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Module 1: Assignments 0-6
on Trigonometric Functions
Test 1 on Module 1 is assigned as part of
Assignment 7 |
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For Assignment 0 click on each of the following and
read the document:
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01 |
6.1 |
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Class Notes
#'s 1-2 |
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Sketching Exercise
Graphing Vertical Position; Effects of Angular Velocity, Radius,
Starting Point
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Class Notes Topics
#01: First Circular Model; Introduction to Radian Measure
·
definition and illustration of
radian
·
number of radians corresponding
to complete circle
·
relationship between radian
measure of angle and arc length on circle, common sense and symbolic
·
dividing the circle into 8 or
12 sectors, labeling angles
#02: Sketching Exercises
·
time rate of change of angle
and peak separation on graph of y vs. t
·
radius of circle at y
coordinates of peaks and valleys
·
period, frequency, amplitude,
angular frequency, wavelength
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sketch
a
graph
of
y
coordinate vs. clock time
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specify
the
effect
on the curve
of the
angular
velocity of
the point moving around the circle
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Consider the
effect
of
changing the radius
of the circle
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Investigate
the effect of the starting point on the graph
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Objectives:
Relate
{theta, x, y} on the unit circle.
Relate
{omega, t, theta(t), x(t), y(t)} for a reference point initially at (1,
0) on the circle, moving with angular velocity omega.
Relate
for a reference point initially at coordinates x = cos(phi), y = sin(phi)) on the
circle, moving with angular velocity omega.
-
period
-
omega
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t
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theta
-
phi
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x
-
y
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h
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k
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v_arc (the
speed of the reference point as it moves around the arc of the
circle)
- `dt
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`ds_arc (the
distance traveled by the reference point along the arc during time
interval `dt)
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02 |
6.2,3 |
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Class Notes
# 3 |
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Class Notes Topics
#03: Modeling Periodic Phenomena with Circles
·
Discussion of sketching exercises from preceding class
·
model daily daylight in hours
vs. clock time in months
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modeling problems: general
cyclical graphs, day length model, daily mean temperature
vs month, tide, buoy
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Objectives:
1. Construct a unit circle showing all standard angular positions
which are multiples of pi/6 or pi/4.
2. Given starting point and angular velocity model motion on the unit
circle.
3. Relate angular displacement on the unit circle to arc distance and
vice versa.
4.
Relate
- reference circle angles which are multiples of pi
/ 6 or pi / 4
- x and y coordinates on the unit circle
- x and y coordinates on a reference circle of
radius A, centered at the origin
- table of y vs. x
5. Using triangles and the Pythagorean Theorem determine
the exact values of the sines and cosines of pi/4, pi/6 and pi/3 and use
these values to construct a table of the sines and cosines of all
angles which are multiples of pi/4 and pi/6.
6.
Relate
for a reference point initially at the point (cos(phi), sin(phi)) on the
circle, moving with angular velocity omega.
-
period
-
omega
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t
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theta
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phi
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x
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y
-
h
-
k
graph of sin(t), cos(t)
v_arc (the
speed of the reference point as it moves around the arc of the
circle)
`dt
`ds_arc (the
distance traveled by the reference point along the arc during time
interval `dt)
7. Apply all the above to applications and
problems.
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qa02: on unit circle: {i * pi/6, x_i, y_i, 0<= i <= 12} U { i *
pi/4, x_i, y_i, 0<= i <= 8}
on unit circle: {theta_i = i * pi/6, x_i, y_i, sin(theta_i),
cos(theta_i), 0<= i <= 12} U { theta_i = i * pi/4, x_i, y_i,
sin(theta_i), cos(theta_i), 0<= i <= 8}
{A, theta_i = i * pi/6, x_i, y_i, sin(theta_i), cos(theta_i),
0<= i <= 12} U { theta_i = i * pi/4, x_i, y_i, sin(theta_i),
cos(theta_i), 0<= i <= 8}
{theta_0, omega, t, theta}
On unit circle: {theta_0, omega, t, theta, x, y}
{A, theta_0, omega, t, theta, x, y}
Table: {A, theta(t)} U {i * pi/6, x_i, y_i, 0<= i <= 12} U
table form: {(t_i, theta(t_i), sin(theta(t_i)), cos(theta(t_i))
, A sin(theta(t_i)), A cos(theta(t_i)) | theta(t_i) = i pi / 6,
n_1 <= i <= n_2} U {graph of sin(theta(t)) vs. t, cos(theta(t))
vs. t} |
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03 |
6.4,5,6 |
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# 4-5 |
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Class Notes Topics
#04: The Sine Function
·
graph of sine function
·
exact values of sine function
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graph constructed from
reference circle
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table constructed from exact
values, resulting graph
#05: Modeling with the Sine Function
·
detailed circular model of
temperature function vs. months from Jan 1
· y = A
sin(`omega * (t - C) ) + D, effects of four
parameters
·
finding historical daily mean
temperature for given day of year
·
day length vs. clock time by
evaluating parameters A, B, C, D
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Objectives:
Estimate the x and y coordinates and angle theta
associated with a given point on a picture of the unit circle.
