Sample U341 Final Problems

  1. Find a vector of length 7 in the same direction as MATH, where MATH and MATH.

  2. Let MATH, MATH, and MATH .

    1. Find the equation of the plane through MATH.

    2. Find the area of triangle $/QTR{bf}{PQR}$.

    3. Find the (parametric) equations of the line joining $/QTR{bf}{P}$ and $/QTR{bf}{R}$.

    4. Find the point where the line $/QTR{bf}{L}:x=2+t$, $y=-1+2t$, $z=-3+5t$ intersects the plane from part a.

      We put these values for $x,y,z$ into the equation for the plane:MATHSolving for $t$ give $t=7/2$, so the point is MATH.

    5. Find the angle (less than $/pi /2$) between the normal vectors to the planes $6x+2y-z=5$ and $3x-y+2z=8$.

  3. Find the equation of the line that is tangent to the curve MATH at the point where $t=0$.

  4. The following are the equations of three surfaces. Give their names and either draw sketches of them, or describe in words what they look like.

    1. $x^{2}=y^{2}+z^{2}$

    2. $z=x^{2}-y^{2}$

    3. MATH (Spherical)

  5. MATH. Verify that the mixed partials $h_{xy}$ and $h_{yx}$ are equal.

  6. The temperature $T(x,y)$ at the point $(x,y)$ on a metal plate is given by MATH, where $T$ is measured in $^{/circ }C$ and $x$ and $y$ are measured in $cm$.

    1. Compute the gradient vector field of $T(x,y)$.

    2. Starting from the point $(-1,2)$, what is the rate of change of temperature in the direction of the vector MATH?

    3. From the point $(-1,2)$, in which direction should one move so that the temperature decreases as fast as possible?

    4. Isothermal curves are the level curves of temperature. Find a normal vector to the isothermal curve of $T$ passing through $(-1,2)$.

    5. Write the equation of the tangent line to the isothermal curve of $T$ passing through $(-1,2)$.

  7. Sketch the contour diagram for $f(x,y)=-x^{2}-y+2$ for the values $c=2,1,-1$.

  8. For each of the surfaces $S$ below, find the equation of the tangent plane at the indicated point $P$.

    1. $S$ is the graph of the function MATH and $P$ is the point where $x=2$ and $y=2$.

    2. $S$ has equation MATH, $P=(2,2,2)$. (Hint: bring all the variables to one side.)

    3. $S$ is the surface parametrized by MATH, $P=/QTR{bf}{r}(1,2)$.

  9. Consider the function MATH.

    1. Find the linearization $/ L(x,y)/ $ of the function $/ f(x,y)/ $ at the point $/ (3,3)/,$.

    2. Use the linearization $/ L(x,y)/ $ in part a to find an approximation to $f(2.8,3.1)/ $

  10. The charge density at the point $(x,y)$ is given to be MATH.

    1. Find $/nabla /beta $, the gradient of the charge density.

    2. Suppose we convert to polar coordinates: $x=r/cos /theta $, $y=r/sin /theta $. Calculate, using the Chain Rule, the derivatives MATH and MATH, expressed in terms of $r$ and $/theta $.

  11. A particle is traveling on the curve $x=3/cos (2/pi t)$, $y=3/sin (2/pi t)$, $z=2t$, where $t$ represents time in seconds. Suppose also that the temperature at a point is given by MATH.

    1. Describe the curve for $0/leq t/leq 1$. Where will the particle be at $t=0.1$?

    2. Use the chain rule to find the rate of change of temperature of the particle at $t=0.1$.

  12. The density (gms per cu cm) of a gas at temperature $T$ (deg K.) and pressure $P$ (atm) is given by MATH ($K$ is a constant).

    1. Use linearization to estimate $/Delta /delta $ in terms of $/Delta P$ and $/Delta T$.

    2. Estimate the percent change in $/delta $ if $P$ changes by $k_{1}$% and $T$ by $k_{2}$%.

  13. Find all critical points of MATH. Then use the second derivative test to classify the critical points (max/min/saddle). Do the same for MATH.

  14. Let MATH. Find the absolute maximum and minimum of $f$ on the triangular region with vertices (0,0), (3,0) and (3,3).

  15. Use Lagrange multipliers to find the point on the plane $-x+4y+3z=2$ which lies closest to the origin.

  16. For each of these integrals, sketch the region of integration and evaluate the integral by changing the order of integration.

    1. MATH

    2. MATH.

  17. Let $R$ be the region between the parabola $y=9-x^{2}$ and the line $y=x+3$. Let $V$ be the solid bounded below by $R$ and above by the plane $z=2y+5$. Calculate the volume of $V$.

  18. Consider the annulus $D$: MATH with density MATH (gms per cm$^{2}$); use polar coordinates to calculate its total mass.

  19. Set up but do not evaluate a triple integral (in rectangular coordinates) for the volume of the solid cut from the cylinder $x^{2}+y^{2}=4$ by the plane $z=0$ and the plane $x+z=3$.

  20. Find the volume of the region bounded by MATH and MATH.

  21. Let $R$ be the pyramid bounded by the $xy$, $yz$ and $zx$ planes, and above by the plane $x/2+y/5+z/10=1$. Suppose its density is given by some function $/delta (x,y,z)$ (grams per cu cm). Set up an iterated integral to calculate its mass.

  22. Use spherical coordinates to calculate the following integral: MATHwhere $B$ is the unit ball MATH.

  23. Let $/QTR{cal}{C}$ be the curve in the $xy$-plane with parametric equation MATH.

    1. Let MATH be the force on a particle at $(x,y)$. Calculate the work done by $/QTR{bf}{F}$ in moving the particle from $(0,0)$ to $(4,8)$ along $/QTR{cal}{C}$.

    2. Suppose, instead, that MATH is the force. $/QTR{bf}{G}$ is the gradient of a function $g$; find $g$ and use $g$ to calculate the work done in moving from $(0,0)$ to $(4,8)$.

  24. Let $/QTR{cal}{C}$ be the closed rectangular curve MATH and let MATH be a planar flow.

    1. Use Green's Theorem to find the circulation of $/QTR{bf}{F}$ around $/QTR{cal}{C}$.

    2. Use Green's Theorem to find the flux of $/QTR{bf}{F}$ through $/QTR{cal}{C}$.

  25. Let $/QTR{cal}{S}$ be the surface of the solid cylinder bounded on the bottom by the $xy$-plane, on the top by $z=3$ and on the sides by the cylinder $x^{2}+y^{2}=4$. Suppose we have the flow given by MATH. Use the Divergence Theorem to calculate the outward flux: MATH.