%begin macros %\magnification=\magstephalf \magnification=\magstep1 \def\g{\bigtriangledown} \def\L{{\cal L}} %\baselineskip=14pt %\parskip=10pt \def\Tilde{\char126\relax} \font\eightrm=cmr8 \font\sixrm=cmr6 \font\eighttt=cmtt8 \def\P{{\cal P}} \def\Q{{\cal Q}} \parindent=0pt \overfullrule=0in \def\frac#1#2{{#1 \over #2}} %end macros \medskip {\bf NAME: (print!)} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \bigskip {\bf Section:} \_\_\_\_ \qquad {\bf E-Mail address:} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \bigskip MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Exam 2, Monday, Nov. 23, 2020, 8:40-10:40am \bigskip Email the completed test, renamed as {\tt mt2FirstLast.pdf } to DrZcalc3@gmail.com no later than 10:40am, (or, in case of conflict, two hours after the start). \bigskip \hrule \bigskip WRITE YOUR FINAL ANSWERS BELOW \medskip \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \bigskip 1. \qquad \bigskip 2. \qquad \bigskip 3. \qquad \bigskip 4. \qquad \bigskip 5. \qquad \bigskip 6. \qquad \bigskip 7. \qquad \bigskip 8. \qquad \bigskip 9. \qquad \bigskip 10. \qquad \bigskip \hrule \bigskip {\bf Types:} Number, Function of $variable$(s), 2D vector of numbers, 3D vector of numbers, 2D vector of functions, 3D vector of functions, equation of a plane, parametric equation of a line, equation of a line, equation of a surface, equation of a line, DNE (does not exist). \bigskip \hrule \bigskip Sign the following declaration: \bigskip I \quad \quad \quad \quad \quad \quad \quad \quad Hereby declare that all the work was done by myself. I was allowed to use Maple, calculators, the book, and all the material in the web-page of this class but {\bf not} other resources on the internet. \bigskip I only spent (at most) 90 minutes on doing the exam. The last 30 minutes were spent in checking and double-checking the answers. \bigskip I also understand that I may be subject to a random short chat to verify that I actually did it all by myself. \bigskip Signed: \vfill\eject \medskip {\bf 1}. (10 pts.) Compute the line integral $$ \int_C \, yz \, dx \, + \, xz \, dy \, + \, xy \, dz \quad, $$ over the path $$ {\bf r}(t) = \langle e^{t^3}, t e^{t^4}, t^4 e^{t^7} \rangle \quad, \quad 0 \leq t \leq 1 \quad . $$ Explain! \bigskip \hrule \medskip The {\bf type} of the answers is: \medskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 2.} (10 points) Change the order of integration. Explain! $$ \int_{0}^{9} \int_{\sqrt{x}}^{3} F(x,y) \, dy \, dx $$ \bigskip \bigskip \hrule \medskip The {\bf type} of the answer is: \medskip \hrule \hrule \bigskip \bigskip {\bf ans.} \bigskip \bigskip \hrule \vfill\eject {\bf 3.} (10 points) Find the equation of the tangent plane at the point $(1,1,1)$ to the surface given parametrically by $$ x(u,v)= u^3 v \quad, \quad y(u,x)=uv \quad \, \quad z(u,v)=uv^3 \quad , \quad -\infty < u < \infty \quad , \quad -\infty < v < \infty \quad . $$ Express you answer in {\bf explicit} form, i.e in the format $z=ax+by+c$. \bigskip \hrule \medskip The {\bf type} of the answer is: \medskip \hrule \hrule \bigskip {\bf ans.} $z=$ \bigskip \hrule \vfill\eject {\bf 4.} (10 points) Let $f(x,y,z)=e^{\cos x^2 \, + \, \sin xyz + \cos xz}$. Let $$ {\bf F}= \langle \frac{\partial f}{\partial x} , \frac{\partial f}{\partial y} , \frac{\partial f}{\partial z} \rangle $$ Let $C$ be the closed curve $$ r(t)= \langle \cos t , t, \sin t \rangle \quad, \quad 0 \leq t \leq 2\pi \quad . $$ Find the value of the line-integral $$ \int_C {\bf F} . d{\bf r} \quad . $$ Explain! \bigskip \hrule \medskip The {\bf type} of the answer is: \medskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 5.} (10 points) Evaluate the triple integral $$ \int_R \, (x^2+y^2+z^2)^3 \, dx \, dy \, dz \quad, $$ where $R$ is the region in 3D space given by $$ \{ (x,y,z) \, | \, x^2+y^2+z^2 \leq 1 \quad, \quad x,y,z \geq 0 \} \quad . $$ \bigskip \hrule \medskip The {\bf type} of the answer is: \medskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 6.} (10 points) Evaluate the double integral $$ \int_{-3}^{0} \int_0^{\sqrt{9-x^2}} (x^2+y^2)^2 \, dy \, dx $$ \bigskip \hrule \medskip The {\bf type} of the answer is: \medskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 7.} (10 points altogether) Decide whether the following limits exist. If it does find them. If it does not {\bf Explain} why not? $$ (a) (2 \,\, points) \, \lim_{(x,y)\rightarrow (\pi/2, \pi/2)} \,\, \frac{\cos x + \sin x}{x+y} \quad, \quad (b) (2 \,\, points) \, \lim_{(x,y)\rightarrow (0,0)} \,\, \frac{(x^2-y^2)}{x-y} \quad, \quad $$ $$ (c) (2 \,\, points) \, \lim_{(x,y)\rightarrow (0,0)} \,\, \frac{x-y}{x^2-y^2} \quad, \quad (d) (4 \,\, points) \,\lim_{(x,y)\rightarrow (1,1)} \,\, \frac{x+y-2}{2x+y-3} \quad, \quad $$ \bigskip \vfill\eject {\bf 8.} (10 points) Compute the line integral $\int_C f \, ds$ where $$ f(x,y,z)=xyz \quad $$ and $C$ is $$ {\bf r}(t) = \langle t, -2t, 3t \rangle \quad, \quad for \quad 0 \leq t \leq 1 $$ \hrule \medskip The {\bf type} of the answer(s) is: \medskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 9.} (10 points) Compute the vector-field surface integral $\int \int_S {\bf F}.d{\bf S}$ for the given oriend surface $$ {\bf F} \, = \, \langle z,z,x \rangle \quad, \quad z=9-x^2-y^2 \quad, x \geq 0, y \geq 0 , z \geq 0 $$ with {\bf downward pointing} normal. \bigskip \hrule \medskip The {\bf type} of the answer is \bigskip \hrule \bigskip \hrule \bigskip {\bf ans.} \bigskip \hrule \bigskip \vfill\eject {\bf 10.} (10 points) Find the {\bf point } on the plane $x+2y+3z=18$ where the function $f(x,y,z)=xyz$ is {\bf as large as possible}. \bigskip \hrule \bigskip The {\bf type} of the answer is: \bigskip \hrule \medskip \hrule \bigskip {\bf ans.} \bigskip \hrule \end