Hand-sketch a unit circle and the points corresponding
to angles which are multiples of pi/4 and pi/6:
- Using the sketch estimate the values of the sine
and cosine of each angle
- graph the sine and cosine functions based on your
estimates
- use periodicity properties to extend the graph.
- For motion of the reference point at constant
angular velocity omega, determine the change in t associated with a
complete cycle and use along with estimates from the hand sketch to
graph y = sin(omega * t) and y = cos(omega * t)
Apply the periodicity property of
trigonometric functions to evaluation at given points, to construction
of graphs and to tables.
Explain and apply the even-odd behavior of given
trigonometric functions.
Relate
for a reference point initially at the point (cos(phi), sin(phi)) on the
circle, moving with angular velocity omega.
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period
-
omega
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t
-
theta
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phi
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x
-
y
-
h
-
k
-
A cos(omega t + phi) + k
-
A sin(omega t + phi) + k
-
A cos(omega ( t – h) +
k
-
A sin(omega (t
– h) ) + k
- graph of
y = A cos(omega t + phi) + k
- graph of
y = A A sin(omega t + phi) + k
- graph of
y = A A cos(omega ( t – h) +
k
- graph of
y = A A sin(omega (t – h) ) + k
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v_arc (the
speed of the reference point as it moves around the arc of the
circle)
- `dt
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`ds_arc (the
distance traveled by the reference point along the arc during time
interval `dt)
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qa_03: {graph of sin(theta), picture of unit circle, given
point, estimate sin(theta), theta}
on unit circle: {theta_i = i * pi/6, x_i, y_i, sin(theta_i),
cos(theta_i), 0<= i <= 12} U { theta_i = i * pi/4, x_i, y_i,
sin(theta_i), cos(theta_i), 0<= i <= 8, approximate values,
graph sin(theta), cos(theta)}
{theta(t), change in t for complete cycle, graph of complete
cycle, inequality corresponding to 0 <= theta <= 2 pi} |
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04 |
7.3,4 |
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Class Notes
# 6 |
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Class Notes Topics
#06: Trigonometric Identities
·
Pythagorean identity based on unit circle
·
Reflection identities
·
Sine and cosine of sum of and
difference of two angles
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Law of Cosines
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Proving identities
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Objectives:
1. Construct a table of the values of y = A sin(x) for a given value
of A, extending for a complete cycle of this function, with x equal to
multiples of pi/6 or pi/4, and using the table construct a graph of one
cycle of y = A sin(x ).
2. Given a function y = A sin(theta) with theta given as a function of
x, construct a table of the values of y = A sin(theta) for a complete
cycle of this function with theta equal to multiples of pi/6 or pi/4,
then determine the x value corresponding to each value of theta. Using
a table of y vs. x construct a graph of one cycle of y = A sin(theta) in
terms of the given function theta of x, clearly labeling the x axis for
each quarter-cycle of the function.
3. Interpret the function and graph corresponding to Goal 2 in terms of
angular motion on a unit circle.
4. Use transformations to construct graphs of A cos(omega t +
theta_0) + k, and similarly translated graphs for the other
trigonometric functions.
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qa_04: {radian, radius, position, arc distance}
{theta, radius, position, arc distance}
{omega, radius, position, velocity on arc}
Exact coordinates at mult of pi/3, pi/4
{theta, radius, position, arc distance, x, y}
{theta_0, omega, t, radius, x, y}
{theta(t), A, table and graph of sin, cos, tan} … relabeling |
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05 |
7.1,2,5 |
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Class Notes
# 10-11 |
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Class Notes Topics
#10: Solving Trigonometric Equations
·
single solution of a
trigonometric equation
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multiple solutions of a
trigonometric equation
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representation of solutions of
trigonometric equations by graphs and/or unit circle
·
graphs of inverse sine and
inverse cosine functions
#11: Inverse Sines, Tangents and
Inverse Tangents
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When do we have 10 hours of daylight? Solution
of the equation, solutions represented by graph and unit circle.
