Ma 231
Recitation 1Exercises-Solutions
January 16, 2014
Exercise 1 Sketch the vector a b c :
Exercise 2 Let a 2,6, 3 and b 2, 1, 2 .
(1) What is the cosine of the angle between a and b?
ab 4 6 6
8
Answer: cos( )
ab
21
49 9
(2) What is the unit vector in
Ma 231
Lecture 5 Outline
January 20, 2014
HW 5 (WebAssign)Due Wednesday, January 22nd at2 pm
Recall:
Parabola:
To make a quick sketch, locate vertex and decide if it open upward, downward, to
the right or to the left
Ellipse:
To make a quick sketch, locat
Ma 231
Lecture 7 Outline
January 24, 2014
HW 7 (WebAssign)due Monday, January 27th at 2 pm
Review Exercises:
(1) Parameterize the curve formed by the intersection of the surfaces y x 2 5x and
2 x 3 y z 7 (Use the first to parameterize x and ythen use seco
Review Problems for Exam ISet 1
Ma 231
Spring 2014
(1) (a) Let v 3,0, 4 and w 4, 7, 4 .
(i)
(ii)
(iii)
(iv)
What is the cosine of the angle between v and w?
What is the unit vector in the same direction as w?
What is the scalar projection of v on w, compw
Ma 231
Lecture 3 Outline
January 15, 2014
HW 3 (WebAssign)Due Friday, January 17th at 2 pm
Cross Product (Chapter 10.4):
i
j
k
v w v2 w3 v3 w2 , v3 w1 v1 w3 , v1w2 v2 w1 or v w v1
v2
v3 .
w1
w2
w3
Example 1 v 2i 2 j 3k, w 2i k. Find v w.
Properties:
v w w
Recitation 2Exercise Set 1-Solutions
Ma 231
January 23, 2014
Exercise 1 For the line through the points (2, 1,3) and (2, 2,5), find the vector,
parametric and symmetric equations.
Solution :
D 2 2, 2 (1),5 3 4,3, 2 .
Vector Parametric: r(t ) 2, 1,3 t 4,3,
Review Problems for Exam 1Set 2
Ma 231
(1)
Spring 2014
Let a 6, 3, 2 and b 2,1, 2 .
(i) a 2b
(ii) What is the unit vector in the same direction as a?
(iii) What is the cosine of the angle between a and b?
(iv) What is the scalar projection of b on a, com
Ma 231
Review of Freshman Calculus Basics
Spring 2014
Differentiation formulas:
f ( x) and g ( x) differentiable, c a constant:
d
d
cf ( x) cf ( x).
f ( x) g ( x) f ( x) g ( x).
dx
dx
d
d f ( x) f ( x) g ( x) f ( x) g ( x)
.
f ( x) g ( x) f ( x) g ( x)
Ma 231
Analytic Geometry ReviewWith Solutions
Spring 2014
Conic sectionscanonical equations and sketches:
Parabola (points equal distance from line and a point):
Ellipse (points so sum of distances to two points is a constant):
Hyperbola (points so differ
Ma 231
Review Problems for Exam ISet 1-Solutions
Spring 2014
(1) (a) Let v 3,0, 4 and w 4, 7, 4 .
(i) What is the cosine of the angle between v and w?
vw
4
Answer: cos( )
v w 45
(ii) What is the unit vector in the same direction as w?
w
4 74
Answer:
, ,
Ma 231
(1)
Review Problems for Exam ISet 2Solutions
Let a 6, 3, 2 and b 2,1, 2 .
(i) a 2b 2,5, 6
(ii) What is the unit vector in the same direction as a?
a
6 32
, ,
a
7 77
(iii) What is the cosine of the angle between a and b?
cos( )
ab
5
5
a b (7)(3) 21
Recitation Week 4Exercise Set 1Solution
Ma 231
February 6, 2014
Exercise Sketch the level curves z c, c 0,1, 2, 3 for the surface z x y 2 .
Using these and the traces x 0 and y 0 make a rough sketch of the graph of the
function f ( x, y) x y 2 .
