MA201 Midterm Test
November 4, 2010
[7 marks ]
1. Find parametric equations for the line which is tangent at the point
t
t
curve r(t) = sin ti + 3e j + 2e k.
[6 marks ]
2. A particle starts at the origin with initial velocity i
6ti + 12t2j 6tk. Find its v
MA201 Lab Report 4 Notes
TOPIC SUMMARY
Iterated Integrals:
If f is continuous on the rectangle R = f(x; y )ja x b; c y
i R hR
i
RR
R b hR d
d
b
f (x; y )dA = a c f (x; y )dydx = c a f (x; y )dxdy
R
dg; then:
The integrals on the right hand side are known
MA201 Lab Report 5 Multiple Integrals
Name:
Winter 2006
Student Number:
Lab
1. [9 marks] For each of the following iterated integrals:
(a) Reverse the order of integration.
(b) Evaluate the integral using the Doubleint(.); command in Maple.
[Note: For eac
MA201 Lab Report 6 Notes
TOPIC SUMMARY
Line Integrals:
With respect to Arc Length:
Used to evaluate an integral over a curve, say C; instead of over an interval [a; b] (as we are used to)
If f is a continuous function and C is a smooth curve, then the lin
MA201 Lab Report 6 Line Integrals
Name:
Winter 2006
Student Number:
Lab
1. [8 marks] Suppose C is the lower half of the circle with radius 3 centered at the origin.
(a) Determine two different parametrizations of C:
Z
(b) Manually evaluate
yds using both
MA201 Information Pertaining to Labs (Fall 2014)
Lab Coordinator:
Katie McGarry
Oce:
Phone:
BA530 (Bricker Academic Building)
884-0710 Ext. 2146 E-mail: cmcgarry@wlu.ca
Lab Format
Each lab (every other week starting September 17) will consist of a number
MA201 - Midterm Page 1 of 9
Student Number: M
(12 marks) 1. Let f(x, y) = 2:569 3g + 1.
Mfr/{e0 Find the equation of the tangent plane to f at the point (5, 0).
EF/ ® : Qs/j => {71(30): Q60 12;
[El/45) Use part (a) to approximate f(4.8,0.25).
- Lei L093)
MA201 Lab Report 4 Multiple Integrals
Name:
Student Number:
Lab
Fall 2013
1. [7 marks] Consider the region D bounded by y = x, x = 3, and y = 0.
(a) Sketch D on the axes provided.
x2 exy dA as an iterated integral.
(b) Integrating with respect to y rst, e
MA201 Lab Report 1 Vectors and The Geometry of Space
Name:
Student Number:
Lab
Fall 2013
x1 + x2 y1 + y2 z1 + z2
,
,
, which all lie on
2
2
2
1. [6 marks] Consider the points P (x1 , y1 , z1 ), Q(x2 , y2 , z2 ), and M
the same straight line.
(a) Prove tha
MA201 Lab Report 2 Vectors and The Geometry of Space; Vector Functions
Name:
Student Number:
Lab
Fall 2013
Equations of Lines and Planes:
1. [4 marks] Determine both the parametric and vector equations of the line of intersection between the planes
2x 3y
MA201 Lab Report 3 Functions of Several Variables
Name:
Student Number:
Lab
Fall 2013
1. [8 marks] In this question, we will be investigating the linear approximation L(x, y) and the quadratic approximation (called the second-degree Taylor polynomial) Q(x
MA201 Lab Report 1 Vectors and The Geometry of Space
Name:
Student Number:
Lab
Winter 2014
1. [5 marks] Prove that an angle inscribed in a semicircle is a right angle. [Hint: Consider a circle with centre
C and radius r. Let QRP be inscribed on the diamet
MA201 Lab Report 4 Notes
TOPIC SUMMARY
The Chain Rule:
Case 1: If z = f (x; y ) is a differentiable function of x and y , where x = g (t) and y = h(t) are both differentiable
functions of t; then:
@z dx @z dy
dz
=
+
dt
@x dt
@y dt
Case 2: If z = f (x; y )
MA201 Lab 3 Report Vector Functions; Partial Derivatives
Name:
Winter 2006
Student Number:
Lab
1
t cos t; cos t + t sin t; t2 .
