Math 200
Solutions to Test #1
1.
a) Sketch the curve in the xy-plane given parametrically by x(t) =
9 cos t, y(t) = 4 sin t, where 0 t .
cos t =
x
9
and
sin t =
y
4
cos2 t + sin2 t = 1 =
x2 y 2
+ ,
92 42
which is an ellipse. Since 0 t , we only get part
#171400 are mar a ’4, Mats ﬂ
1. (9.) Find an equation for the plane through the points (1,0, 1), (0, w2, 1), (5,5, 0).
(b) Find parametric equations for the line of intersection of the two planes:
:c+z=-—1 and 2x—2y—z=2.
2. At a given point (to, go) a f
Math 200, Midterm 1, Section 104
10 October 2002
The duration of the exam is 90 minutes. Answer all questions. No calculators, notes are not allowed.
Maximum score 50.
1. Find the parametric equations of the line of intersection of the planes:
x + y z = 2
Suggested Problems from Text
Selected sections from chapters 10,12,13, and 14 in the textbook will be covered in math 200. The
following table gives a list of suggested problems (not to be handed in for grading) for each section
covered in chapters 10 and
Math 200, Midterm 1, Section 105
10 October 2002
The duration of the exam is 90 minutes. Answer all questions. No calculators, notes are not allowed.
Maximum score 50.
1. Show that u(x, t) = e
2 2
k t
sin kx is a solution of the heat equation
u
2u
= 2 2
t
Math 200, Midterm 2, Section 107
November 1998
Instructions. The duration of the exam is 90 minutes. Answer all questions. Calculators are allowed.
Maximum score 50.
1. You have been asked to make a cylindrical container out of tin without a lid which mus
Math 200, Midterm 2, Section 102
2 November 2001
Instructions. The duration of the exam is 50 minutes. Answer all questions. Calculators are not allowed.
Maximum score 40.
2
1
1. Show that u(x, y, z) = x + y 2 + z 2 2 is a solution to Laplaces equation
DIFFERENTIATION RULES
Chain Rule - Dierentiation of multi-variable functions
Let F (x1 , x2 , . . . , xn ) be a function of the m variables x1 , x2 , . . . , xm and let each of these variables
be a function of the n variables t1 , t2 , . . . , tn . Then
F
Math 200
Practice Test #2
1. A ball moves along a table in the direction (0,1,0) at constant speed = 2 m/s.
a) Assume that gravity accelerates the ball downwards at 10 m/s2 (i.e. =
a
(0, 0, 10). If the ball rolls o the edge at (0,0,1), at what point P wil
Math 200 Practice Test #3
1.
a) Suppose an ant starting at (1, 0) is crawling at unit speed along the x-axis and feels
a temperature increase of 2o /s at the point (0, 0). Suppose a second ant crawling along
the y = x line at unit speed starting at (1, 1)
Math 200
Practice Test #2
Solutions
1. A ball moves along a table in the direction (0,1,0) at constant speed
= 2 m/s.
a) Assume that gravity accelerates the ball downwards at 10 m/s2
(i.e. = (0, 0, 10). If the ball rolls o the edge at (0,0,1), at
a
what p
Solutions to Practice Test #3
1.
a) Suppose an ant starting at (1, 0) is crawling at unit speed along the x-axis
and feels a temperature increase of 2o /s at the point (0, 0). Suppose a second
ant crawling along the y = x line at unit speed starting at (1
Math 200, Test 1, Solutions
1. [12]
Let
(a)
u = 3i + 5j 4k and v = 4i 3j + 5k.
u v =12 15 20 = 23
(b)
|u| = 9 + 25 + 16 = 5 2
(c) The cosine of the angle between u and v;
uv
23
23
=
cos = =
l 0.5
|u|v|
50
(5 2)(5 2)
o
arccos(.5) = 120
(d) The uni
Math 200, Test 2 Solutions
1. [12] Consider the following surface:
z = f (x, y) = x2 + x sin y.
(a) Find the equation of a plane that is tangent to the surface at the
point x = 1, y = 0.
(b) At what rate is the function increasing in the x-direction at th
Math 200 Test #2
Name:
Student Number:
1. [10 marks] A surface S consists of all points P = (x, y, z) such that the distance
between P and the plane y = 1 is equal to the distance between P and the line
(1, 0, 1) + t(0, 2, 0).
(a) Find the equation that d
Math 200 Test #1
Name:
Student Number:
1.
a) Sketch the curve in the xy-plane given parametrically by x(t) = 9 cos t,
y(t) = 4 sin t, where 0 t .
b)
i) Consider the region inside r = 4 cos and to the right of r = 2 sec
(i.e. r 2 sec ). Write down the int
Math 200
Solutions to
Practice Test #1
1.
a) One way to sketch the curve is to eliminate the parameter t from the parametric equations: x(t) = 3 + 2 cos t, and y(t) = 5 2 sin t to get an equation in just x and y:
x(t) = 3 + 2 cos t
x 3 = 2 cos t
and
y(t)
Math 200 Problem Set V
1) Write out the chain rule for each of the following functions.
a)
b)
c)
h
x
dh
dx
h
x
for h(x, y) = f x, u(x, y)
for h(x) = f x, u(x), v(x)
for h(x, y, z) = f u(x, y, z), v(x, y), w(x)
2) Use two methods (one using the chain rule)
Math 200 Problem Set IV
1) Show that the function z(x, y) =
x+y
xy
obeys
z
z
x x + y y = 0
2) Let f be any dierentiable function of one variable. Dene z(x, y) = f (x2 + y 2 ). Is
the equation
z
z
y x x y = 0
necessarily satised?
3) Four positive numbers,
Math 200 Problem Set III
1) A projectile falling under the inuence of gravity and slowed by air resistance proportional to
its speed has position satisfying
d2 r
dt2
= g k dr
dt
where is a positive constant. If r = r0 and
u(t) =
et dr (t)
dt
and substitut
Math 200 Problem Set II
1) Find the equation of the sphere which has the two planes x + y + z = 3, x + y + z = 9
as tangent planes if the centre of the sphere is on the planes 2x y = 0, 3x z = 0.
2) Find the equation of the plane that passes through the p
Math 200, Midterm 1, Section 107
8 October 1998
Instructions. The duration of the exam is 90 minutes. Answer all questions. Calculators are allowed.
Maximum score 50.
1. Find the angle between a diagonal of a cube and a diagonal of one of its faces.
[6 ma