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Chapter 4: Problem 80
Find \(F^{\prime}(x)\). $$ F(x)=\int_{-x}^{x} t^{3} d t $$
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(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). $$ \frac{d y}{d x}=x^{2}-1, \quad(-1,3) $$
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).
In Exercises 7 -12, use sigma notation to write the sum. $$ \left[1-\left(\frac{1}{4}\right)^{2}\right]+\left[1-\left(\frac{2}{4}\right)^{2}\right]+\cdots+\left[1-\left(\frac{4}{4}\right)^{2}\right] $$
Use \(a(t)=-32\) feet per second per second as the acceleration due to gravity. Show that the height above the ground of an object thrown upward from a point \(s_{0}\) feet above the ground with an initial velocity of \(v_{0}\) feet per second is given by the function $$ f(t)=-16 t^{2}+v_{0} t+s_{0} $$.
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let \(n=4\) and round your answers to four decimal places. Use a graphing utility to verify your result. $$ \int_{2}^{6} \ln x d x $$
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