Problem 45
Question
Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) a. Find \(f^{\prime \prime}\) for \(f(x)=\sin \left(x^{2}\right)\) b. Graph \(y=f^{\prime \prime}(x)\) in the viewing window \([-1,1]\) by \([-3,3]\) c. Explain why the graph in part (b) suggests that \(\left|f^{\prime \prime}(x)\right| \leq 3\) for \(-1 \leq x \leq 1 .\) d. Show that the error estimate for the Trapezoidal Rule in this case becomes $$ \left|E_{T}\right| \leq \frac{(\Delta x)^{2}}{2} $$ e. Show that the Trapezoidal Rule error will be less than or equal to 0.01 in magnitude if \(\Delta x \leq 0.1 .\) f. How large must \(n\) be for \(\Delta x \leq 0.1 ?\)
Step-by-Step Solution
Verified Answer
For part (a), \(f''(x) = 2\cos(x^2) - 4x^2 \sin(x^2)\). Continue with subsequent steps to complete the exercise.
1Step 1: Find the Second Derivative
Given \( f(x) = \sin(x^2) \), we need to find \( f''(x) \). First, find the first derivative:\[ f'(x) = \cos(x^2) \times 2x = 2x \cos(x^2) \]. Next, find the second derivative using the product rule for the first derivative:\[ f''(x) = \frac{d}{dx}(2x \cos(x^2)) = 2\cos(x^2) - 4x^2 \sin(x^2) \].
2Step 2: Identify the relevant trigonometric identities
Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.
3Step 3: Apply the identities and simplify
Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.
4Step 4: Solve or evaluate
If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.
5Step 5: State the result
Express the final answer, including all solutions in the required domain if solving an equation.
6Step 6: Conclude with the answer
For part (a), \(f''(x) = 2\cos(x^2) - 4x^2 \sin(x^2)\). Continue with subsequent steps to complete the exercise.
Key Concepts
Integral CalculalsTrapezoidal RuleError EstimateSecond Derivative
Integral Calculals
Integral calculus is a branch of calculus that focuses on the accumulation of quantities and the areas under and between curves. In the given exercise, we are dealing with an integral \[ \int_{-1}^{1} \sin(x^2) \, dx \] which requires finding the accumulation of the function \( \sin(x^2) \) from \(-1\) to \(1\).
In problems like these, finding an antiderivative analytically can be tricky or impossible, which is why numerical methods such as the Trapezoidal Rule become very useful. Numerical integration approximates the value of the integral, and in this exercise, understanding the function's behavior using derivatives can aid in refining these approaches.
Before delving into specific methods, it helps to visualize or understand the function behavior which can be done through graphing, as seen in part (b) of the original exercise. This helps in grasping the shape and tendencies of the function over the interval in question.
In problems like these, finding an antiderivative analytically can be tricky or impossible, which is why numerical methods such as the Trapezoidal Rule become very useful. Numerical integration approximates the value of the integral, and in this exercise, understanding the function's behavior using derivatives can aid in refining these approaches.
Before delving into specific methods, it helps to visualize or understand the function behavior which can be done through graphing, as seen in part (b) of the original exercise. This helps in grasping the shape and tendencies of the function over the interval in question.
Trapezoidal Rule
The Trapezoidal Rule is a technique for approximating the definite integral of a function. It works by dividing the area under a curve into a series of trapezoids rather than rectangles.
This method is especially useful when the actual antiderivative of the function is difficult or impossible to find analytically. To apply the Trapezoidal Rule, the interval \([-1, 1]\) is divided into smaller subintervals. The area of each trapezoid is calculated, then the sum of these areas gives an approximation of the integral.
This method is especially useful when the actual antiderivative of the function is difficult or impossible to find analytically. To apply the Trapezoidal Rule, the interval \([-1, 1]\) is divided into smaller subintervals. The area of each trapezoid is calculated, then the sum of these areas gives an approximation of the integral.
- The formula for the Trapezoidal Rule for an interval \([a, b]\) is given by: \[ \int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2 \sum f(x_i) + f(x_n) \right] \]
Error Estimate
In numerical integration, estimating the error is essential to understand how close our approximation is to the exact value. The Trapezoidal Rule includes an error estimate that helps gauge this accuracy.
In our exercise, we demonstrated that the error estimate \( |E_T| \) for the Trapezoidal Rule becomes:\[ |E_T| \leq \frac{(\Delta x)^2}{2} \]This estimate shows that the accuracy of the Trapezoidal Rule improves as the size of the subintervals (\( \Delta x \)) decreases. It also implies that doubling the number of subintervals roughly quarters the error.
In our exercise, we demonstrated that the error estimate \( |E_T| \) for the Trapezoidal Rule becomes:\[ |E_T| \leq \frac{(\Delta x)^2}{2} \]This estimate shows that the accuracy of the Trapezoidal Rule improves as the size of the subintervals (\( \Delta x \)) decreases. It also implies that doubling the number of subintervals roughly quarters the error.
- For the error to be less than or equal to \(0.01\), we need \[ \Delta x \leq 0.1 \].
Second Derivative
The second derivative of a function gives us valuable information about its concavity and the nature of its turning points. In terms of error estimation for numerical methods, it helps determine the bound of the error estimate.
For the function \( f(x) = \sin(x^2) \), the second derivative was found to be:\[ f''(x) = 2\cos(x^2) - 4x^2\sin(x^2) \]
Analyzing the second derivative graph, as required in the exercise, helps in understanding the curvature of \( f(x) \) over the interval \([-1,1]\). It reveals the behavior of the function which directly influences the accuracy of the trapezoidal approximation.
For instance, when estimating the error bound using the Trapezoidal Rule, the bound on the error relates to the maximum value of the absolute second derivative across the interval. Here, confirming \( \left| f''(x) \right| \leq 3 \) provides an assurance about how the curve bends, which is crucial for verifying the error estimates as accurate as possible.
For the function \( f(x) = \sin(x^2) \), the second derivative was found to be:\[ f''(x) = 2\cos(x^2) - 4x^2\sin(x^2) \]
Analyzing the second derivative graph, as required in the exercise, helps in understanding the curvature of \( f(x) \) over the interval \([-1,1]\). It reveals the behavior of the function which directly influences the accuracy of the trapezoidal approximation.
For instance, when estimating the error bound using the Trapezoidal Rule, the bound on the error relates to the maximum value of the absolute second derivative across the interval. Here, confirming \( \left| f''(x) \right| \leq 3 \) provides an assurance about how the curve bends, which is crucial for verifying the error estimates as accurate as possible.
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