·
fullness of the Moon vs. clock
time
·
field of vision and flagpole
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Objectives:
1. Construct a table of the values of y = A sin(x) for a given value
of A, construct a graph of a single cycle of this function, then
extending the graph forward and/or backward for any specified number of
complete cycles.
2. Given a function y = A sin(theta) with theta given as a function of
x, construct a y vs. x graph of a single cycle of this function, then
extending the graph forward and/or backward for any specified number of
complete cycles.
3. Interpret the function and graph corresponding to Goal 2 in terms of
angular motion on a circle of appropriate radius.
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qa_05: Extending graph of periodic function
{theta(t), sin\cos\tan, a, b, number of cycles a <= t <= b, dist
between peaks}
{theta(t), slope of sin(theta(t)) at beginning of cycle, max
slope of sin(theta(t)) both as multiple of x = 0 slope of sin(x)}
t intercepts of y = sin(theta(t)) for first cycle, or
generalized |
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06 |
7.7,8 |
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Objectives:
1. Construct the basic triangles corresponding to angles pi/6, pi/4,
pi/3, as appropriate, in order to find the sines and cosines of these
angles.
2. Using exact values construct tables and graphs for the basic
trigonometric functions.
3. Using exact values construct y vs. x graphs of y = A sin(theta) or y
= A cos(theta), where theta is given as a function of x.
4 . Use the technique of reversing columns and restricting domain to
construct tables and graphs for the arcsin and arccos functions.
5. Given an equation in which the argument of the sine or cosines
function is a function of x, solve the equation for all values of x
within a given interval.
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qa_06: {30-60-90 and 45-45-90 triangles}
exact coord mult pi/4, pi/6, once more
table for sine, reverse to get table for arcsin, restrict to get
fn
values of arcsine
soln of eqns for trig fns and their inverses
… incl multiple solns |
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Module 2: Assignments 7-9
on Triangle Trigonometry, Trigonometric Equations and Additional
Trigonometric Functions
Test 2 on Module 2 is to be completed as
part of Assignment 10 |
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07 |
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Objectives:
1. Understand and explain how the vertical asymptotes of
the graph of the tangent function occur.
2. Using exact values construct y vs. x graphs of y = A sin(theta) or y
= A cos(theta), where theta is given as a function of x.
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qa_07: tan, unit circle, graphs, tables, inverse fn,
approaching y axis on unit circle thru given quadrant,
tan(theta(t))
1. Understand and be able to explain how the vertical
asymptotes of the graph of the tangent function occur.
2. Using exact values construct y vs. x graphs of y = A
sin(theta) or y = A cos(theta), where theta is given as a
function of x. |
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08 |
8.1,2 |
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# 7-8 |
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Class Notes Topics
#07: Tangents and Other Trigonometric Ratios
·
height measured by similar
triangles
·
tangent as ratio of rise to run
·
inverse tangent to find angle
#08: Sines and Cosines as Ratios
·
sine and cosine in terms of
opposite side, adjacent side and hypotenuse of right triangle
·
Law of Sines
·
Law of Cosines
·
solving triangles
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Objectives:
1. Given an identity involving sine, cosine, tangent,
cosecant, secant and cotangent functions prove or disprove it using the
Pythagorean identities and the definitions of these functions.
2. Find the trigonometric functions of arbitrary angles by first
finding the reference angle, then applying the sign appropriate to the
quadrant.
3. Given a side of a right triangle and another side or angle,
determine all of its angles and sides.
4. Given a side of a triangle, another side or and angle, and
still another side or angle of a triangle, find all of its sides and
angles.
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09 |
8.3,5 |
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Class Notes
# 9-10 |
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Class Notes Topics
#09: Miscellaneous Problems
·
high and low point of a
pendulum
·
review: graphing a sine
function by transformations
#10: Solving Trigonometric Equations
·
single solution of a
trigonometric equation
·
multiple solutions of a
trigonometric equation
·
representation of solutions of
trigonometric equations by graphs and/or unit circle
·
graphs of inverse sine and
inverse cosine functions
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Objectives:
1. Given the magnitude and direction of a vector determine its
components.
2. Given the components of a vector determine its magnitude and
direction.
3. Given two or more vectors determine the magnitude and angle of their
sum.
4. Model simple harmonic motion using sine or cosine functions
and appropriate transformations.
