Question

Refer to Theorem 2 and let \(f(x)=\sin e^{x}\) a. Find a Trapezoid Rule approximation to \(\int_{0}^{1} \sin \left(e^{x}\right) d x\) using \(n=40\) subintervals. b. Calculate \(f^{\prime \prime}(x)\) c. Explain why \(\left|f^{\prime \prime}(x)\right|<6\) on \([0,1],\) given that \(e<3\). (Hint: Graph \(\left.f^{\prime \prime} .\right)\) d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 2.

Short answer

Question: Find an upper bound on the absolute error in the Trapezoid Rule approximation of the integral \(\int_{0}^{1}\sin(e^x)dx\) using 40 subintervals, given that exponential function is less than 3. Answer: The upper bound on the absolute error in the Trapezoid Rule approximation is \(\frac{1}{3200}\).

Step-by-step solution

Trapezoid Rule approximation formula

To approximate the integral using the Trapezoid Rule with \(n=40\) subintervals, we first need to calculate the width of each subinterval: \(\Delta x = \frac{1-0}{40} = \frac{1}{40}\) Then, we can use the Trapezoid Rule formula to approximate the integral: \(\int_{0}^{1}\sin(e^x)dx \approx \frac{\Delta x}{2}\left[f(0)+2f\left(\frac{1}{40}\right)+2f\left(\frac{2}{40}\right)+\cdots+2f\left(\frac{39}{40}\right)+f(1)\right]\)

Calculate the second derivative of the function

To calculate the second derivative \(f^{\prime \prime}(x)\), we first need to find the first derivative: \(f'(x) = \frac{d}{dx}(\sin(e^x)) = \cos(e^x)e^x\) Now, we differentiate again to find the second derivative: \(f^{\prime \prime}(x) = \frac{d}{dx}(\cos(e^x)e^x) = -\sin(e^x)e^{2x}+e^x\cos(e^x)\)

Prove the bounds of the second derivative

We are given that \(e<3\). We want to prove that \(|f^{\prime \prime}(x)|<6\) on the interval \([0, 1]\). Considering that \(\left|\cos(x)\right| \le 1\), we can rewrite the inequality as follows: \(\left|-sin(e^x)e^{2x} + e^x \cos(e^x)\right| < 6\) Since \(x\in[0,1]\), we can safely say that \(e^x < e^1 = e < 3\). Therefore, we can bound the term \(-sin(e^x)e^{2x}\) as follows: \(\left|-sin(e^x)e^{2x}\right|\leq e^{2x} < (3)^{2} = 9\) We also know that \(\left|\cos(e^x)e^x\right|\leq e^x < 3\). Since \(\left|-sin(e^x)e^{2x}\right| < 9\) and \(\left|\cos(e^x)e^x\right|<3\), it follows that \(|f^{\prime \prime}(x)| < 6\) in the interval \([0, 1]\).

Find an upper bound on the absolute error

Using Theorem 2, we can determine an upper bound on the absolute error of the Trapezoid Rule approximation as follows: Error \(\le \frac{(1-0)^3}{12\times 40^2} \max_{x\in[0, 1]} \left|f^{\prime \prime}(x)\right| \le \frac{1}{12\times 40^2} \times 6 = \frac{1}{12\times 1600} \times 6 = \frac{1}{3200}\) Thus, the upper bound on the absolute error in the estimate found in part (a) is \(\frac{1}{3200}\).

Key concepts

Trapezoid Rule

The Trapezoid Rule is a simple technique for approximating the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles. Each trapezoid spans a short interval and the sum of all trapezoids gives an approximation of the integral.

In the exercise, we used the Trapezoid Rule to approximate the integral \( \int_{0}^{1} \sin(e^x)dx \) using 40 subintervals. The width of each subinterval, \( \Delta x \), was calculated as \( 1/40 \). The approximation formula is:

• \( \int_{0}^{1}\sin(e^x)dx \approx \frac{\Delta x}{2}\left[f(0)+2f\left(\frac{1}{40}\right)+2f\left(\frac{2}{40}\right)+\cdots+2f\left(\frac{39}{40}\right)+f(1)\right] \)

The Trapezoid Rule gives a simple method to compute integrals, especially when the function is too complicated for analytical methods.

Derivative Calculation

Calculating derivatives is a fundamental skill in calculus. It helps in understanding the behavior and tendencies of functions. In this exercise, the function \( f(x) = \sin(e^x) \) required us to calculate its derivatives.

First, we find the first derivative \( f'(x) \):

• \( f'(x) = \frac{d}{dx}(\sin(e^x)) = \cos(e^x)e^x \)

After finding the first derivative, the second derivative \( f''(x) \) is computed by differentiating \( f'(x) \):

• \( f^{\prime \prime}(x) = \frac{d}{dx}(\cos(e^x)e^x) = -\sin(e^x)e^{2x}+e^x\cos(e^x) \)

The calculation of derivatives provides insights into the rate of change and curvature of the function.

Error Bound

An error bound estimates how far an approximation can be from the actual result. It is crucial in numerical methods like the Trapezoid Rule to estimate and control potential errors.

In part (d) of the exercise, we used Theorem 2 to find the upper bound on the absolute error of our Trapezoid Rule approximation. The error formula is:

• Error \( \le \frac{(b-a)^3}{12n^2} \max_{x\in[a, b]} \left|f^{\prime \prime}(x)\right| \)

Given \( b-a = 1 \), \( n = 40 \) and \( \left|f^{\prime \prime}(x)\right| < 6 \), the error was computed as:

• \( \frac{1\times 6}{12\times 40^2} = \frac{1}{3200} \)

This error bound assures us that our approximation is reasonably close to the actual value.

Exponential Function

Exponential functions, such as \( e^x \), are a critical part of calculus. They are functions where a constant base is raised to a variable exponent, and they frequently occur in natural processes and theoretical mathematics.

In the exercise, the differentiation and integration involved the function \( e^x \). This made the problems interesting due to the unique properties of exponential growth.

• The derivative of \( e^x \) is itself because it establishes \( e^x \) as a self-similar function with a unique rate of growth.
• \( e \) is a constant approximately equal to 2.718, and hence \( e < 3 \), which was crucial in proving the boundedness of the second derivative.

Understanding exponential functions allows us to solve complex problems involving growth and decay across various disciplines.