First Examination. Thursday, February 11, 1999.
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Problem1. % latex2html id marker 276 \( (\alph{enumi}) \) The speed of a braking car decreases steadily from 75 ft/sec to 0 ft/sec over a period of 8 seconds. The speed at two second intervals is given in the following table:

\begin{tabular} {\vert c c\vert\vert c\vert c\vert c\vert c\vert c\vert} \hline... ... \hline \( v(t) \) & (ft/sec) & 75 & 45 & 25 & 10 & 0 \ \hline \end{tabular}
1.
Sketch a reasonable graph of velocity \( v(t) \) as a function of time \( t \) in the figure. Calculate the Riemann sum for integral \( {\displaystyle \int_{0}^{8}v(t)dt} \) with \( n=4 \) subintervals using the right-hand endpoint rule. Sketch the four rectangles in the Figure.








  
Figure: \( v(t) \) versus \( t \)
\begin{figure} \centering \includegraphics{182S99E1gr1.eps} \end{figure}

2.
What does the integral \( {\displaystyle \int_{0}^{8}v(t)dt} \) tell us? Is the approximating Riemann sum an underestimate or an over estimate for the value of the integral? Explain.







Problem2. % latex2html id marker 286 \( (\alph{enumi}) \) The graph of \( f \) is shown in the Figure. Calculate the following integrals by interpreting them as areas: \( {\displaystyle \int_{0}^{2}f(x)dx=\mbox{\qquad}} \), \( {\displaystyle \int_{2}^{6}f(x)dx=\mbox{\qquad}} \), \( {\displaystyle \int_{-2}^{2}f(x)dx=\mbox{\qquad}} \).
  
Figure: \( y=f(x) \)
\begin{figure} \centering \includegraphics{182S99E1gr2.eps} \end{figure}

Problem3. % latex2html id marker 292 \( (\alph{enumi}) \) Evaluate the integral \( {\displaystyle \int_{0}^{2}\vert 1-x^{2}\vert dx} \). Show all work.






























Problem4. % latex2html id marker 294 \( (\alph{enumi}) \) Given \( {\displaystyle F(x)=\int_{0}^{x}\sqrt{t^{2}+9}dt} \), find each of the following quantities.
1.
\( F'(x) \)




2.
\( F'(4) \)




3.
\( F'(0) \)




Problem5. % latex2html id marker 299 \( (\alph{enumi}) \)
1.
Evaluate the integral \( {\displaystyle \int e^{-x/7}dx} \).
2.
Evaluate the integral \( {\displaystyle \int \frac{x^{3}}{\sqrt{1+x^{4}}}dx} \).






















Problem6. % latex2html id marker 302 \( (\alph{enumi}) \)
1.
Evaluate \( {\displaystyle \int x\cos\left(\frac{x}{3}\right)dx} \). Show all work.






















2.
Evaluate \( {\displaystyle \int_{0}^{2}x\sqrt{4-x^{2}}dx} \). Show all work. Be sure to change the limits of integration when performing substitutions.
Problem7. % latex2html id marker 305 \( (\alph{enumi}) \) In each of the following cases determine whether the improper integral converges by expressing the integral as a limit of proper integrals. If the integral converges, find its value.
1.
\( {\displaystyle \int_{0}^{\infty}\frac{1}{\left(x+\frac{1}{2}\right)^{3/2}}dx} \).





































2.
\( {\displaystyle \int_{0}^{1}\frac{1}{x^{2/3}}dx} \).