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
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
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 (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/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
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=
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
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
Topics 11.6; 12.10; 13.1
11.6. Conic Sections in Polar Coordinates
The conics are parabola, ellipse, and hyperbola. Parabola can be
deﬁned in terms of a focus and a directrix. Ellipse and hyperbola can
be deﬁned in terms of two foci.
Here we give a uniﬁed
13.5. Equations of lines and planes
Vector equation of a line. Consider a line L with a point P0 (x0 , y0 , z0 )
from the line and a vector v parallel to the line. Let P (x, y, z) be an
arbitrary point from the line L. Denote by r0 the position vector cor
14.1. Vector Functions and Space Curves
Vector function and component functions.
r(t) = f (t), g(t), h(t) = f (t)i + g(t)j + h(t)k,
t ∈ [a, b]
The domain is a set of numbers but the range is a set of vectors.
Example 1.
r(t) = cos(t), sin(t), t = cos(t)i
14.3. Arc Length and Curvature of Space Curves
The length of a space curve is deﬁned in the same way as a
length of a planar curve. Given a space curve
r(t) = f (t), g(t), h(t) ,
a≤t≤b
or equivalently with parametric equations
x = f (t),
y = g(t),
z = g(t
15.1 Multivariable Functions
Problem 6/page 902. Let
f (x, y) = ln(x + y − 1).
(a) Evaluate f (1, 1), f (e, 1), f (−4, 6).
(b) Find and sketch the domain of f .
(c) Find the range of f .
Solution. (a)
f (1, 1) = ln(1 + 1 − 1) = ln(1) = 0; f (e, 1) = ln(e
15.6. Directional Derivatives and Gradient Vector
Problem 6/page 956. Find the directional derivative of
f (x, y) = x sin(xy)
at the point (2, 0) in the direction θ = π/3.
Solution.
Du f (a, b) =
f (a, b) · u = fx (a, b), fy (a, b) · u1 , u2 .
We compute:
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
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
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: pawel.gora@concordia.ca
Class Schedule:
Wednesday,