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Chapter 12: Problem 34
Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq e^{-x}, y \geq 0, x \geq 0 $$
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Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x $$
Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=2, n=4 $$
Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$
Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} e^{x^{3}} d x $$
Marginal Analysis In Exercises 27 and 28, use a program similar to the Simpson's Rule program on page 906 with \(n=4\) to approximate the change in revenue from the marginal revenue function \(d R / d x .\) In each case, assume that the number of units sold \(x\) increases from 14 to 16 . $$ \frac{d R}{d x}=5 \sqrt{8000-x^{3}} $$
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