\documentclass[12pt, twocolumn]{article} \pagestyle{empty} \usepackage{fullpage} \usepackage{graphicx} \usepackage{amssymb,amsmath} \usepackage{amsbsy} \usepackage{psfrag} \usepackage{amscd} \begin{document} %\maketitle \begin{itemize} % section 14.1 \item The domain of the function $$\mathbf{r}(t) = \langle t^2, \sqrt{t-1}, \sqrt{5-t} \rangle$$ \vspace{2in} \item $$\lim _{t\rightarrow 1} \left( \sqrt{t+3}\mathbf{i} +\frac{t-1}{t^2 -1} \mathbf{j} + \frac{\tan t}{t} \mathbf{k} \right)$$ \vspace{2in} \item A parametric equation for the line segment that joins (0,0,0) to (1,2,3). \vspace{2in} \item A Daily Double! Sketch the curve with the equation $\mathbf{r}(t) = \langle t, \cos 2t, \sin 2t \rangle$. \vspace{2in} \item The time at which the following two particles collide, if they do indeed collide; for $t\geq 0$, their paths are given by \begin{displaymath} \mathbf{r}_1(t) = \langle t^2, 7t -12, t^2 \rangle \quad \mathbf{r}_2(t) = \langle 4t -3, t^2, 5t -6 \rangle \end{displaymath} % section 14.2 \vspace{2in} \item The derivative of $\mathbf{r}(t) = \langle \cos t, \sin t \rangle$. \vspace{2in} \item The derivative of $\mathbf{r}(t) = e^{t^2} \mathbf{i} - \mathbf{j} +\ln (1+3t) \mathbf{k}$. \vspace{2in} \item The unit tangent vector of the path $\mathbf{r}(t) = \cos t \mathbf{i} +3t \mathbf{j} +2 \sin 2t \mathbf{k}$ when $t=0$. \vspace{2in} \item The parametric equations for the tangent line to the curve with the parametric equations \begin{displaymath} x = e^{-t} \cos t,\ y = e^{-t} \sin t,\ z = e^{-t} \end{displaymath} at the point (1,0,1). \vspace{2in} \item $\int (e^t \mathbf{i} +2t \mathbf{j} + \ln t \mathbf{k}) dt$ % section 15.1 \vspace{2in} \item $f(2,-1,6)$ where $f(x,y,z) = e^{\sqrt{z-x^2 -y^2}}$. \vspace{2in} \item The approximate value for $f(-3,3)$ given the contour map for $f$ below: \vspace{2in} \item (A Daily Double!) Sketch the graph of $f(x,y) = 1-x^2$. \vspace{2in} \item The contour map of $f(x,y) = x-y^2$ showing the level curves $z = -2, -1, 0, 1, 2$. \vspace{2in} \item A thin metal plate, located in the $xy$-plane, has temperature $T(x,y)$ at the point $(x,y)$. The level curves of $T$ are called \textit{isothermals} because at all points on an isothermal the temperature is the same. Sketch some isothermals if the temperature function is given by \begin{displaymath} T(x,y) = 100/(1+x^2 +2y^2) \end{displaymath} % section 15.2 \vspace{2in} \item $$\lim _{(x,y) \rightarrow (0,0)} \frac{x^2}{x^2 + y^2}$$ \vspace{1.5in} \item $$ \lim _{(x,y,z) \rightarrow (3,0,1)} e^{-xy} \sin (\pi z/2)$$ \vspace{2in} \item The points at which the function $F(x,y) = \sin(xy)/(e^x -y^2)$ is continuous. \vspace{2in} \item The value of $$\lim _{(x,y) \rightarrow (0,0)} (x^3 +y^3)/(x^2 + y^2)$$ (hint: use polar coordinates). \vspace{2in} \item The value of $$\lim _{(x,y,z) \rightarrow (0,0,0)} \frac{xyz}{x^2 +y^2 +z^2}$$ (hint: use spherical coordinates). % section 15.3 \vspace{2in} \item The partial derivatives $f_x(x,y)$ and $f_y(x,y)$ of the function $f(x,y) = xe^{3y}$. \vspace{2in} \item