Solution:
Ma 231
Determinants
Spring 2014
A determinant is a way to assign a certain number to an n-by-n square matrix
2-by-2 Determinant:
Example:
12
a11
a12
a21
a22
a11a22 a12 a21 .
1(4) 2(3) 4 6 2.
34
Larger matrices can be computed by a process known as expan
Ma 231
Lecture 12 Outline
February 5, 2014
HW 11 (WEBassign): Due Friday, February 7th at 2 pm
Chapter 11.4Linearization and Tangent Plane:
Linear approximation of a function f ( x, y) near (a, b) :
f
f
L( x, y) f (a, b) (a, b) x a (a, b) y b
x
y
Geo
Ma 231
Lecture 4 Outline
January 17, 2014
HW 4 (WebAssign) due Monday, January 20th at 2 pm
Lines and planes (Chapter 10.5)
Position vector: Vector with origin as initial pointthese can be identified with points.
Lines: Identify using point and direction.
Ma 231
Lecture 6 Outline
January 22, 2014
HW 6 (WebAssign): Due Friday, January 24th at 2 pm
Review Example: Sketch 4 x2 3 y 2 12 z 2 12 0
Chapter 10.7Vector-valued functions and curves
Vector-valued functions: r(t ) x(t ), y(t ), z (t )
Parameterized cur
Ma 231
Lecture 8 Outline
January 27, 2014
HW 8 (Webassign): Due Wednesday, January 29th at 2 pm
Arc length and Curvature (Chapter 10,8)
b
r(t ) a parameterized curve, a t b, arclength r (t ) dt.
a
Example 1 Find the length of the helix r (t ) 4cos(2t ), 4
Ma 231
Lecture 1 Outline
January 10, 2014
Homework 1 (WebAssign): Due Tuesday, January 14th, 1 pm
Chapter 10.1: Three-dimensional Coordinate Systems
Note that the positive x-axis comes out of the page.
First octantwhere all coordinates are nonnegative.
Mi
Ma 231
Lecture 2 Outline
January 13, 2014
HW 2 (on WebAssign) is due Wednesday, January 15th at 2 pm
Vectors (Chapter 10.2):
Scalar multiplication; vector addition/subtraction (done component-wise)
Example 1: v 3,6, 2 , w 4,0, 3 . Find: 2v 5w
Geometric in
Ma 231
Lecture 10 Outline
January 31, 2014
HW 9 (WEBAssign) Due Monday, February 3rd at 2 pm
Chapter 11.1: Functions of two and three variables
For a relation with two variables f ( x, y) to be a function, it must be that whenever you
enter the same value
Recitation 3Exercise Set 1-Solutions
Ma 231
January 30, 2014
Exercise 1 Find a parameterization for the curve formed by the intersection of the
y2
2
surfaces x
1 and x2 2 x y z 6 .
4
Solution:
y2
For x 2
1: use x cos(t ), y 2sin(t )
4
x 2 2 x y z 6 z
Ma 231
Lecture 13 Outline
February 7, 2014
HW 12 (WEBassign): Due Monday, February 10th at 2:00 pm
Chapter 11.5Chain Rule
Recall: Chain rule for functions of several variables:
dz dx z dy z
If z f ( x, y), x x(t ), y y(t ), then
.
dt dt x dt y
z x z y z
Lecture 9Review Examples for Exam 1
Ma 231
January 29, 2014
Exam IThursday, January 30, 8:30-9:45 pm
Lecture section 1SC160
Lecture section 2SC360
Lecture section 3Last names A-K in SC 162, last names L-Z in SC 362
(1)
(a) Let a 2,6, 3 and b 2, 1, 2 .
(i)
Ma 231
Lecture 11 Outline
February 3, 2014
HW 10 (WEBAssign)due Wednesday, February 5th at 2 pm
Chapter 11.3Partial Derivatives
Concept; definition as a limit of a difference quotient lim
x 0
Notations:
Examples 1 Find
f
f
df
and f x . Note:
not same as
The Case of the Crossed Vectors
A rap on the famous Baker Street door
Come in Watson, the door is unlocked.
Watson enters. Holmes. . . How did you know it was I?
Very elementary. I was looking out the window and saw you coming in.
Im afraid I have some ba