91Ó°ÊÓ

In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{i=1}^{5}(2 i+1) $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{1}^{2} \frac{2}{x^{2}} d x, \quad n=4 $$

In Exercises 7 -12, use sigma notation to write the sum. $$ \left[1-\left(\frac{1}{4}\right)^{2}\right]+\left[1-\left(\frac{2}{4}\right)^{2}\right]+\cdots+\left[1-\left(\frac{4}{4}\right)^{2}\right] $$

Verify the identity. \(\sinh 3 x=3 \sinh x+4 \sinh ^{3} x\)

Find the indefinite integral. $$ \int \frac{1}{x^{2 / 3}\left(1+x^{1 / 3}\right)} d x $$

Use the error formulas in Theorem 4.19 to estimate the error in approximating the integral, with \(n=4\), using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} \sin (\pi x) d x $$

Find the derivative of the function. \(g(x)=\operatorname{sech}^{2} 3 x\)

The table lists several measurements gathered in an experiment to approximate an unknown continuous function \(y=f(x)\). (a) Approximate the integral \(\int_{0}^{2} f(x) d x\) using the Trapezoidal Rule and Simpson's Rule. \begin{tabular}{|c|c|c|c|c|c|} \hline\(x\) & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 \\ \hline\(y\) & 4.32 & 4.36 & 4.58 & 5.79 & 6.14 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\(x\) & 1.25 & 1.50 & 1.75 & 2.00 \\ \hline\(y\) & 7.25 & 7.64 & 8.08 & 8.14 \\ \hline \end{tabular} (b) Use a graphing utility to find a model of the form \(y=a x^{3}+b x^{2}+c x+d\) for the data. Integrate the resulting polynomial over [0,2] and compare your result with your results in part (a).

(a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point.(b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). $$ \frac{d y}{d x}=x^{2}-1, \quad(-1,3) $$

Use the table of values to find lower and upper estimates of $$\int_{0}^{10} f(x) d x$$ Assume that \(f\) is a decreasing function. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline f(x) & 32 & 24 & 12 & -4 & -20 & -36 \\ \hline \end{array} $$