% exams from m222 spring 87 \newcount\probno \def\prob{\advance\probno by 1 \par\smallskip\noindent\the\probno . } \def\der{^{\prime}} \def\derder{^{\prime\prime}} \def\del{\triangle} \def\i{\bf i} \def\j{\bf j} \def\k{\bf k} \def\blankline{$\underline{\phantom{12345}}$} \def\a{{\bf A}} \def\b{{\bf B}} \def\o{{\bf 0}} \def\x{{\bf X}} \def\0{{\bf 0}} \def\c{{\bf C}} \def\k{{\bf k}} \def\ut{{\bf U_{\theta} }} \def\ur{{\bf U_r }} \def\a{{\bf A}} \def\b{{\bf B}} \def\o{{\bf 0}} \probno=0 \noindent Exam 1\hfill A. Miller \hfill Spring 87 \hfill Math 222 \par\bigskip \prob Find the equation of the parabola with directrix $y=-1$ and focus F(0,0). \prob Given that the graph is nondegenerate, use the Discriminant Test to determine whether the graph is a parabola, ellipse, or hyperbola. $$9x^2+6xy+4y^2+6x-22=0$$ \prob Find the volume of the solid generated by rotating the curve $y=1/x$, $1\leq x<\infty$, about the x-axis. Show that the surface area is infinite. \prob Calculate the integral: $$\int\;{{dx}\over{(1+x)\sqrt{x}}}$$ \prob Calculate the integral: $$\int\;\sin(\theta)\cos(\theta)\cos(\sin(\theta))\;d\theta$$ \prob Calculate the integral: $$\int \;\tan^6(\theta)\;d\theta$$ \prob Calculate the integral: $$\int\;{ {x^3}\over{\sqrt{1+x^2}} }\;dx$$ \prob Find the area inside the spiral $r=\theta$, $0\leq \theta\leq 2\pi$. \prob Find the surface area generated by rotating the curve $$ r=1+\cos(\theta),\;\;0\leq\theta\leq {\pi\over 2}$$ about the x-axis. \prob Compute the standard part of $$ {H+1+\epsilon}\over{2H-1+3\epsilon} $$ where $H$ is infinite and $\epsilon$ is infinitesimal. \bigskip \probno=0 \noindent Exam 2\hfill A. Miller \hfill Spring 87 \hfill Math 222 \par\bigskip \prob Find all solutions of the equation: $${dy\over{dx}}=e^{x-y}$$ \prob Calculate the following limit: $$\lim_{x\rightarrow 0^+} x^x$$ \prob Find the derivitive with respect to x of: $$\log_x(2)$$ \prob Find $\int{ 1 \over {x^2+x^4} } \;dx$. \prob Find $\int \sqrt{e^x +1}\;dx$. \prob Find $\int {1\over {1+\sqrt{x}} }\;dx$ \prob Find $\int {\log(x) \over {(1+x)^2} }\;dx$ \prob For each of the following series fill in the blank with exactly one of the answers a,b,c,d or e which stand for: a) divergent b) conditionally convergent c) absolutely convergent d) absolutely and conditionally convergent e) none of above $\Sigma {1\over n}$ is \blankline. $\Sigma {(-1)^n\over n}$ is \blankline. $\Sigma {(-1)^n\over n^2}$ is \blankline. $\Sigma{(-1)^n \over \sqrt{n}}$ is \blankline. \prob Determine whether the following series converges or diverges, and state your reason: $$\Sigma\;{1 \over n(\log(n))^2}$$ \prob Find the radius of convergence and the interval of convergence: $$\Sigma\; {(-1)^n\over n2^n} \;x^{2n+1}$$ Suppose that $f(x)=\Sigma^{\infty}_{n=0} { {n+1}\over 3^n}x^n$. \prob Find the power series for $f^{\prime}(x)$. \prob Find the power series for $\int_0^x\;f(t)\;dt$ where f(x) is given at the top of this page. \prob Find the polynomial $$p(x)=a_0+a_1x+a_2x^2+a_3x^3$$ which best approximates $\tan(x)$ for x's near zero. \prob Find a power series converging to $\sqrt{1-x^2}$. \bigskip \probno=0 \noindent Final Exam\hfill A. Miller \hfill Spring 87 \hfill Math 222 \par\bigskip \prob Find the volume of the solid generated by rotating the curve $y=1/x$, $1\leq x<\infty$, about the x-axis. Show that the surface area is infinite. \prob Find $$\int \sqrt{1-x^2}\; dx$$ \prob Use four terms of a Taylor series to approximate $$\int^{1\over 