Sample Practice Exam - M261

This is just a SAMPLE; not all topics can be on the final exam, and just because a topic is not on this sample final does not mean that it will not be included on the real final.

Colorado State University
Department of Mathematics
4/28/2000 5:47PM

(50 points, 10 points each part.) Consider the vector function $\mathbf{r}(t) = \langle e^t,-e^t,te^t \rangle,$which describes a particle moving along a curve in space.
Find the velocity v and the acceleration a at time t=0.
Find the speed of the particle at time t=0.
Write a parametric form of the tangent line to the curve at the point (1,-1,0).
Compute the work done in moving along the curve above from the point (1,-1,0) to the point (e,-e,e) in the force field $\mathbf{F}(x,y,z) = \langle y, x, 0 \rangle$.
Set up, but do not evaluate, the integral that computes the distance the particle travels from time t=0 to time t = 1.

(10 points) Use the multivariable chain rule to find $\frac{\partial w}{\partial s}$ and $\frac{\partial w}{\partial t}$, where $w = x^2 + y^2 + z^2, x = st, y = s \cos t, z = s \sin t.$(There is no need to simplify your answer.)

(40 points; 10 points each part) The temperature at a point (x,y,z) in space is $T = \frac{1}{x^2+y^2+z^2},$ with some appropriate choice of units.
Compute the gradient vector $\mathbf{\nabla}T$, at any point (x,y,z).
Find the rate of change (i.e., the directional derivative) of the temperature at the point P = (1,0,2) in the direction towards the point (2,1,2).
Specify a vector in whose direction the temperature increases the fastest at P.
What is this maximum rate of increase at P?

(10 points) Find $\frac{\partial z}{\partial x}$ if z is implicitly defined by the equation xy2z3 + x3y2z = x + y + z. (Your answer should be in the form $\frac{\partial z}{\partial x}
= ??$.)

(20 points) Find and classify all the critical points of the function $f(x,y) = \frac{3}{2}x^4 -6xy + 2y^3.$

(15 points) Set up completely, but do not evaluate, the double integral in polar coordinates that computes the mass of the plane lamina D in the first quadrant enclosed by the curves y=0,y= x, and the circle x2 + y2 = 9; use the density function $\rho(x,y) = (x^2 + y^2)^{3/2}.$ Sketch the plane region D as well.

(15 points) Set up completely, but do not evaluate, the triple integral in spherical coordinates that computes the yz-moment of the solid H, where H is the part of the sphere of radius 2, centered at the origin,lying in the first octant; use the density function $\rho(x,y,z) = z \sqrt{x^2 + y^2 + z^2}$. (Note that the yz-moment is the integral that is the ``numerator" of the formula computing the x-coordinate of the center of mass.)

(20 points; 10 points each part.) Consider the closed surface S, where S consists of the part of the paraboloid z=1-(x2 + y2) lying above the xy-plane, and the disk $x^2 + y^2 \leq 1 $ in the xy-plane.
Draw a careful sketch of the surface S.

Let F(x,y,z) be the vector field $\mathbf{F}(x,y,z) = \langle x^3, xz^2+zx^2, 3y^2z \rangle.$Use the divergence theorem to set up a triple integral that computes the flux integral $\int \int_S \mathbf{F} \cdot d\mathbf{S},$ using outward pointing normal vectors to the surface.

Use cylindrical coordinates for your final answer. Do not evaluate the triple integral.

(20 points; 10 points each part.) Let C be the boundary curve of the part of the surface S, given by the equation z=1-x2-y2, lying in the first octant. Note that the curve C lies entirely on the surface S.
Draw a careful sketch of C and the surface. Draw a sample outward pointing normal vector to S. Indicate with arrows a direction of travel around C that positively orients C with respect to this outward pointing normal vector.
Let $\mathbf{F}(x,y,z) = \langle y,y,x^2 + y^2 \rangle$ be a vector field. Use Stokes' theorem to compute the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$.