Problem 13
Question
Technology Draw the graphs of \(F(x)=\sin x\) and $$F(x)=\sin x \quad \text { and } \quad P_{5}(x)=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$$ on the range \(0 \leq x \leq \pi\). Compute the relative error in \(P_{5}(\pi / 4)\) as an approximation to \(F(\pi / 4)\) and in \(P_{5}(\pi / 2)\) as an approximation to \(F(\pi / 2)\).
Step-by-Step Solution
Verified Answer
Plot both functions, evaluate at given points, and compute relative errors.
1Step 1: Plotting the Graphs
We have two functions to graph: \(F(x)=\sin x\) and \(P_{5}(x)=x-\frac{x^3}{6}+\frac{x^5}{120}\). Using graphing software or a graphing calculator, plot both functions over the interval \([0, \pi]\). The function \(\sin x\) is the standard sine function, while \(P_{5}(x)\) is a polynomial approximation of \(\sin x\) using the first few terms of its Taylor series.
2Step 2: Evaluating Functions at Specific Points
To compute the relative error, first evaluate both functions at the specific points \(x=\frac{\pi}{4}\) and \(x=\frac{\pi}{2}\).Calculate:- \(F(\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2}\).- \(P_{5}(\pi/4) = \left(\frac{\pi}{4}\right) - \frac{\left(\frac{\pi}{4}\right)^3}{6} + \frac{\left(\frac{\pi}{4}\right)^5}{120}\).Calculate:- \(F(\pi/2) = \sin(\pi/2) = 1\).- \(P_{5}(\pi/2) = \frac{\pi}{2} - \frac{\left(\frac{\pi}{2}\right)^3}{6} + \frac{\left(\frac{\pi}{2}\right)^5}{120}\).
3Step 3: Computing the Relative Error
The relative error at a point is given by:\[ \text{Relative Error} = \frac{|P_{5}(x) - F(x)|}{|F(x)|} \]For \(x = \frac{\pi}{4}\):- Substitute the values computed in previous steps to find the relative error: \[ \text{Relative Error at } \frac{\pi}{4} = \frac{|P_{5}(\pi/4) - \frac{\sqrt{2}}{2}|}{|\frac{\sqrt{2}}{2}|} \]For \(x = \frac{\pi}{2}\):- Substitute the values computed in previous steps to find the relative error: \[ \text{Relative Error at } \frac{\pi}{2} = \frac{|P_{5}(\pi/2) - 1|}{|1|} \]
4Step 4: Interpreting Results
Interpret the results by evaluating the numerical values of the relative errors obtained from the calculations. Compare these errors to assess how well \(P_5(x)\) approximates \(\sin x\) at these points.
Key Concepts
Polynomial ApproximationGraphing FunctionsRelative ErrorSine Function
Polynomial Approximation
Polynomial approximation is a technique used to estimate more complex functions with simpler polynomial functions. These approximations are especially useful when a direct calculation is complicated or impossible to evaluate easily.
A Taylor series is a common method for creating polynomial approximations. It expands a function into an infinite sum of terms calculated from the values of its derivatives at a single point.
The exercise uses a Taylor series to approximate the sine function, \[ P_{5}(x) = x - \frac{x^3}{6} + \frac{x^5}{120} \]This polynomial is designed to mimic the sine function within a specific range, namely from 0 to \(\pi\).
Polynomial approximations like this one are particularly helpful in computational mathematics, where they reduce the complexity of calculations while maintaining reasonable accuracy over an interval.
A Taylor series is a common method for creating polynomial approximations. It expands a function into an infinite sum of terms calculated from the values of its derivatives at a single point.
The exercise uses a Taylor series to approximate the sine function, \[ P_{5}(x) = x - \frac{x^3}{6} + \frac{x^5}{120} \]This polynomial is designed to mimic the sine function within a specific range, namely from 0 to \(\pi\).
Polynomial approximations like this one are particularly helpful in computational mathematics, where they reduce the complexity of calculations while maintaining reasonable accuracy over an interval.
Graphing Functions
In mathematics, graphing functions helps us visually understand and compare the behavior of different functions over a specified range.
For this exercise, we need to graph two functions:
When graphing manually or using software, plot both functions on the same set of axes to visually compare their shapes and how they align. The approximation should closely follow the true sine curve, particularly where the polynomial's degree offers an accurate representation.Graphs enable students to see discrepancies between the original function and its approximation, and analyze how good the approximation is within the specified range.
For this exercise, we need to graph two functions:
- The sine function: \( F(x) = \sin x \)
- The polynomial approximation: \( P_{5}(x) = x - \frac{x^3}{6} + \frac{x^5}{120} \)
When graphing manually or using software, plot both functions on the same set of axes to visually compare their shapes and how they align. The approximation should closely follow the true sine curve, particularly where the polynomial's degree offers an accurate representation.Graphs enable students to see discrepancies between the original function and its approximation, and analyze how good the approximation is within the specified range.
Relative Error
Relative error is a critical concept in assessing the accuracy of approximations. It provides a measure of how much the estimated value deviates from the true value, relative to the true value itself.
The formula for relative error is:\[\text{Relative Error} = \frac{|P_{5}(x) - F(x)|}{|F(x)|}\]
In our problem, we calculate the relative error at specific points:
The formula for relative error is:\[\text{Relative Error} = \frac{|P_{5}(x) - F(x)|}{|F(x)|}\]
In our problem, we calculate the relative error at specific points:
- For \(x = \frac{\pi}{4}\), the error tells us how well the polynomial \(P_{5}(x)\) approximates \(\sin(\frac{\pi}{4})\).
- For \(x = \frac{\pi}{2}\), it measures the discrepancy between \(P_{5}(x)\) and \(\sin(\frac{\pi}{2})\).
Sine Function
The sine function, denoted as \( \sin x \), is a fundamental part of trigonometry and appears frequently in various mathematical and physical contexts. This periodic function maps angles, in radians, to a value between -1 and 1, describing wave-like shapes which repeat every \(2\pi\).
The sine function is defined using right-angled triangles or the unit circle. In the unit circle, the sine of an angle corresponds to the y-coordinate of a point on the circle's circumference.
In the context of this problem, we aim to approximate \( \sin x \) using a polynomial, which provides useful insights into the function's behavior within a specific interval.
Understanding the sine function's properties, such as its symmetry and periodicity, is useful for both graphing and polynomial approximation. Mastery of these concepts is crucial for effectively solving trigonometric problems, modeling oscillating systems, and applying them in real-world scenarios.
The sine function is defined using right-angled triangles or the unit circle. In the unit circle, the sine of an angle corresponds to the y-coordinate of a point on the circle's circumference.
In the context of this problem, we aim to approximate \( \sin x \) using a polynomial, which provides useful insights into the function's behavior within a specific interval.
Understanding the sine function's properties, such as its symmetry and periodicity, is useful for both graphing and polynomial approximation. Mastery of these concepts is crucial for effectively solving trigonometric problems, modeling oscillating systems, and applying them in real-world scenarios.
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