Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Technology. Let \(F(x)=\sqrt[3]{x}\). The polynomials $$P_{2}(x)=\frac{5}{9}+\frac{5}{9} x-\frac{1}{9} x^{2} \quad \text { and } \quad P_{3}(x)=P_{2}(x)+\frac{5}{81}(x-1)^{3}$$ closely approximate \(F\) near the point (1,1) of \(F\). a. Draw the graphs of \(F\) and \(P_{2}\) on the range \(0 \leq x \leq 3\). b. Compute the relative error in \(P_{2}(2)\) as an approximation to \(F(2)=\sqrt[3]{2}\). c. Draw the graphs of \(F\) and \(P_{3}\) on the range \(1 \leq x \leq 3\). d. Compute the relative error in \(P_{3}(2)\) as an approximation to \(F(2)=\sqrt[3]{2}\).

Step-by-Step Solution

Answer

The relative errors are approximately 0.03 for \( P_2 \) and 0.0191 for \( P_3 \).

Step 1: Graphing F(x) and P_2(x)

To produce the graph of the function \( F(x) = \sqrt[3]{x} \) and the polynomial \( P_2(x) = \frac{5}{9} + \frac{5}{9}x - \frac{1}{9}x^2 \) within the range \( 0 \leq x \leq 3 \), plot both equations on the same set of axes using graphing technology or software. This will allow us to visually compare how closely \( P_2 \) approximates \( F \) around \( x = 1 \).

Step 2: Calculating Relative Error for P_2 at x=2

First, evaluate \( F(2) \), which is \( \sqrt[3]{2} \approx 1.2599 \), then compute \( P_2(2) \) using the polynomial: \( P_2(2) = \frac{5}{9} + \frac{5}{9}(2) - \frac{1}{9}(2)^2 = \frac{5}{9} + \frac{10}{9} - \frac{4}{9} = \frac{11}{9} \approx 1.2222 \). The relative error is given by \( \left| \frac{1.2222 - 1.2599}{1.2599} \right| \approx 0.03 \).

Step 3: Graphing F(x) and P_3(x)

For graphing \( F(x) = \sqrt[3]{x} \) and \( P_3(x) = P_2(x) + \frac{5}{81}(x-1)^3 \) within the range \( 1 \leq x \leq 3 \), use a graphing tool to plot both equations. This comparison helps us determine the improved approximation of \( P_3 \) near \( x = 1 \).

Step 4: Calculating Relative Error for P_3 at x=2

Evaluate \( P_3(2) = P_2(2) + \frac{5}{81}(2-1)^3 = \frac{11}{9} + \frac{5}{81} \approx 1.2839 \). The relative error when using \( P_3 \) as an approximation for \( F(2) \) is \( \left| \frac{1.2839 - 1.2599}{1.2599} \right| \approx 0.0191 \).

Key Concepts

Taylor Polynomials

Taylor Polynomials are an essential tool in approximating complicated functions with simpler polynomials. Essentially, they provide a way of estimating a function by considering its value and derivatives at a specific point. In this exercise, the goal is to approximate the cube root function, \( F(x) = \sqrt[3]{x} \), using finite polynomial expressions. The polynomial \( P_2(x) = \frac{5}{9} + \frac{5}{9} x - \frac{1}{9} x^2 \) is a quadratic polynomial, meaning it focuses on approximating the function around a point with three terms.
Taylor Polynomials are constructed by using terms derived from the function's derivatives at a particular point. The more terms we include, the more accurate the approximation, particularly close to this point. Including terms like \( \frac{5}{81}(x-1)^3 \) in \( P_3(x) \) improves this approximation further, making the polynomial more reflective of the actual function's behavior near \( x = 1 \).
This idea is crucial because it allows engineers and scientists to use simpler expressions to predict complex behavior, reducing computational time while still retaining significant accuracy.

Relative Error

Relative error is a valuable concept for understanding the accuracy of an approximation. This error is a measure of the difference between an approximate value and the true value, placed in context by dividing by the true value. It helps in identifying how significant the approximation error is relative to the true value.
To compute the relative error for the polynomial approximation at a given point, such as \( x = 2 \), you take the difference between the function value \( F(2) \) and its approximation \( P_2(2) \) or \( P_3(2) \).
The computation is shown like this:

• Calculate \( F(2) = \sqrt[3]{2} \approx 1.2599 \)
• For \( P_2(2) \), the error is \( \left| \frac{1.2222 - 1.2599}{1.2599} \right| \approx 0.03 \)
• For \( P_3(2) \), the error is \( \left| \frac{1.2839 - 1.2599}{1.2599} \right| \approx 0.0191 \)

This analysis shows that \( P_3(x) \) offers a better approximation at \( x = 2 \) than \( P_2(x) \), providing a smaller relative error.

Graphing Functions

Graphing functions provides a visual method to compare the behavior of different mathematical expressions. In this particular exercise, graphing the cube root function \( F(x) = \sqrt[3]{x} \) alongside the polynomial approximations \( P_2(x) \) and \( P_3(x) \) allows us to see how closely these polynomials match the actual function.
When you graph these, you'll notice that near \( x = 1 \), where the polynomials are centered, the approximations are quite close to \( F(x) \). The graph changes slightly as \( x \) moves away from this center point, reflecting why polynomials may deviate at farther values.
Using graphing technology can help demonstrate this visual learning. Viewing the graph helps cement understanding of how well our polynomial approximates the function over a range. Keeping your eye on key shared points, like \( (1,1) \), can help highlight where approximations begin to decline. By seeing the lines diverge, students grasp how important choosing the number of terms in a Taylor expansion is for accuracy.