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(MATH100)midtermex.pdf
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Here are some preparatory exercises for the midterm

Sample 1 Ex 1 Let L be the line (x,y,z)=(-2,0,1)+t(3,2,2) and let M be the line (x,y,z)=(3,5,4)+t(1,-1,1).
(a)
Where do the two lines intersect?

(b)
Find the equation of a plane which is parallel to the two lines and exactly one


unit distance away from the lines. Ex 2 Consider the vector function r(t)=(t,t2)
(a)
Find the tangent line to the graph of r(t) at point (2,4)

(b)
Find the arc length parametrization of curve r(t) ,0t1


Ex 3 Find the limit in 2-space if it exists. Otherwise prove it does not existcos xy .1
(a)
lim
( x,y)(0,0) x
ex . ey
(b) 22
lim
( x,y)(0,0) x + y
Ex 4 Consider the surface z=x2+y2.
(a)
The plane y=3 intersects the surface in a curve. Find the equations of the tangent line to this curve at x=2.

(b)
The plane x=2 intersects the surface in a curve. Find the equations of the tangent line to this curve at y=3.


Ex 5 If w=f(x2-y2,y2-x2), show that
.w .w
y + x = 0.
.x .y
Ex 6 (a) In what direction is the directional derivative of
x2 . y2
f (x, y) = 22
x + y
at (1,1) equal to zero?
(c) Find the direction in which f decreases most rapidly at (1,1).
Sample 2 Ex 1 A plane P contains the points (1,1,1), (2,3,0) and (2,5,-1). Find the equation of a parallel plane containing the origin.
Ex 2 Find the equation of the plane that contains the intersection of the planes x-2y-5z=3 and 5x+y-z=1 and is parallel to the plane 4x+3y+4z+7=0.
Ex 3 (a) Express the general equation of the elliptic paraboloid in spherical coordinates.
(b) Prove that if z=f(x,y) is the equation of the elliptic paraboloid, then f satisfies the one-dimensional wave equation:
.2 f 2 .2 f
2 = c 2 where c is a positive constant.
.x .y
(c) Rewrite the one-dimensional wave equation using the polar coordinates: x=rcos, y=rsin.
Ex 4 Find the limit if it exists
lim (x2 + y2)sin 21 2
( x,y)(0,0) x + y

Ex 5 Find the arc length of r(t)=ti+(lnt)j+2
2t k for t2.