MTH 243

1.   Find an equation of the sphere having the points  A(2,-1,1) and  B(-2,1,5) as the opposite points of the diameter.

2.   Which of the tables below represent a linear function  f(x,y)? For those that represent a linear function find an equation for the function.

(a)
 
 

x \ y	-1	0	1	2
0	1.5	1	0.5	0
10	3.5	3	2.5	2
20	5.5	5	4.5	4
30	7.5	7	6.5	6

(b)

x \ y	-3	-2	-1	0
-3	18	13	10	9
-2	13	8	5	4
-1	10	5	2	1
0	9	4	1	0

3.   Find an equation of the linear function represented by a contour map below. Make sure that your equation is as accurate as possible by choosing points where values of the function are shown on the diagram exactly.

4.   Consider two vectors  a = (-2, 1, 2) and  b = (1, c, 4).

(a)  For what value(s) of c  is the angle between a and b  equal to 45^o   ?

(b)  For what value(s) of c  is the area of parallelogram formed by these two vectors equal to 19 ?
 

 5.  (a)  Find an equation of the plane containing the points  A(4,1,1),  B(-2,2,-1)  and  C(2,1,5).

(b)  Find an equation of the line through the points A and C.
 

6.   Let f(x,y) = ln(1 + y² +  x³  ). Find f_x  and f_y  and the differential df =   f_xdx +  f_y dy.  Using this and the linear approximation formula find the approximate value of f(0.2, 0.1).
 

7.  (a)  Find the directional derivative of  f(x,y) = 4x sin( x + y)  + cos( x - y) at  (0,p/4)  in the direction toward (p/4,p/2).

(b)  Find an equation for the tangent plane to the surface given by the equation z = f(x, y) at the point (p/4, p/4).

(c)  In what direction does f  increase the fastest at the point (2, 2) ? Give your answer as a unit vector. Find the maximum rate of change at this point.
 

8.   (a)   Find the gradient of the surface  x y⁵ + y z⁵  + z x⁵  - 3 = 0  at the point  (1,1,1).

(b)  Find an equation for the tangent plane to this surface at this point. What angle does this gradient vector make with this tangent plane ? Explain.