Exam 1, MTH 241 M, Summer 2012
Name:
.
Carefully read all instructions and problems. Calculators are allowed, but must have
the memory cleared. Draw a kitten at the top of this page. The use of notes,
MTH 241T
Exam 1
10/2/14
Solutions
The grade ranges were: A (4.005.00): 85100, B (3.003.99): 7484, C (2.002.99): 5673, D
(1.001.99): 4655, F (.00.99): 045. A description of how the number gra
The Dot Product
Algebraic Definition
Dot Product
If ~a = ha1 , a2 , a3 i and ~b = hb1 , b2 , b3 i, then the dot product of
~a and ~b is the number ~a ~b given by:
~a ~b = a1 b1 + a2 b2 + a3 b3
The Dot
Functions of Several Variables
Functions of Two Variables
A function f of two variables is a rule that assigns to each
ordered pair of real numbers (x, y) in a set D a unique real
number denoted by f
Arc Length and Curvature
Length of a Curve
bp
L=
[f 0 (t)]2 + [g 0 (t)]2 + [h0 (t)]2 dt
a
bp
L=
[f 0 (t)]2 + [g 0 (t)]2 dt
a
Arc Length and Curvature
Example 1. Find the length of the arc of the cir
Cylinders and Quadric Surfaces
Cylinders
A cylinder is a surface that consists of all lines (called rulings) that
are parallel to a given line and pass through a given curve.
Cylinders and Quadric Sur
Equations of Lines and Planes
Equations of Lines and Planes
Describing a Line
vector equation
parametric equations
symmetric equations
Equations of Lines and Planes
Example 1.
(a) Find a vector equati
Derivatives and Integrals of Vector
Functions
Derivatives
The Derivative
~r(t + h) ~r(t)
d~r
= ~r 0 (t) = lim
h0
dt
h
tangent vector
tangent line
unit tangent vector
Derivatives and Integrals of Ve
Vectors
Introduction
The term vector is used to indicate a quantity that has both a
magnitude and direction.
Important Terminology
Vector
Initial Point
Terminal Point
Zero Vector
Vectors
Equivalen
Vector Functions and Space Curves
Vector Functions
A vectorvalued function, or vector function, is a function whose
domain is a set of real numbers and whose range is a set of vectors.
~r(t) = hf (t)
The Cross Product
Algebraic Definition
Cross Product
If ~a = ha1 , a2 , a3 i and ~b = hb1 , b2 , b3 i, then the cross product
of ~a and ~b is the vector ~a ~b given by:
~a ~b = ha2 b3 a3 b2 , a3 b1 a1
Math 241 W
College Calculus III
Spring 2018
Instructor:
Kim Javor Mathematics Building 101
Office Hours: Tuesday 12:30 1:45 and Wednesday 12:30 1:30
Office: 6458826 ; [email protected]
Textbook:
J
MTH 241 Formula Sheet
EXAM 1
Scalar Projection of b onto a: compab =
Vector Projection of b onto a: projab =

 
Vector Equation of the Line L through P0 = (x0, y0, z0) with position vector r0 and
Line Integrals
Introduction
Suppose we have a function f of two variables with f (x, y) 0
defined on a smooth, plane curve C
x = x(t)
y = y(t)
atb
The Area Problem: Find the
area of one side of the ve
ThreeDimensional Rectangular Coordinate
Systems
3D Space z
x
y
Important Terminiology
Origin
Coordinate Axes
Coordinate Planes
Octants
First Octant
Right Hand Rule
ThreeDimensional Rectangular
Mathematics 241T
Exam 2
11/6/14
SOLUTIONS
The grade ranges were: A (4.005.00): 88100, B (3.003.99): 7887, C (2.002.99): 6077, D (1.001.99): 5059, F (.00.99): 049. A description of how the num
PRACTICE EXAM 2 241
This is a closed book exam. SHOW YOUR WORK OR POINTS WILL BE
TAKEN OFF!
1. (a) Evaluate
2
0
3
1
(2xy 3y 2 )dy dx .
(b) Find f x , f xy and fxyz if f (x, y, z) = z2 ln(3x + 2y).
f
PRACTICE EXAM 3 241
This is a closed book exam. SHOW YOUR WORK OR POINTS WILL BE TAKEN OFF!
1. Let D be the region in the left half plane (the set of all (x, y) with x 0 ) that is between the
circles
The Divergence Theorem
The Divergence Theorem
The Divergence Theorem
~=
F~ dS
S
divF~ dV
E
Hypotheses of The Divergence Theorem
E is a simple solid region
S is the boundary surface of E, given with
Curl and Divergence
Curl
Curl
curl F~ =
P
Q P ~
R Q ~
R ~
i+
j+
k
y
z
z
x
x
y
curl F~ = F~
Curl and Divergence
Example 1.
Find curl F~ for the vector field:
F~ (x, y, z) = xz~i + xyz~j y 2~k
Curl and
Greens Theorem
Greens Theorem
Greens Theorem
Let C be a positively oriented, piecewisesmooth, simple
closed curve in the plane and let D be the region bounded by
C.
If P and Q have continuous partia
Stokes Theorem
Stokes Theorem
Stokes Theorem
Stokes Theorem
~
curlF~ dS
F~ d~r =
S
C
Hypotheses of Stokes Theorem
S is an oriented piecewisesmooth surface that is
bounded by a boundary curve C
C is
Surface Integrals
Parametric Surfaces
Suppose that a surface S has a vector equation
~r(u, v) = x(u, v)~i + y(u, v)~j + z(u, v)~k
f (~r(u, v)~ru ~rv  dA
f (x, y, z) dS =
S
D
Surface Integrals
Exampl
The Fundamental Theorem for Line
Integrals
The Fundamental Theorem
The Fundamental Theorem of Line Integrals
Let C be a smooth curve given by the vector function ~r(t),
a t b. Let f be a differentiabl
Vector Fields
Introduction
Let D be a set in R2 . A vector field on R2 is a function F~ that
assigns to each point (x, y) in D a twodimensional vector
F~ (x, y).
Let E be a set in R3 . A vector field
Parametric Surfaces and Their Areas
Parametric Surfaces
Parametric Curves
Parametric Surfaces
Parametric Surfaces and Their Areas
Examples of Parametric Surfaces
Plane containing:
point P0
two nonp
Cylinders
A cylinder is a surface that consists of all lines (or rulings) that are parallel to a given
line and pass through a given plane curve.
A right circular cylinder with radius r and height h.