MATH 241- REVIEW Problems In this file I'll collect the bare-bones statements of review problems as they are given in class. Plug in your own numerical data and try them again! ******************************************************** 1. A quadrilateral has one pair of opposite sides parallel and of equal length. Show that the other pair of opposite sides is parallel, of equal length. 2. Decompose a vector V into components parallel and orthogonal to a given vector W. 3. Find the angle between a diagonal of a cube and a diagonal of one of its faces. 4. Given only the LENGTHS of two vectors V and W, find the largest and smallest possible values of: (i)the length of V+W; (ii)the dot product V.W; (ii)the length of the component of V in the direction W. 5. Let A be a 2X2 matrix. Given A(1,1) and A(1,-1), find A(2,3). 6. Let A be the 2X2 matrix which applied to any vector in the plane rotates it by 45 degrees counterclockwise. Find the entries of A. 7. Let A be the 2X2 matrix which sends each vector in the plane to its vector component on the direction (1,1). Find the entries of A. 8. Find all unit vectors perpendicular to two given vectors. 9. Given four points in space, check whether they are coplanar. 10. Given two planes, find their line of intersection and the angle between them. 11. Identify the motion described by a given parametrized curve 12. Show that a parametrized curve lies in a cone (or cylinder,etc.) 13. Given the definitions of angular momoentum and torque, and Newton's law of motion, show that the time derivative of angular momentum equals torque. *************************************************************** The following file contains the first exam for Math 241 when I taught it in the spring of 1995. The file is in LaTeX. (For instance, \frac{z+1}3 means (z+1)/3). When reading it, keep in mind that the text and topics covered were different. In particular, problem 4 deals with material not yet covered in 1997. **************************************************************** \documentstyle[12pt]{article} \begin{document} \begin{center} MATH 241-FIRST EXAM-FEBRUARY 8,1995 \end{center} {\bf 1.}[20]The vector $(1,0,2)$ may be written as a sum: $$ (1,0,2)=\vec{v}+\vec{w},$$ where $\vec{v}$ has the same direction as $(1,1,1)$ and $\vec{w}$ is perpendicular to $(1,1,1)$. Find $\vec{v}$ and $\vec{w}$. \vspace{1cm} {\bf 2.}[20](i) Consider the line $(L): \frac{x-1}2=y=\frac{z+1}3$ and the plane $(\alpha): 3x-2z=0$. Show that they don't intersect. (ii) Find a vector perpendicular to the direction vector of $L$ and to the normal vector of $(\alpha)$. \vspace{1 cm} {\bf 3.}[20](i)Let $w=(a,b)$ be the tangent vector to the circle $(2\cos t,2\sin t)$ at the point $(\sqrt{3},1)$. Find $a$ and $b$; (ii)Show that the line through the point $(\sqrt{3},1,0)$, with direction vector $(-1,\sqrt{3},2)$ is contained in the surface: $x^2+y^2-z^2=4$. \vspace{1 cm} {\bf 4.}[40]A particle of mass $5 kg$ moves under the effect of a force depending on position: $$\vec{F}(\vec{r})=-5 \frac{\vec{r}}{|\vec{r}|^2}$$ (in Newtons). At a certain point during the motion $(t=1)$, its position vector is given by $\vec{r}=(1,0,0)$ and the velocity vector is $\vec{v}=(1,1,1)$. (i)Find the acceleration vector $\vec{a}$ at $t=1$. (ii)Find the tangential component of the acceleration vector at $t=1$ (this is the vector component of $\vec{a}$ along $\vec{v}$). (iii)Find the normal component of the acceleration vector at $t=1$. (iv)Find the unit normal to the trajectory at $t=1$. \end{document} ***************************************************************************** The following file contains the second exam for Math241 when I taught in in 1995. This will give you a feeling for what you should expect on Wednesday, Oct 22. Remark: file in LaTeX. ***************************************************************************** \documentstyle[12pt]{article} \begin{document} \begin{center} MATH 241- SECOND EXAM- MARCH 6, 1995 \end{center} {\bf Instructions.} Show all work and erase any work you do not want graded. THIS EXAM CONSISTS OF FOUR PROBLEMS. \vspace{1cm} {\bf 1.} Suppose $F(x,y,z)=x^4+y^4+x^2z^2$ gives the concentration of salt in a fluid at $(x,y,z)$, and that you are at $(1,1,1)$. (a) [15]In which direction should you move if you want the concentration to change at the fastest possible rate? (b)[10] Suppose you start to move in the direction found in (a) with unit speed . How fast is the concentration changing at the starting point $(1,1,1)$? \vspace{1cm} {\bf 2.} In a pendulum experiment, $g$ is determined using the formula: $$g=\frac{4\pi ^2l}{T^2},$$ where $l$ is the length and $T$ is the period. (a)[15] Find the differential of $g$ (in terms of $l,T,dl$ and $dT$); (b)[10]Find an expression for the accuracy $\frac{dg}g$ in the computed value of $g$ in terms of the accuracies $\frac{dl}l$ and $\frac{dT}T$ in the measurements of $l$ and $T$. \vspace{1cm} {\bf 3.}[25] When you are diving in the ocean, the pressure $P$ you experience is a function of water density $x$ and depth below the surface, $y$: $$P=P(x,y).$$ Suppose you are descending into water (so density and depth are functions of time: $x=x(t)$, $y=y(t)$)). At a given moment: $$ \frac{dx}{dt}=5,\frac{dy}{dt}=2, \frac{\partial P}{\partial x}=1, \frac{\partial P}{\partial y}=2. $$ How fast is the pressure changing at that moment? \vspace{1cm} {\bf 4.} An international airline has a regulation that the sum of width $x$ and length $y$ of an item of carry-on luggage may not exceed 70 cm. (i)[10]Sketch in the same graph the region: $$D=\{(x,y)| x\geq 0, y\geq 0, x+y\leq 70\}$$ in the plane, and some level sets of the function: $$V(x,y)=10xy.$$ (ii)[15]Find the largest volume of a briefcase of height 10cm that fits these requirements. \end{document}