Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Technology. Let \(F(x)=\sqrt{x}\). The polynomials $$P_{2}(x)=\frac{3}{4}+\frac{3}{8} x-\frac{1}{64} x^{2} \quad \text { and } \quad P_{3}(x)=\frac{5}{8}+\frac{15}{32} x-\frac{5}{128} x^{2}+\frac{1}{512} x^{3}$$ closely approximate \(F\) near the point (4,2) of \(F\). a. Draw the graphs of \(F\) and \(P_{2}\) on the range \(1 \leq x \leq 8\). b. Compute the relative error in \(P_{2}(2)\) as an approximation to \(F(2)=\sqrt{2}\). c. Draw the graphs of \(F\) and \(P_{3}(x)\) on the range \(1 \leq x \leq 8\). d. Compute the relative error in \(P_{3}(2)\) as an approximation to \(F(2)=\sqrt{2}\).

Step-by-Step Solution

Answer

P_2(2) is 1.4375 with a 1.66% error; P_3(2) is 1.40625 with a 0.55% error.

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

To graph both \(F(x) = \sqrt{x}\) and the polynomial \(P_2(x) = \frac{3}{4} + \frac{3}{8} x - \frac{1}{64} x^2\), plot the functions on the interval \(1 \leq x \leq 8\). Use graphing software or a graphing calculator. Notice how closely \(P_2(x)\) follows \(F(x)\) near \(x = 4\).

Step 2: Calculating F(2)

Calculate \(F(2) = \sqrt{2}\) by using a calculator to find \(\sqrt{2} \approx 1.414\).

Step 3: Evaluating P_2(2)

Substitute \(x = 2\) into \(P_2(x) = \frac{3}{4} + \frac{3}{8} x - \frac{1}{64} x^2\). Calculate: \(P_2(2) = \frac{3}{4} + \frac{3}{8} \cdot 2 - \frac{1}{64} \cdot 2^2\). This results in \(P_2(2) = \frac{3}{4} + \frac{3}{4} - \frac{4}{64} = 1.4375\).

Step 4: Computing Relative Error for P_2(2)

Find the relative error using the formula: \( \text{Relative Error} = \left| \frac{P_2(2) - F(2)}{F(2)} \right| \). Substitute \(P_2(2) = 1.4375\) and \(F(2) \approx 1.414\), then compute: \( \left| \frac{1.4375 - 1.414}{1.414} \right| \approx 0.0166 \).

Step 5: Graphing Instructions for F(x) and P_3(x)

To graph both \(F(x) = \sqrt{x}\) and the polynomial \(P_3(x) = \frac{5}{8} + \frac{15}{32} x - \frac{5}{128} x^2 + \frac{1}{512} x^3\), plot the functions on the interval \(1 \leq x \leq 8\). Observe how closely \(P_3(x)\) follows \(F(x)\) near \(x = 4\).

Step 6: Evaluating P_3(2)

Substitute \(x = 2\) into \(P_3(x) = \frac{5}{8} + \frac{15}{32} x - \frac{5}{128} x^2 + \frac{1}{512} x^3\). Calculate: \(P_3(2) = \frac{5}{8} + \frac{15}{32} \cdot 2 - \frac{5}{128} \cdot 4 + \frac{1}{512} \cdot 8 = 1.40625\).

Step 7: Computing Relative Error for P_3(2)

Find the relative error using the formula: \( \text{Relative Error} = \left| \frac{P_3(2) - F(2)}{F(2)} \right| \). Substitute \(P_3(2) = 1.40625\) and \(F(2) \approx 1.414\), then compute: \( \left| \frac{1.40625 - 1.414}{1.414} \right| \approx 0.0055 \).

Key Concepts

Relative Error

When approximating values using polynomials, it's essential to know how closely these approximations match our actual functions. The relative error provides a measure of this accuracy, indicating the difference between the approximated value and the exact value in relation to the exact value.
If you have an approximation value, say from a polynomial, the relative error is calculated using the formula:
\[ \text{Relative Error} = \left| \frac{\text{Approximated Value} - \text{Exact Value}}{\text{Exact Value}} \right|\]

• "Approximated Value" is the value calculated by the polynomial.
• "Exact Value" is the value of the function you are approximating, such as \( \sqrt{2} \) in this context.

This helps gauge the precision of the polynomial function, giving insight into how well it performs, especially around critical points like \(x=4\) in this problem.
In our steps, when using \( P_2(2) \) and \( P_3(2) \) to approximate \( F(2) = \sqrt{2} \), we computed relative errors of approximately 0.0166 and 0.0055, respectively. This indicates that \( P_3(x) \) is a more accurate approximation at \( x=2 \) than \( P_2(x) \).

Understanding relative error is vital in fields where precision is crucial, such as engineering and physics. Small errors can lead to significant consequences.

Graphing Functions

Graphing functions provides a visual representation of mathematical equations and can reveal insights into how functions behave. It allows you to see the relationships and intersections between different functions and helps identify where approximations like polynomial expressions match or diverge from the original function.

In this exercise, graphing \( F(x) = \sqrt{x} \) alongside the polynomial approximations \( P_2(x) \) and \( P_3(x) \) within the range \( 1 \leq x \leq 8 \) is crucial to understanding their accuracy. With the help of graphing software or a graphing calculator, you can identify:

• How closely each polynomial follows the actual square root function.
• Specific points, such as \(x=4\), where the approximation is near perfect.
• Areas where the approximation deviates significantly.

Graphing not only confirms numeric calculations but also assists in qualitative assessments of function behavior. By visually inspecting these functions, you can comprehensively understand their performance across different values of \(x\). This process is foundational in mathematical analysis, economics, and various sciences to support conclusions and predictions.

Square Root Function

The square root function \( F(x) = \sqrt{x} \) is a type of irrational function characterized by its smooth, continuous curve that only outputs positive numbers for non-negative inputs. It is a cornerstone in both algebra and calculus, often used to model real-world phenomena where growth diminishes over time, such as sound intensity over distance.

The function is non-linear but fairly simple to understand and graph. Its primary properties include:

• Domain and Range: The domain is all non-negative numbers, \( x \geq 0 \), while the range consists of all non-negative numbers as well.
• Increasing Nature: As \(x\) increases, \(\sqrt{x}\) increases but at a decreasing rate.
• Common Values: Easy reference points like \(\sqrt{1} = 1\), \(\sqrt{4} = 2\), and \(\sqrt{9} = 3\).

In the given exercise, we examine polynomials \(P_2(x)\) and \(P_3(x)\) for approximating \(\sqrt{x}\) near \(x=4\). Understanding the behavior of the square root function is key to appreciating the efficiency and limits of these polynomial approximations.
Studying this function helps build a strong foundation for exploring more complex non-linear functions and can be applied to solve real-world problems involving rates and exponential decay.