\input epsf \input graphicx \magnification 1200 \vsize 9.5truein \nopagenumbers \parskip 30pt \parindent 0 pt \def\f{\vskip 10pt} \def\h{\hskip 10pt} \def\u{\vskip -8pt} \def\bh{\hfil \break} \def\N{{\bf N}} \def\S{\Sigma_{i=1}^n} \def\0{{\bf 0}} \def\Z{{\bf Z}} \def\R{{\bf R}} \def\C{{\bf C}} \def\x{{\bf x}} \def\a{{\bf a}} \def\b{{\bf b}} \def\c{{\bf c}} \def\F{{\bf F}} \def\y{{\bf y}} \def\e{{\bf e}} \def\i{{\bf i}} \def\j{{\bf j}} \def\k{{\bf k}} \def\p{{\bf p}} \def\s{{\bf s}} Final Exam PART A, May 13, 2008 \hfil SHOW ALL WORK \hfil \vskip -30pt Math 28 Calculus III \hfil Either circle your answers or place on answer line. \vskip -20pt All problems required on this part of the exam. \vskip -20pt [13]~ 1.) Let $f(x) = (2x, e^{3x - 3})$. Let $g(x, y) = \sqrt{x^2y - 4}$. Use the chain rule to calculate $D(f \circ g)(1, 4)$ and $D(g \circ f)(1)$ \vfill \vfill \vfill \vfill $D(f \circ g)(1, 4)$ = $ \underline{\hskip 1in}$ ~~~~~~~\hfil~~~~~~~~ $D(g \circ f)(1)$ = $ \underline{\hskip 1in}$ \eject [13]~ 2.) {\bf Evaluate} the following integral by transforming this integral in Cartesian coordinates to one in polar coordinates. {\bf Sketch} the region of integration for the integral in Cartesian coordinates and the region of integration for the integral in polar coordinates. \f \vskip -10pt $\int_{-2}^2 \int_0^{\sqrt{4- x^2}} e^{-(x^2 + y^2)} ~dy dx$ = $ \underline{\hskip 3in}$. \vfill \eject [13]~ 3.) Let $S$ denote the surface of the cylinder $x^2 + y^2 = 9$, $ -1 \leq z \leq 1$. A parametrization of $S$ is $ \underline{\hskip 3in}$ Use this parametrization to calculate $\int \int_S 1 dS = $ $ \underline{\hskip 1in}$. The surface area of $S$ is $ \underline{\hskip 1in}$. \vfill \eject [13]~ 4.) Use a Lagrange multiplier to find the largest sphere centered at the origin that can be inscribed in the ellipsoid $3x^2 + 2y^2 + z^2 = 6$. \vfill \eject Final Exam PART B, May 13, 2008 \hfil SHOW ALL WORK \hfil \vskip -30pt Math 28 Calculus III \hfil Either circle your answers or place on answer line. \vskip -20pt Choose 4 out of the following 7 problems: {\bf Clearly indicate which 4 problems you choose.} Each problem is worth 12 points You may do more than 4 problems for up to five points extra credit. I have chosen the following 4 problems: $\underline{\hskip 3in}$ A.) Find the following limit if it exists. If it doesn't exist, state why you know it doesn't exist. $$lim_{(x, y) \rightarrow (0, 0)} = { xy - 2x^2 \over x^2 + y^2}$$ \vfill \eject B.) Let $\a, \b, \c \in \R^n$. \vskip -20pt Is the scalar product associative (i.e., does $\a \cdot (\b \cdot \c) = (\a \cdot \b) \cdot \c)?$ Is the cross product associative (i.e., does $\a \times (\b \times \c) = (\a \times \b) \times \c)?$ Prove that $\a \cdot (\b + \c) = \a \cdot \b + \a \cdot \b$. \vfill \eject C.) Show that the vector field $\F(x, y) = (y^2 + 2x + 4) \i+ (2xy + 4y - 5)\j $ is conservative. \vfill Find a scalar potential function for $\F$ \vfill \vfill Evaluate $\int_X \F \cdot d\s$ along the path $\x: [2, 5] \rightarrow \R^2$, $\x(t) = (t\sqrt{t^2 + 1}, 2t^2 + 3)$ \vskip 60pt \eject D.) Find the arclength parameter $s = s(t)$ for the path $\x(t) = (t^3, t^2)$, $0 \leq t \leq 10$ \vskip 60pt \vskip 60pt \vfill The length of this path is $ \underline{\hskip 1in}$. Express the original parameter $t$ in terms of $s$: $\underline{\hskip 2in}$. Reparametrize $\x$ in terms of $s$: \vfill \eject E.) Let $f(x, y, z) = x^2 sin(yz)$. Calculate the directional derivative of $f$ at $\a = ( 3, 0, 2)$ in the direction parallel to the vector (3, 4, 0). \vfill \eject F.) Let $\x(t) = (ln(t), 2t, e^{3t})$. The velocity of this path when $t = 1$ is $ \underline{\hskip 1in}$ \vfill The speed of this path when $t = 1$ is $ \underline{\hskip 1in}$ \vfill The acceleration of this path when $t = 1$ is $ \underline{\hskip 1in}$ \vfill The tangential component of acceleration of this path when $t = 1$ is $ \underline{\hskip 1in}$ \vfill The normal component of acceleration of this path when $t = 1$ is $ \underline{\hskip 1in}$ \vfill \eject G.) \end