This examination has 3 pages.
The University of British Columbia
Final Examination 12 December 2006
Mathematics 217
Multivariable and Vector Calculus
Closed book examination
Time: 150 minutes
Special Instructions: To receive full credit, all answers must
FINAL EXAM
Math 217 Section 101
December 12, 2008
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The exam is worth a total of 100 points with a duration of 2.5 hours. No books, notes or calculators
are allowed. Justify all answers, show all work and ex
December 2005
[12] 1.
MATH 217
Consider the surface
UBC ID:
Page 2 of 11 pages
cos(x) x2 y + exz + yz = 4.
S:
(a)
(b)
Suppose (0.03, 0.96, z) lies on S. Give an approximate value for z.
(c)
[12] 2.
Find the plane tangent to S at (0, 1, 2).
Suppose a > 0 i
The University of British Columbia
Final Examination - December 16, 2011
Mathematics 217
Time: 2.5 hours
LAST Name
First Name
Signature
Student Number
Special Instructions:
One formula sheet allowed. No communication devices allowed. One calculator allowe
Math 217: Partial Derivatives (Ch. 14)
Lecture 6 (Sep. 19)
Functions of Several Variables (reading: 14.1)
Functions of 2 variables
f : D R2 ! R
(x, y ) 7! f (x, y )
D is the domain of f
The range of f is
Ran(f ) := cfw_f (x, y ) | (x, y ) 2 D R
Example:
a
Lecture 9 (Sep. 26)
Partial Derivatives (reading: 14.3)
Idea: a partial derivative of a function of several variables is obtained by treating all
but one variable as a constant and dierentiating in the usual way with respect to
the remaining variable.
Den
Math 217: Vectors and the Geometry of Space (Ch. 12)
Lecture 1 (Sep. 5)
3D Coordinate Systems (reading: 12.1)
R = cfw_x |
1 < x < 1
R2 = cfw_(x, y ) | x, y, 2 R
R3 = cfw_(x, y, z ) | x, y, z 2 R
Coordinate planes:
The xy -plane in R3 is cfw_(x, y, 0) | x,
Lecture 10 (Oct. 1)
Linear Approximation
Recall that the tangent plane to the graph z = f (x, y ) at the point (a, b, f (a, b)
is given by the equation
z = f (a, b) + fx (a, b)(x
a) + fy (a, b)(y
b) .
This is the plane which best approximates the surface
Math 217 (101): Practice Mid-Term Test 2 Solutions
1. The region (you should have sketched it) is above the graph of y = x, below y = 2,
and between x = 0 and x = 4. To do the integral, we change the order of the iterated
integral:
4
2
y2
2
3
0
x
2
sin(y
Lecture 7 (Sep. 24)
Limits (reading: 14.2)
Example: Consider the two functions
f (x, y ) =
x4 y 4
,
x2 + y 2
g (x, y ) =
x2
xy
+ y2
which are dened for all (x, y ) 2 R2 except the origin (0, 0). How do they behave as
(x, y ) ! (0, 0)?
Denition: Suppose f
Lecture 5 (Sep. 17)
Arc Length and Curvature (reading: 13.3)
What is the length of the space curve C described by the vector function r(t) =
h f ( t ) , g ( t ) , h( t ) i , a t b .
Denition: So we dene the length of C to be
L :=
Z
b
a
0
|r (t)|dt =
Z
b
a
Math 217: Some solutions to Assignment 2
13.1 # 40:
For any xed z , the cylinder x2 + y 2 = 4 is a circle, and can be parameterized by
x = 2 cos(t), y = 2 sin(t), 0 t < 2 . So on the intersection with z = xy , we have
z = 2 cos(t) 2 sin(t). Thus the vecto
Math 217: Selected Solutions to Assignment 1
12.1 # 12: The sphere with centre (2, 6, 4) and radius 5 has equation
(x 2)2 + (y + 6)2 + (z 4)2 = 25.
Its intersection with the xy -plane (z = 0) is
(x 2)2 + (y + 6)2 = 25 16 = 9
is the circle of radius 3 cent
Math 217: Assignment 3 selected solutions
14.2 # 6: f (x, y ) = exy cos(x + y ) is continuous for all (x, y ) (being a product of
compositions of continuous functions), in particular at (1, 1), so
lim
(x,y )(1,1)
exy cos(x + y ) = e(1)(1) cos(1 + (1) = e1