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.
|x \ y||-1||0||1||2|
|x \ y||-3||-2||-1||0|
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 45o ?
(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 fx
and fy and the differential
df = fxdx
+ fy dy. Using
this and the linear approximation formula find the approximate value of
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.