2}_0 {dx\over 1+x^3}$$ Give an estimate for the error. It is not neccessary for you to simplify your answer by using a calculator. \prob Find a power series converging to $\sqrt{1-x^2}$. \prob What is the equation, in the form $ax+by+cz=d$, of the plane which contains the three vectors: $\i +\j+2\k$, $2\i+\j+\k$, and $\i +\k$? \prob Find the position vector ${\bf X}(t)$ of a particle with the velocity vector $${\bf V}(t)=t\i +t^2\j +t^3\k $$ and initial position (time t=0) $${\bf X}(0)=\i +2\j +3\k $$ Where will the particle be at time t=1? What is the speed the particle is moving at time t=2? What is the acceleration vector at time t=3? \prob Find the angle between the vectors: $3\i + \j -\k$ and $\i -\j$. Find a unit vector which is orthogonal to both of these vectors. \prob Find the vector equation of the line containing the vectors: $\i + \j -\k$ and $\i -\j$. \bigskip \probno=0 \noindent Practice Final Exam \hfill A. Miller \hfill Spring 87 \hfill Math 222 \prob Find $$\int \sqrt{1+x^2} dx$$ \prob Evaluate the limit $$ \lim_{x\mapsto\infty} { (\ln x)^2 \over\sqrt{x} }$$ \prob Find $\underline{all}$ solutions of the differential equation $$ {dy \over dx}=\sqrt{xy}$$ \prob Use four terms of a Taylor series to approximate $$\int^{1\over 2}_0 {dx\over 1+x^3}$$ Give an estimate for the error. It is not neccessary for you to simplify your answer by using a calculator. \prob What is the equation, in the form $ax+by+cz=d$, of the plane which contains the three points: $(1,0,1)$, $(-2,1,-3)$, and $(1,2,3)$? \prob Let $$f(x)=\sum^{\infty}_{n=0} {x^{n}\over (n!)^2}$$ What is the radius of convergence? What is the power series of the function $f^{(2)}(x)$? What is the power series (up to a constant) of $\int f(x)\, dx $? What is $f^{(48)}(0)$? \prob Find $dy \over dx$ if $y=x^{ln(x)}.$ \prob Find $$\lim_{n \to \infty} n^{\raise8pt\hbox{$\sqrt{1 \over n}$}}$$ \prob Find $$\int {{2x^3 + 4x + 2}\over {x^4-1}} dx$$ \prob Calculate the series: $${{2\times3}\over{4}}+{{2\times3\times3}\over{4\times4}} +{{2\times3\times3\times3}\over{4\times4\times4}}+\cdots$$ \prob Find $$\int {{\sqrt x dx}\over{1 + \root 4 \of x}}$$ \prob Define sequence. Define what it means for a sequence to converge. Define what it means for a sequence to diverge. What is the difference between a sequence and a series? ( Your answer to this question should be about four or five sentences.) \prob Are the following series absolutely convergent, conditionally convergent, or divergent? Say why your answer is correct. $$ \eqalign{ &a) \sum^{\infty}_{n=1} {{(-1)^n} \over {n(n+1)} } \cr &b) \sum _{n=2}^{\infty} {{{(-1)^n} \over {n}}} ({{1} \over {log(n)}} )^{3 \over 2} \cr } $$ \prob Starting with the geometric series show how to obtain the Taylor series of $f(x) = {\rm arctan}(x^3)$. What is $f^{(47)}(0)$? \prob Find the intervals of convergence for the following series: $$ \eqalign { &a) \sum^{\infty}_{n=0} {{n} \over { 2^n3^{n+1}}}\, x^{2n} \cr &b) \sum^{\infty}_{n=0} n^2 (x+1)^{n} \cr }$$ \prob Let $\a = i - 2j + k$ and $\b = -i + j. $ Find the following: $$ \eqalign { &a) \a + 4 \b \cr &b) \a \cdot \b \cr &c) ||\a|| \cr &d) {\rm the\,line\,containing}\,\a\,{\rm and}\, \b \cr &e) \a \times \b \cr &f) {\rm the\,plane\,containing}\, \a,\,\b,\,{\rm and}\, \o \cr &g) {\rm the\,point\,on\,the\,line\,in\,d)\, closest\,to\,}\, \o \cr } $$ \prob Let ${\bf S}=2\cos(2\pi t)i+3\sin(2\pi t)j + tk$. Find the velocity, speed, and acceleration. \end