2
(a) Determine the arc length of the curve from t = 0 to t = :
1. [13 marks] Consider the vector function ~(t) =
r
L
=
Z
sin t
MA201 Midterm Test
November 1, 2011
Page 1 of 7
(1, 1, 1) which is parallel
to the tangent line of the space curve r(t) = (sin t)i + (sin 2t)j + tk at the origin
when t = 0.
[10 marks ] 1. Find parametric equations for the straight line through the point
February 3, 2005
MA201 Test I
Page 1 of 1
[10 marks ] 1. Find an equation of the plane that passes through the line of intersection of the planes xz = 1
and y + 2z = 3 and is perpendicular to the plane x + y 2z = 1.
[6 marks ] 2. Change (1, 1, 6) from rec
March 10, 2005
MA201 Test II
Page 1 of 1
[5 marks ] 1. Find an equation of the tangent plane to the surface z = y ln x at the point (1, 4, 0).
[5 marks ] 2. Find the dierential of the function
w = xyexz .
[5 marks ] 3. Find the directional derivative of f
WILFRID LAURIER UNIVERSITY
WATERLOO, ONTARIO
Winter Term, 2003
Course ID: MA201 B
Name:
Student ID:
Course Title: Multivariable Calculus
Professor: Dr. Y. Chen
Number of Pages: 3 plus cover page
Length of Examination: 2 hours
INSTRUCTIONS:
Please place yo
MA201 Final Exam, Fall Term 2007
Page 1
Disclaimer: Tests and exams from previous o erings of mathematics courses are posted to the on-line Exam
Bank, administered by the Mathematics Assistance Centre, as a courtesy to WLU students. Past exams
should be u
MA201 Midterm, Fall Term 2007
Page 1
Disclaimer: Tests and exams from previous o erings of mathematics courses are posted to the on-line Exam
Bank, administered by the Mathematics Assistance Centre, as a courtesy to WLU students. Past exams
should be used
MA 201 Week 1 Lab Notes
Differentiation Techniques Review:
1. Rules of Differentiation:
Power Rule:
dn
(x ) = nxn
dx
Product Rule:
d
(f (x)g (x) = f 0 (x)g (x) + f (x)g 0 (x)
dx
Quotient Rule:
d
dx
Chain Rule:
1. Given f (x) and g (x); by the chain rule:
MA 201 Lab Report 1 -Differentiation/Integration Techniques Review
Name:
Winter 2006
Student Number:
Lab
dy
1. [12 mark ] Determine
for each of the following. Then evaluate the derivative at the point indicated,
dx
simplifying as much as possible.
!
3=2 4
MA 201 Lab Report 2 Notes
Topic Summary: (Chapter 12-Vectors and the Geometry of Space)
Dot Products and Cross Products:
Given ~ =< x; y; z > ~ =< m; n; p >:
a
b
~ ~ = xm + yn + zp = j~ j ~ cos ; where is angle between ~ and ~
ab
ab
a
b
~
ijk
xyz
mnp
~=
b
MA201 Lab Report 2 Vectors and The Geometry of Space
Name:
Winter 2006
Student Number:
Lab
1. [4 marks ] Dot Products and Cross Products:
Using the with(VectorCalculus): package in Maple, de ne the vectors ! =< 1; 3; 5 >; ~ =<
u
v
and ! =< 2; 7; 4 >. Calc
MA201 Lab Report 3 Notes
TOPIC SUMMARY
Vector Functions and Space Curves:
For t 2 [a; b], a vector function is de ned by ~(t) =< f (t); g (t); h(t) > and is a function whose domain is a set
r
of real numbers while the range is a set of vectors.
Given ~(t)
MA201 Lab Report 2 Vectors and The Geometry of Space; Vector Functions
Name:
Student Number:
Lab
Winter 2014
Equations of Lines and Planes:
1. [4 marks] Determine the angle (in radians) at which each of the following pairs of planes intersect (if at all).