MATH 182
Final Examination. Wed., Dec. 16, 1998, 10:00-11:50am.

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Problem1. % latex2html id marker 105 \( (\alph{enumi}) \) Find a substitution \( u=g(x) \) and the associated change of differential \( du=g'(x)dx \) for evaluating the integral \begin{displaymath} \int\frac{x+1}{x^{2}+2x}dx\mbox{\,.} \end{displaymath} Then use the substitution to find the integral. Show all steps.






















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Problem2. % latex2html id marker 140 \( (\alph{enumi}) \) Find a substitution \( u=g(x) \) and the associated change of differential \( du=g'(x)dx \) for evaluating the integral \begin{displaymath} \int_{-a}^{a}x\sqrt{x^{2}+a^{2}}\,dx\mbox{\,,} \end{displaymath} where \( a \) is a positive constant. Then use the substitution to evaluate the integral. Show all steps.
Problem3. % latex2html id marker 174 \( (\alph{enumi}) \) Use integration by parts to evaluate the integral \begin{displaymath} \int_{0}^{\pi/2}x\cos(x)\,dx\mbox{\,.} \end{displaymath} Show all steps.






















Problem4. % latex2html id marker 218 \( (\alph{enumi}) \) Determine whether each integral is convergent or divergent. If the integral converges, evaluate it. Show all steps. To receive credit you must express the improper integrals as limits of proper integrals: e.g., \begin{displaymath} \int_{0}^{\infty}f(x)\,dx= \lim_{b\rightarrow\infty}\int_{0}^{b}f(x)\,dx \end{displaymath}
1.
\( {\displaystyle \int_{2}^{\infty}\frac{1}{(x+3)^{3/2}}\,dx} \).






















2.
\( {\displaystyle \int_{2}^{\infty}\frac{1}{\sqrt{x+3}}\,dx} \).
Problem5. % latex2html id marker 252 \( (\alph{enumi}) \) Find the volume of the solid obtained when the region in the \( (x,y) \)-plane bounded by the \( y \)-axis, the parabola \( y=x^{2} \) and the horizontal line \( y=4 \) is rotated around the \( y \)-axis (see Fig. 1). Sketch a typical approximating cylinder, showing the \( \Delta x \) or \( \Delta y \) (you must decide whether to use \( \Delta x \) or \( \Delta y \)).























  
Figure 1: Parabolic Segment
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Problem6. % latex2html id marker 297 \( (\alph{enumi}) \) Find the exact arclength of the curve given by the equations \( {\displaystyle x=t^{3}} \) and \( {\displaystyle y=t^{2}} \), for \( 0\leq t\leq 1 \). Your answer must show how you set up the integral for the arclength, and all steps used in evaluating the integral.
Problem7. % latex2html id marker 330 \( (\alph{enumi}) \) Find the solution of the differential equation \begin{displaymath} \frac{dy}{dx}=-xy \end{displaymath} which satisfies the initial condition \( y(0)=1/2 \). Show all work.
Problem8. % latex2html id marker 359 \( (\alph{enumi}) \) A bacteria culture starts with 500 bacteria and grows at a rate proportional to its size (as measured by the number of bacteria in the colony at any time). After 3 hours there are 8000 bacteria.
1.
Let \( P(t) \) denote the number of bacteria in the colony at time \( t \) (measured in hours from the start of the experiment). Write down the differential equation which models the population \( P(t) \) as a function of the time \( t \).














2.
Find a formula for \( P(t) \), and use the given data to find all of the constants in this formula.














3.
Find a formula for the time \( t_{2} \) at which the population reached twice its initial size (the doubling time ).
Problem9. % latex2html id marker 415 \( (\alph{enumi}) \) A sequence is recursively defined by the formula \begin{displaymath} a_{n+1}=\frac{2\cdot a_{n}}{3}+\frac{1}{(a_{n})^{2}}\mbox{\,,} \end{displaymath} where \( a_{0}=1.4 \). The first few values correct to 12 places are \begin{eqnarray*} a_{0} & = & 1.40000000000 \\ a_{1} & = & 1.44353741497 \\ a_... ...{3} & = & 1.44224957031 \\ a_{4} & = & 1.44224957031\mbox{\,.} \end{eqnarray*} Assuming that the sequence \( \{a_{n}\} \) has a limit \( L \), find the exact value of \( L \).














Problem10. % latex2html id marker 460 \( (\alph{enumi}) \) Show that the infinite series \begin{displaymath} \sum_{n=0}^{\infty}\frac{2^{n+2}}{3^{n}}=4+\frac{8}{3}+ \frac{16}{9}+\frac{32}{27}+\cdots \end{displaymath} is convergent and find the exact value of its sum. Show all work.














Problem11. % latex2html id marker 494 \( (\alph{enumi}) \) Determine whether the series \( {\displaystyle \sum_{n=1}^{\infty} \frac{5^{n}}{n^{3}}=5+\frac{25}{8}+\frac{125}{27}+\cdots} \) is convergent or divergent. Name any test you used in drawing your conclusion: divergence test; alternating series test; integral test; comparison test; limit comparison test; or ratio test.
Problem12. % latex2html id marker 544 \( (\alph{enumi}) \) Find the radius of convergence of the power series \begin{displaymath} \sum_{n=1}^{\infty}\frac{x^{n}}{n^{2}2^{n}}= x+\frac{x^{2}}... ...^{3}}{3^{2}2^{3}}+ \frac{x^{4}}{4^{2}2^{4}}+\cdots\mbox{\,.} \end{displaymath} Name any tests you used in drawing your conclusion: divergence test; alternating series test; integral test; comparison test; limit comparison test; or ratio test.






















Problem13. % latex2html id marker 578 \( (\alph{enumi}) \) Find the power series representation \( {\displaystyle \sum_{n=0}^{\infty}a_{n}x^{n}} \) for the function \( f \) defined by the equation \( {\displaystyle f(x)=\frac{1}{(x+1)^{2}}} \). You may begin with the infinite geometric series, or you may use Taylor's method which is based on evaluating the successive derivatives of \( f \). Tell us which method you are using. For full credit you should give a formula for the \( n \)th coefficient \( a_{n} \) in the power series for the function \( f \).