Sample Questions for Exam 1
Math 2374, Spring 2002
rogness@math.umn.edu

Problems marked by an asterisk (*) may be harder than the actual exam.

1. Find the equation for the following piece of a single cone.  Note that the vertex is at (0,0,200), and the bottom is on the circle x^2 + y^2 = 100in the xy-plane.

[Graphics:HTMLFiles/index_2.gif]

2.  (a) Find the equation for a plane containing the points (1,2,3), (2,4,9), and (-1,3,-4).

(b)  Find the parametric equation for the plane given by the cartesian equation  2x - 3y + 3z = 9.

(c)  Find the cartesian equation for the plane given by the parametric equation p(s, t) = (0, 1, 1) + s(-1, 1, 0) + t(2, 4, 9).  

(d) Which, if any, of the following vectors is normal to the plane p(s, t) used in part (c)?
        (0,1,1), (2,2,2), (-3,-3,2), (0,1,0).

[Note: you should be able to answer this question either before or after finishing part (c), i.e. you should be able to do it using the information in either the cartesian or the parametric equations to the plane.]

3.  Suppose we have the following linear transformations:

T(x, y) = (3x, 2x + y, x - 9y)

S(x, y, z) = (-4z, 2y + z, x + z)

(a) Find T(S(Overscript[v, ⇀])), if possible.  If it is not possible, explain why.

(b) Find S(T(Overscript[v, ⇀])), if possible.  If it is not possible, explain why.

4.  (a) using the limit defintiion of partial derivatives, find the value of ∂f/∂x(0,0) for the function

f (x, y) = { 3 3 x - y -------, if (x, y) ≠ (0, 0) ; ... 0,             if (x, y) = (0, 0) .

(b) Using any valid method, find a general formula for ∂f/∂y at any point away from (x, y) = (0, 0) .

5.  (a) Plot the level curves for c=0,1,4,9 of the function

f (x, y) = x^2/4 + y^2/9

(b) Plot the level surfaces for c=1,4 of the function

f (x, y, z) = x^2 + y^2 + z^2

6.* Given a three dimensional vector Overscript[a, ⇀], show that T(Overscript[x, ⇀])=(Overscript[a, ⇀].Overscript[x, ⇀])Overscript[a, ⇀] is a linear transformation.  (Hint: write the right hand side using matrix notation and use the fact that kOverscript[a, ⇀]=Overscript[a, ⇀]k for a scalar k.)  [This is problem 22 in section 2.3]

7. Parametrize the intersection of the cylinder x^2 + y^2 = 4 and the plane z = x - y.

8.  Give a parametrization for the line segment from the point (1,0,0) to the point (2,3,-4).  How would you change this to give a parametrization for the entire line containing these two points?

9.  Working in ^4, consider the points A=(0,0,1,0), B=(1,0,2,0), and C=(3,2,0,-1).

(a) Find the vectors Overscript[AB, ⇀] and Overscript[AC, ⇀].

(b) Find the angle between Overscript[AB, ⇀] and Overscript[AC, ⇀].

(c) Find the area of the parallelogram spanned by Overscript[AB, ⇀] and Overscript[AC, ⇀].  Warning: the cross product is only defined in three dimensions, so you cannot use the cross product to answer this question.  Draw a few good pictures and use a little trigonometry.  Remember the area of a triangle is (1/2)(base)(height), and the area of a parallelogram is (base)(height).

10. Find the linear approximation to the given function at the indicated point.

(a) f(x, y) = x^2 + y^2, a=(2,3).
(b) g(x, y) = (x^2 - y^2, 2xy), b=(3,1)

11.  Use your approximations to estimate the value of FormBox[RowBox[{f, (, RowBox[{2.1, ,, 3}], )}], TraditionalForm], and FormBox[RowBox[{g, (, RowBox[{2.9, ,, 0.99}], )}], TraditionalForm].  Compare these to the true values of the functions at these points.  How good are the approximations?

12.  Find the total derivative for f(x, y) and g(x, y) at the points a and b.  (Note that you should have already done most of the work for this in #10, i.e. finding the Jacobians.)

13*. Find a function F : ^2⟶^2whose Jacobian matrix is ( 2 ) y 2xy y cos x sin x

14.  Use the chain rule to find the derivative of g(f(x))at the indicated point.

(a) g(x, y) = (x^2y^3, 3x - y^2), f(x, y) = (-y, x), a=(3,2).
(b) g the same as above, f(x_1, x_2, x_3) = (x_1x_3, x_2/(x_1x_3)), a=(3,1,-1).

15.  Find the partial derivatives of u with respect to s and t for the following functions:

(a) u = x^2y^3 + x - 3y, x = t^2 - s, y = t + s^2.
(b) u = g(s^2 - t^2, 2s t).

Created by Mathematica  (February 16, 2004)