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Math 223 Name
Fall 1997
Test 1

Directions: Please read each question carefully and show all your work in doing each question. No partial credit will be given if you do not show your work. If you have any questions about the test, please ask me. Your work should of course be your own. Good Luck!

1.
(5 points) Consider the function z = f(x,y) given by the table below. Determine whether or not the function is linear and explain your reasoning.
x \( \setminus \) y -2   2  
-2 6 3    
  4 1 -2  
2 2 -1    

2.
(10 points) Consider the surface given by \( z = f(x,y) = \sqrt{y-x^{2}} \). Make a contour plot showing contours for z = 0,1,2,3.

3.
(20 points) Consider the vector \( \vec{u} = \vec{i} - \vec{j} + 2 \vec{k} \).

(a)
Find a unit vector in the direction of \( \vec{u} \).
(b)
Find a nonzero vector \( \vec{w} \) such that \( \vec{w} \)is perpendicular to \( \vec{u} \).

(c)
Given the vector \( \vec{v} = 6 \vec{i} + \vec{j} + 0 \vec{k} \),find the component of \( \vec{v} \) parallel to \( \vec{u} \).

4.
(20 points) You are given vectors \( \vec{v} = \vec{i} + \vec{j} -2 \vec{k} \) and \( \vec{w} = 2 \vec{i} - \vec{j} - 2 \vec{k} \). Let \( \theta \)be the smallest angle between \( \vec{v} \) and \( \vec{w} \).

(a)
Use the definitions of the dot product to find \( \theta \).
(b)
Use the definitions of the cross product to find \( \theta \).

5.
(20 points) Find an equation of the plane containing the points
\( (1,2,3) \mbox{, } (0,4,7) \) and (-1,6,1).

6.
(15 points)

=3.5ingraph6.ps

Consider the surface z = f(x,y) above.

(a)
Sketch a cross-section of the surface with x = 2.
(b)
Is fx(1,2.5) positive or negative? Explain your reasoning.

(c)
Is fx(1,2.5) larger or smaller than fy(3,0.75)? Explain your reasoning.

7.
(10 points) Consider \( f(x,y) = xy^{2} - \sin{2y} + e^{x} \).

(a)
Find \( \frac{\partial f}{\partial x} \).
(b)
Find \( \frac{\partial f}{\partial y}\vert _{(1,0)} \).

Vector Calculus
8/21/1998