13.2. VECTORS
Geometrical Approach to Vectors. The notion vector is used to indicate a quantity
(like velocity, acceleration, force) that has a magnitude and a direction. It is represented
by an arrow or a directed line segment. The length of the arrow is
Topics 11.1 and 11.2, Problems and Solutions
11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS
Suppose that a particle moves along a curve C given below
Because C fails the vertical test there is no equation of the form y =
f (x) describing the curve C (the tra
15.3. Partial Derivatives
15.4. Tangent Planes and Linear Approximation
Partial derivative of z = f (x, y) with respect to x at (a, b) is denoted by fx (a, b) and it is
fx (a, b) =
∂f
f (a + h, b) − f (a, b)
(a, b) = lim
.
h→0
∂x
h
Partial derivative of z
MATH-264/MAST-218 (MULTIVARIABLE CALCULUS)
SAMPLE MIDTERM TEST 2
SOLUTIONS
1. Find the length of the parametric curve
x=e
0.1t
cos t ,
y=e
0.1 t
sin t ,
z=e
0.1t
( 0 t 2 ).
Solution.
We use the following formula: if a curve is defined parametrically as
x=
Exercise Problems for the Midterm
MAST 218 / MATH 264, Winter 2016
Three pages !
Problem 1:
Sketch the curve given parametrically
x(t) = et sin t , y(t) = et cos t , t [0, ] .
Find its length. Find the area of the region bounded by the curve and the y-axi
Mast 218 Midterm Practice Test
Professor:
Instructions:
Richard Hall
Please answer all 4 questions which carry equal marks.
Explain your work clearly.
1.
(a) Find the equation of the plane P L which contains the three points
A = (1, 0, 1), B = (1, 2, 5),
Math 264 Sec A Midterm Test 30 th October 2012
Professor:
Instructions:
Richard Hall
Please answer all 3 questions which carry equal marks.
Duration: 1 hour. Please explain your work clearly.
1. In a pyramidal molecular model for ammonia NH3 , the hydroge
MATH-264/MAST-218 (MULTIVARIABLE CALCULUS 1)
SAMPLE MIDTERM TEST 2
Each problem is 20 pt. worth. Only calculators are permitted.
1. Find the length of the parametric curve
0.1
y=e
0.1 t
sin t ,
z=e
0.1 t
( 0 t 2 ).
sh is
ar stu
ed d
vi y re
aC s
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ou urc
MATH-264/MAST-218 (MULTIVARIABLE CALCULUS 1)
SAMPLE MIDTERM TEST 1
Solve as many problems as you can. Each problem is worth 20%.
No books and notes. Only calculators are permitted.
7
5
3
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Prob
MATH 264 Assignment 6
Name:
ID number:
Score (/50):
TJ 2:45 (Section A)
Due Thursday, Dec. 1
MW 2:45 (Section B) Due Monday, Dec. 5
Part 1: Answer-only problems [5 points each]
For the following problems, provide only the final answer. Write your answer i
page 1 of 1
Math 264 Secs A & B Final Exam December 2010
Professors:
Instructions:
Richard Hall & Dimiter Dryanov
Please answer all 5 questions which carry equal marks.
Explain your working carefully.
Calculators are permitted. [Lined booklets]
1. Conside
Midterm Test, MATH 264/Mast 218, Winter 2016
Only Faculty Approved Calculators are allowed during this test. Problems 1,2,3,4 are worth 10 marks each, Problems 5 and Bonus are 5 marks
each. Show all your steps. Write the solutions in the examination bookl
Midterm Test, MATH 264/Mast 218, Winter 2016
Only Faculty Approved Calculators are allowed during this test. Problems 1,2,3,4 are worth 10 marks each, Problems 5 and Bonus are 5 marks
each. Show all your steps. Write the solutions in the examination bookl
MATH-264/MAST218 (MULTIVARIABLE CALCULUS-1)
SAMPLE MIDTERM TEST-1
SOLUTIONS
Problem 1. Find the length of the curve x=15t
0 t 2 .
