%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!)} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \quad\quad {\bf RUID: (print!)} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\ \bigskip {\bf SSC:} (circle) None / I / II / I and II \bigskip \bigskip MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Final Exam , Tue., Dec. 15, 2020 \bigskip Email the completed test, renamed as {\tt finalFirstLast.pdf } to DrZcalc3@gmail.com no later than 3:30pm, (or, in case of conflict, three and half 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 11. \qquad \bigskip 12. \qquad \bigskip 13. \qquad \bigskip 14. \qquad \bigskip 15. \qquad \bigskip 16. \qquad \bigskip 17. \qquad \hfill\eject 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 (unless specifically told not to), 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) $3$ hours 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: \bigskip {\bf Possibly useful facts from school Geometry} (that you are welcome to use) : (i) The area of a circle radius $r$ is $\pi r^2$. (ii) The circumference of a circle radius $r$ is $2\pi r$ (iii) The parametric equation of an ellipse with axes $a$ $b$ and parallel to the $x$ and $y$ axes respectively is $x=a \cos \theta$, $y=b\cos \theta$, $0<\theta<2\pi$. (iv) The area of an ellipse with axes $a$ and $b$ is $\pi a b$ (v) The volume and surface area of a sphere radius $R$ are $\frac{4}{3} \pi R^3$ and $4\pi R^2$ respectively (vi) The volume of a cone is the area of the base times the height over $3$. (vii) The volume of a pyramid is the area of the base times the height over $3$. (viii) The area of a triangle is base times height over $2$. \bigskip {\bf Formula that you may (or may not) need} \bigskip If the surface $S$ is given in {\bf explicit} notation $z=g(x,y)$, above the region of the $xy$-plane , $D$, then $$ \int\int_S {\bf F} \cdot d {\bf S}= $$ $$ \int\int_D \left ( -P {{\partial g} \over {\partial x}} -Q {{\partial g} \over {\partial y}} +R \right ) \, dA \quad . $$ \vfill\eject \medskip {\bf 1}. (12 pts.) {\bf Without using Maple (or any software)} Compute the {\bf vector-field line integral} $$ \int_C \, (cos\, ( e^{\sin x}) +5y) \, dx \, + \, ( sin\, ( e^{\cos y} ) \,+ \, 11 x\,)\, dy \, \quad, $$ over the path consisiting of the five line segments (in that order) $$ (1,0) \rightarrow (-1,0) \rightarrow (-1,1) \rightarrow (0,2) \rightarrow (1,1) \rightarrow (1,0) \quad . $$ Explain! \bigskip \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 2.} (12 points) Change the order of integration $$ \int_{\frac{1}{4}}^{1} \int_{0}^{\sqrt{x}} \, f(x,y)\, dx \, dy \quad \quad . $$ \bigskip \hrule \bigskip \bigskip \bigskip {\bf ans.} \bigskip \bigskip \bigskip \hrule \vfill\eject {\bf 3.} (12 points) Find the equation of the tangent plane at the point $( \frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{6} )$ to the surface given implicitly by $$ 2 \cos (x+y) + 4 \cos (x+z)+ 8 \cos (y+z) \, = \, 7 \quad \quad . $$ Express you answer in {\bf explicit} form, i.e in the format $z=ax+by+c$. \bigskip \hrule \hrule \bigskip {\bf ans.} $z=$ \bigskip \hrule \vfill\eject {\bf 4.} (16 points) Let ${\bf a},{\bf b},{\bf c}$ be three vectors such that $$ {\bf a} \times {\bf b}\,= {\bf i} \,+ \,{\bf j} - {\bf k} \quad , \quad {\bf b} \times {\bf c} \,= {\bf i} \,- \,{\bf j} + {\bf k} \quad , \quad {\bf a} \times {\bf c} \,= 2{\bf i} \,+ \,{\bf j} + 2{\bf k} \quad . $$ What is $$ ({\bf a}+{\bf b}+{\bf c}) \times (2{\bf a}-{\bf b}+3 {\bf c}) \quad \quad ? $$ \bigskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 5.} (12 points) Find the three angles of the triangle $ABC$ where $$ A=(0,0,0) \quad, \quad B=(1,0,1) \quad, \quad C=(1,1,0) \quad \quad . $$ \hrule \hrule \bigskip \bigskip {\bf ans.} The angle at $A$ is: \qquad \qquad radians \quad;\quad \bigskip The angle at $B$ is: \qquad \qquad radians \quad;\quad \bigskip The angle at $C$ is: \qquad \qquad radians \quad;\quad \bigskip \hrule \vfill\eject {\bf 6.} (12 points) Find the directional derivative of $$ f(x,y,z)=x^3+y^3+z^3+xyz \quad , $$ at the point $(1,1,1)$ in a direction pointing to the point $(-1,-1,-1)$ \quad . \bigskip \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 7.} (12 points) Using the Chain Rule (no credit for other methods), find $$ \frac{\partial g}{\partial u} $$ at $(u,v)=(0,1)$, where $$ g(x,y)=3x^2-3y^2 \quad , $$ and $$ x=e^u \cos v \quad, \quad y=e^u \sin v \quad . $$ \bigskip \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 8.} (12 points) Without using Maple (or any other software), compute the vector-field surface integral $\int_S {\bf F}.d{\bf S}$ if $$ {\bf F}= \langle \, 3x +cos(y^3+yz) \, , \, -2y+ e^{x+z^2} \, , \, 5z+\sin(xy^3+e^x) \, \rangle \quad \quad , $$ and $S$ is the closed surface in 3D space bounding the region $$ \{(x,y,z): x^2+y^2+z^2<4 \quad and \quad x>0 \quad and \quad y<0 \quad and \quad z>0 \} \quad . $$ \bigskip \hrule \hrule \bigskip {\bf ans.} \bigskip \hrule \vfill\eject {\bf 9.} (12 points) Compute the vector-field surface integral $\int \int_S {\bf F}.d{\bf S}$ if ${\bf F}$ is $$ {\bf F} \, = \, \langle \, 3z \, , \, 2x \, , \, y+z \, \rangle \quad, $$ and $S$ is the oriented surface $$ z=2x+3y \quad, \quad 0