The partial derivative $f_z (x,y,z)$ of the function $f(x,y,z) = xy^2 z^3 +3yz$ \vspace{2in} \item The partial derivative $\partial z / \partial x$ of the implicitly defined function $x^2 + y^2 +z^2 =3xyz$. \vspace{2in} \item The second partials $u_{xx}, u_{yy},$ and $u_{zz}$ of the function $u = 1/\sqrt{x^2 + y^2 + z^2}$. (Hint: they satisfy the three-dimensional Laplace equation $u_{xx} + u_{yy} + u_{zz} = 0$.) \vspace{2in} \item (A Daily Double!) The kinetic energy of a body with mass $m$ and velocity $v$ is $K = \frac12 mv^2$. Show that $$ \frac{\partial K}{\partial m} \frac{\partial ^2 K}{\partial v^2} = K$$ % section 15.4 \vspace{2in} \item The equation of the tangent plane to the surface $z = 4x^2 - y^2 +2y$ at the point (-1,2,4). \vspace{2in} \item The equation of the tangent plane to the surface $z = y \cos (x-y)$ at the point (2,2,2). \vspace{2in} \item The differential of $u = e^t \sin \theta$. \vspace{2in} \item The differential of $w = \ln \sqrt{x^2 + y^2 + z^2}$. \vspace{2in} \item The estimated maximum error in calculating the area of a rectangle of length 30 cm and width 24 cm (with an error at most 0.1 cm in each), using differentials. % section 15.5 \vspace{2in} \item The derivative $dz/dt$ where \begin{displaymath} z = x^2 y + xy^2,\ x = 2+t^4,\ y = 1-t^3 \end{displaymath} \vspace{2in} \item The derivative $dw/dt$ where \begin{displaymath} w = xe^{y/z},\ x = t^2,\ y=1-t,\ z=1+2t \end{displaymath} \vspace{2in} \item The partial derivative $\partial z/ \partial s$ where $z = e^r \cos \theta,\ r=st,\ \theta =\sqrt{s^2 + t^2}$. \vspace{1.5in} \item The partial derivative $\partial u/ \partial \theta $, where \begin{displaymath} u = x^2 +yz,\ x =pr \cos \theta,\ y = pr \sin \theta \end{displaymath} and $z=p+r$ when $p=2,\ r=3,\ \theta = 0$. \vspace{2in} \item (A Daily Double!) The rate at which the temperature is rising on the bug's path after 3 seconds, where the bug is crawling along the path \begin{displaymath} x = \sqrt{1+t},\ y = 2+\frac13 t \end{displaymath} ($x$ and $y$ are measured in cm) starting at $t=0$ seconds, and the temperature $T(x,y)$ at the point $(x,y)$ satisfies $T_x(2,3) = 4$ and $T_y(2,3) = 3$. % section 15.6 \vspace{2in} \item The directional derivative of $f(x,y) = \sqrt{5x -4y}$ at the point (4,1) in the direction $\theta = -\pi /6$. \vspace{2in} \item The gradient of $f(x,y,z) = xe^{2yz}$. \vspace{2in} \item The directional derivative of the function $f(x,y) = 1+2x\sqrt{y}$ at the point (3,4) in the direction of the vector $\langle 4, -3 \rangle$. \vspace{2in} \item The equation of the tangent plane the surface $x^2 +2y^2 +3z^2 =21$ at the point (4,-1,1). \vspace{2in} \item The points on the hyperboloid $x^2 -y^2 +2z^2 =1$ where the normal line is parallel to the line that joins the points (3,-1,0) and (5,3,6). % final jeopardy \vspace{2.5in} \item A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter $P$, find the lengths of the sides of the pentagon that maximize the area of the pentagon. \end{itemize} \end{document}