7,
y=42t
5,
3
z=70 t ,
Solution. The length is defined by the formula
2
L= x (t )+ y (t )+ z (t )dt
2
2
2
0
In our problem,
2
CONCORDIA UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
MULTIVARIABLE CALCULUS (MAST-218)
SAMPLE EXAM 2
Fall semester 2015-2016; instructor prof. A. Shnirelman
Solve as many problems as you can; each problem is 10% worth. No books
and notes are perm
MATH-264/MAST-218 (MULTIVARIABLE CALCULUS 1)
SAMPLE MIDTERM TEST 2
Each problem is 20 pt. worth. Only calculators are permitted.
1. Find the length of the parametric curve
0.1
x=e cos t ,
y=e
0.1 t
sin t ,
z=e
0.1 t
( 0 t 2 ).
2. Find the equation of the
CONCORDIA UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
MULTIVARIABLE CALCULUS (MATH-264)
SAMPLE EXAM 1
Fall semester 2015-2016; instructor prof. A. Shnirelman
Solve as many problems as you can; each problem is 10% worth. No books
and notes are perm
Midterm, MAST 218 / MATH 264, Winter 2017
SOLUTIONS:
Problem 1:
The curve is given parametrically
x(t) = 8e2t , y(t) = 4e4t 4t , t [1, 1] .
(a) Find the length of the curve. Hint: The expression You obtain under the square
root should be a full square.
(b
Midterm, MAST 218 / MATH 264, Winter 2017
SOLUTIONS:
Problem 1:
The curve is given parametrically
x(t) = e2t 3t , y(t) = 24et , t [1, 1] .
(a) Find the length of the curve. Hint: The expression You obtain under the square
root should be a full square.
(b)
Exercise Problems for the Midterm
MAST 218 / MATH 264, Winter 2016
SOLUTIONS !
Problem 1:
Sketch the curve given parametrically
x(t) = et sin t , y(t) = et cos t , t [0, ] .
Find its length. Find the area of the region bounded by the curve and the y-axis.
MATH-264/MAST-218 (MULTIVARIABLE CALCULUS 1)
SAMPLE MIDTERM TEST 1
Solve as many problems as you can. Each problem is worth 20%.
No books and notes. Only calculators are permitted.
7
5
Problem 1. Find the length of the curve x=15t , y=42 t , z=70 t
(0 t 2
Department of Mathematics & Statistics
Concordia University
MAST 218 (MATH 264)
Multivariate Calculus I
Winter 2016
Instructor:
Dr. P. Gora, Office: LB 901-17 (SGW), Phone: 514-848-2424, Ext. 3257
Email: [email protected]
Class Schedule:
Wednesday,
MATH 264 Assignment 3
Name:
ID number:
Score (/65):
TJ 2:45 (Section A)
Due Tuesday, October 25
MW 2:45 (Section B) Due Monday, October 24
Part 1: Answer-only problems [5 points each]
For the following problems, provide only the final answer. Write your a
MATH 264 Assignment 1
Due in class Sept. 27 (TJ 2:45 class), Sept. 28(MW 2:45 class)
Name:
ID number:
Score (/65):
Part 1: Answer-only problems [5 points each].
For the following problems, provide only the final answer. Write your answer in the boxes
prov
Topics 11.3, 11.4, 11.5
11.3 Polar Coordinates
Polar coordinate system introduced by Newton: It has a pole
that is the origin O of a Cartesian coordinate system. It has
a polar axis that coincides with the right-half of the x-axis in
the Cartesian coordin
TheGeometryofa Tetrahedron
Footnote 18:Section 10.4 Mark Jeng Professor Brewer
Whatisatetrahedron?
A tetrahedron is a solid with 4 vertices: P, Q, R, and S. There are also 4 triangular faces opposite the vertices as shown in the figure.
Problem1
1. Let v1
15.7. Maximum and Minimum Values. Part I.
Critical points. Classifying of a critical point as a point of loc max, a point of loc min,
or a saddle point.
Problem 7/ page 967. Find the loc maximum and loc minimum values; and the saddle points of
the functio
15.7. Max and Min Values. Part III. Extremal Problems.
Problem 40/ page 968. Find the point on the plane x − y + z = 4 that is closest to the point
(1, 2, 3).
Solution.
min d = (x − 1)2 + (y − 2)2 + (z − 3)2
subject to
x − y + z = 4.
Now, instead of looki