Problem 10 - Calculus for Biology and Medicine

Question

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=\sqrt{1+x}, n=3 $$

Step-by-Step Solution

Answer

The Taylor polynomial is \( P_3(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 \).

Step 1: Recall the Taylor series formula

The Taylor series for a function \( f(x) \) about \( a = 0 \) is given by \( f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k \). Since we need a polynomial of degree 3, we will compute terms up to \( x^3 \).

Step 2: Calculate the first few derivatives

First, compute \( f(x) = \sqrt{1+x} \). The first derivative \( f'(x) = \frac{1}{2(1+x)^{1/2}} \).The second derivative \( f''(x) = -\frac{1}{4(1+x)^{3/2}} \).The third derivative \( f'''(x) = \frac{3}{8(1+x)^{5/2}} \).

Step 3: Evaluate the derivatives at \( x=0 \)

Now, find the value of each derivative at \( x=0 \):\( f(0) = \sqrt{1+0} = 1 \).\( f'(0) = \frac{1}{2(1+0)^{1/2}} = \frac{1}{2} \).\( f''(0) = -\frac{1}{4(1+0)^{3/2}} = -\frac{1}{4} \).\( f'''(0) = \frac{3}{8(1+0)^{5/2}} = \frac{3}{8} \).

Step 4: Substitute into the Taylor series equation

Plug these values into the Taylor series formula up to \( n=3 \):\[ P_3(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 \]\[ = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 \].

Step 5: Simplify the expression

Combine all of the terms to get the Taylor polynomial:\[ P_3(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 \].

Key Concepts

Derivatives

In mathematics, derivatives represent the rate at which a function is changing at any given point. When you see the notation \( f'(x) \), this indicates the first derivative of the function \( f(x) \). This concept tells us how steep the curve of \( f(x) \) is at any point \( x \). Now, first derivatives are just the beginning. The second derivative, \( f''(x) \), tells us about the concavity of the function; in other words, it tells us whether the function is curved upwards or downwards. And the third derivative, \( f'''(x) \), provides information on how the rate of change of the slope itself is changing.

In the context of creating a Taylor polynomial, especially when it's derived from the Taylor series, calculating these derivatives is crucial. When approximating a function such as \( \sqrt{1+x} \) about \( x=0 \), each derivative adds a layer of intricacy to our approximation. By evaluating these derivatives at a specific point, we can approximate the behavior of \( f(x) \) around that area using a polynomial.

The Taylor polynomial is built not only on knowing these derivatives but also understanding their role as the coefficients in our polynomial expression.

Maclaurin Series

The Maclaurin series is a special type of Taylor series. Instead of expanding a function around any point \( a \), it specifically expands around \( x=0 \). This makes calculations simpler because all terms are computed at \( x=0 \).

The Maclaurin series of a function \( f(x) \) is expressed as:

• \( f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k \)

Each term of this series \( \frac{f^{(k)}(0)}{k!} x^k \) involves determining the \( k \)-th derivative evaluated at zero. The Maclaurin series becomes increasingly accurate as more terms are added, allowing for a robust approximation over a range of values.

For \( \sqrt{1+x} \), using only a few terms of the series can give you a feel for the function near \( x=0 \). Since we're interested in a third-degree polynomial, the Maclaurin series will provide an approximation up to the \( x^3 \) term. This tells us not only about the slope but also higher-order behaviors around the origin, painting a detailed picture of the function in polynomial form.

Polynomial Approximation

Polynomial approximation involves expressing a complex function in terms of a polynomial. This makes it easier to analyze and compute, especially around specific points. A Taylor polynomial, particularly, serves as a powerful tool for this purpose.

Using derivatives, we craft a polynomial that closely mirrors the behavior of the original function around a certain point. For example, approximating \( \sqrt{1+x} \) up to the third degree involves recognizing the function's nature and fitting it into a simple polynomial format:

• \( P_3(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 \)

This polynomial provides a simplified version of \( \sqrt{1+x} \) that is easy to work with, particularly for small values of \( x \).

Polynomial approximation is widely used in numerical analysis because it offers a balance between simplicity and precision. By approximating complex functions with polynomials, we can perform calculations that might otherwise be too complex or time-consuming.