \approx 0.10033\).- \(P_2(0.1) = 0.1\).

Step 5: Compare Values

We see that the Taylor polynomial value \(P_2(0.1) = 0.1\) is very close to the actual function value \(f(0.1) \approx 0.10033\). This suggests that the polynomial is a good approximation near \(x=0\).

Key Concepts

Taylor series

A Taylor series is a powerful tool in mathematics used to express a function as an infinite sum of terms. These terms are calculated from the values of the function's derivatives at a specific point. The basic idea is to approximate complex functions with polynomials, which are easier to compute and understand. For a function, the Taylor series centered around a particular point (like zero) is constructed using:

• The function's value at that point.
• All derivatives of the function at that point.

The beauty of a Taylor series is that as more terms are added, the approximation becomes more accurate. For example, in the given exercise, we calculate the second-degree Taylor polynomial of the function \(f(x) = \tan x\) about \(x=0\). By calculating just up to the second derivative, we achieve a polynomial that closely follows the behavior of \(f(x)\) near the center point.

derivatives

Derivatives are fundamental in calculus, representing the rate at which a function changes at a certain point. For the Taylor series, derivatives are crucial because they form the coefficients of the polynomial's terms. In the exercise, calculating derivatives of \(f(x) = \tan x\) gives us:

• First derivative: \(f'(x) = \sec^2(x)\), which indicates the slope or rate of change of the tangent function.
• Second derivative: \(f''(x) = 2\sec^2(x)\tan(x)\), which provides information about the curvature or concavity of the function.

By evaluating these derivatives at the point \(x=0\), we derive the coefficients for the Taylor polynomial. This process highlights how derivatives help simplify complex functions into more straightforward polynomial expressions.

approximation

In mathematics, approximation allows us to represent complicated expressions in a simpler form. Taylor polynomials serve as approximations to the actual functions. These polynomials give insights into the function's behavior near a specific point.

• In the exercise, the second-degree polynomial \(P_2(x) = x\) approximates \(f(x) = \tan x\).
• Such approximation is particularly useful when a quick estimate is needed, or when evaluating the precise function is cumbersome.

The Taylor polynomial provides a balance between simplicity and accuracy, often giving a good measure of the function's value with only a few terms. As demonstrated, \(P_2(x) = x\) comes remarkably close to \(\tan(0.1)\), showing the effectiveness of this approximation.

function evaluation

Function evaluation is an essential concept that involves calculating the output of a function for a given input. In the case of Taylor polynomials, it entails comparing the values of the function and the polynomial at a specific point.

• For this exercise, we evaluated both \(f(0.1) = \tan(0.1) \approx 0.10033\) and the polynomial \(P_2(0.1) = 0.1\).
• This comparison helps us gauge how well the polynomial approximates the function near the point of expansion.

Despite being a simple polynomial, \(P_2(x)\) performs well in matching the function's value, which is the crux of function evaluation. It reassures us of the polynomial's validity as a representation of the function within a small range of the center point.