Problem 46 - Thomas Calculus

Question

Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) a. Find \(f^{(4)}\) for \(f(x)=\sin \left(x^{2}\right) .\) (You may want to check your work with a CAS if you have one available.) b. Graph \(y=f^{(4)}(x)\) in the viewing window \([-1,1]\) by \([-30,10] .\) c. Explain why the graph in part (b) suggests that \(\left|f^{(4)}(x)\right| \leq 30\) for \(-1 \leq x \leq 1\) d. Show that the error estimate for Simpson's Rule in this case becomes $$ \left|E_{S}\right| \leq \frac{(\Delta x)^{4}}{3} $$ e. Show that the Simpson's Rule error will be less than or equal to 0.01 in magnitude if \(\Delta x \leq 0.4 .\) f. How large must \(n\) be for \(\Delta x \leq 0.4 ?\)

Step-by-Step Solution

Answer

The fourth derivative is \(f^{(4)}(x) = 8\cos(x^2) - 64x^2\cos(x^2) - 48x\sin(x^2)\). For \(\Delta x \leq 0.4\), \(n\) must be at least 5.

Step 1: Find the fourth derivative of f(x)

The function given is \(f(x) = \sin(x^2)\). We will differentiate it four times. First, recall that the derivative of \(\sin(x)\) is \(\cos(x)\) and the chain rule formula \((u(x))^n\):First derivative: \( f'(x) = 2x \cos(x^2) \).Second derivative: Use product rule on \(2x \cos(x^2)\):- \( u = 2x \), \( u' = 2 \)- \( v = \cos(x^2) \), \( v' = -2x \sin(x^2) \)\( f''(x) = 2\cos(x^2) + 4x^2\sin(x^2) \).Third derivative involves deeper application of product and chain rules:Fourth derivative, \( f^{(4)}(x) = 8 \cos(x^2) - 64x^2 \cos(x^2) - 48x \sin(x^2) \).This derivative should be confirmed with computational tools to ensure accuracy.

Step 2: Graph the fourth derivative

Using a graphing calculator or graphing software, plot the function \(y = f^{(4)}(x) = 8\cos(x^2) - 64x^2\cos(x^2) - 48x\sin(x^2)\) in the window \([-1,1]\) by \([-30,10]\). This allows us to visually inspect the behavior of the function over the given interval.

Step 3: Analyze the graph for magnitude

Upon examining the graph from Step 2, note that its maximum value does not exceed 30 in magnitude. This visual check supports the idea that \( |f^{(4)}(x)| \leq 30 \) for \(-1 \leq x \leq 1\). This is crucial information for later error estimation steps.

Step 4: Derive the error estimate for Simpson's Rule

Recall the error formula for Simpson’s Rule: \( |E_S| \leq \frac{K(b-a)^5}{180n^4} \). Here, \(K\) is the maximum value of the fourth derivative magnitude over \([a, b]\), in this case, 30. Substitute this value into the formula:\[ |E_S| \leq \frac{30(2)^5}{180n^4} = \frac{960}{180n^4} = \frac{16}{3n^4} \]This simplifies to \( |E_S| \leq \frac{(\Delta x)^4}{3} \) when \(\Delta x = \frac{2}{n}\).

Step 5: Show error ≤ 0.01 for Δx ≤ 0.4

Substitute \(\Delta x = 0.4\) into the error formula:\[ |E_S| \leq \frac{(0.4)^4}{3} \]Calculating the above gives \( |E_S| \leq 0.001067 \), ensuring the error is indeed below \(0.01\).

Step 6: Determine n for Δx ≤ 0.4

Determine \(n\) such that \(\Delta x \leq 0.4\), knowing \(\Delta x = \frac{2}{n}\). Therefore:\[ \frac{2}{n} \leq 0.4 \implies n \geq \frac{2}{0.4} = 5 \]Thus, \( n \) must be at least 5 to ensure \(\Delta x \leq 0.4\).

Key Concepts

Error Estimation

When using numerical methods like Simpson's Rule to approximate integrals, understanding error estimation is crucial. Error estimation helps us determine how close the numerical result is to the actual integral. For Simpson's Rule, the error formula is: \[ |E_S| \leq \frac{K(b-a)^5}{180n^4} \]Here,

• \(E_S\) is the error in the Simpson's Rule approximation.
• \(K\) is the maximum value of the fourth derivative of the function on the interval \([a, b]\).
• \(n\) is the number of subintervals.

To minimize error, we need to choose an appropriate number of subdivisions, \(n\), which impacts \(\Delta x\), the width of each subinterval. The finer the partition, the more accurate the approximation.

Integral Calculus

Integral calculus is all about finding areas under curves. When you evaluate an integral like \(\int_{-1}^{1} \sin(x^2)\, dx\), you're calculating the area between the curve and the x-axis from \(-1\) to \(1\).
This particular integral, involving \(\sin(x^2)\), does not have an elementary antiderivative, making it ideal for a numerical method.
Simpson's Rule is a powerful tool here because it provides a way to approximate these integrals by fitting parabolas through points defined by the function.
This method gives more accurate results than simpler rules like the Trapezoidal Rule, especially for smooth functions. Integral calculus, therefore, ties closely with numerical methods when dealing with complicated functions.

Numerical Methods

Numerical methods are techniques used to approximate mathematical analysis that might be too complex to solve analytically. In the field of integral calculus, methods like Simpson's Rule are invaluable.
Simpson’s Rule is particularly effective because it approximates the integral of a function by dividing the area under the curve into an even number of intervals, building parabolas over them.
When applying Simpson's Rule, be mindful of the interval count and width. The error and accuracy depend on these choices. This method requires calculating with precision the value of each function at specified points, ensuring calculations account for the behavior of the function's higher-order derivatives.

Higher-Order Derivatives

Higher-order derivatives, like the fourth derivative in our exercise, reflect how quickly and in what way a function changes. The fourth derivative, \(f^{(4)}(x)\), is especially crucial in error estimation for Simpson's Rule.
Finding \(f^{(4)}(x)\) for \(f(x)=\sin(x^2)\) requires applying the product and chain rules repeatedly, a process that underscores the power of derivative calculus.
The magnitude of \(f^{(4)}(x)\) provides the bound \(K\) for the error formula, giving us a means to limit the potential error in numeric approximations.
Understanding these derivatives is key to precisely estimating potential inaccuracies in numerical methods, indicating just how a function fluctuates.