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Module 3: Assignments
10-13 on Vectors and Analytic Geometry
Test 3 on Module 3 is to be completed as
part of Assignment 14 |
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10 |
9.1,2 |
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Class Notes
# 16 |
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Class Notes Topics
#16: Polar Coordinates
·
pole, polar axis, coordinates
of a point
·
plotting given points
·
‘grid’ for polar coordinates
·
converting polar to rectangular
coordinates
·
polar coordinates of points
given in rectangular coordinates
·
sketching r = 7, r = theta,
theta = pi/4
·
sketching r =
cos(theta), verifying shape of graph
·
sketching r = sin(2 theta)
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Objectives:
Relate
- v, a vector in 2 or 3 dimensions
- w,
a vector in 2 or
3 dimensions
- theta, the angle between v and w
- v dot w, the dot product of v and
w
- || v ||
- || w ||
- scalar projection of v on w
- v / || v ||
- w / || w ||
- vector projection of v on w
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11 |
9.4,5 |
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Objectives:
1. Express the distance from general point (x, y) from a specific
point (x_0, y_0), or from a line y = c, or from a line x = c (where c is
constant).
2. Express the definitions of the conic sections as equations
in terms of distances from a point (x, y) on the conic section to
appropriate points and/or lines, and simplify the equations.
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12 |
10.1,2,3 |
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# 12-13 |
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Class Notes Topics
#12: Introduction to Conic Sections
·
midpoint and distance between
points of the plane
·
(x, y) equidistance from two
given points: equation of straight line
·
sum of distances from two given
points to (x, y) is constant: equation of an ellipse
·
basic equation of ellipse with
semiaxes a and b
#13: Circles, Ellipses, Hyperbola
·
(x, y) at set distance from
(x_0, y_0)
·
equation of circle from
condition on inscribed triangle with two vertices on a diameter, third
vertex (x, y)
·
(x, y) at proportion p of the
way from (x1, y1) to (x2, y2)
·
ellipse as deformed circle
·
graphing the ellipse inscribed
in the 2a x 2b rectangle centered at the origin
·
difference of distances of (x,
y) from two given points is constant (hyperbola)
·
graphing a hyperbola using a
basic rectangle
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Objectives: 1. Know the equations of parabolas with
vertex at the origin, and of ellipses and hyperbolas centered at the
origin.
2. Sketch the graph of the equation of an
ellipse or a hyperbola centered at the origin, using the 'basic
rectangle'.
3. Find, as appropriate, the vertices, foci,
eccentricities and asymptotes of the above.
4. Use shifting and stretching transformations
and the process of completing the square to find the equation of and
graph a general conic section of form A x^2 + B x + C y^2 + D y + E = 0.
5. Relate the equation of an ellipse to its properties (center,
vertices, foci, semimajor and semiminor axes) and to its geometric
definition.
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13 |
10.4, 10.7 |
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Class Notes
#14, 16 |
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Class Notes Topics
#14: Conic Sections at General Positions; Parametric Equations
·
transformations and conic
sections
·
distance of (x, y) from
directrix equal to distance of (x, y) from
focus (equation of parabola)
·
parametric equations for a line
through (x1, y1) and (x2, y2)
·
parametric equations for a
circle based on sine and cosine functions
·
parametric equations for an
ellipse based on sine and cosine functions
·
completing the square to define
transformations used to graph a conic section
#15: Parametric Equations
·
using a table to graph (x(t),
y(t) ), a <= t <= b
·
eliminating the parameter to
expression y in terms of x
·
review: putting the equation of
a conic section into standard form
#16: Polar Coordinates
·
pole, polar axis, coordinates
of a point
·
plotting given points
·
‘grid’ for polar coordinates
·
converting polar to rectangular
coordinates
·
polar coordinates of points
given in rectangular coordinates
·
sketching r = 7, r = theta,
theta = pi/4
·
sketching r =
cos(theta), verifying shape of graph
·
sketching r = sin(2 theta)
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Objectives:
1. Perform the operations of addition or multiplication of two
compatible matrices, and multiplication of a matrix by a scalar.
2. Relate the equation of a hyperbola to its
properties (center, vertices, foci, semimajor and semiminor axes) and to
its geometric definition.
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qa_13: sane, demented, limiting populations, symbolized,
multiplication, powers of stochastic matrix; sane, demented,
borderline |
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Module 4: Assignments
14-19 on Matrices, Sequences, Probability
Test 4 on Module 4 is to be completed as
part of Assignment 19 |
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14 |
11.1,2 |
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Class Notes
# 17, 18, 19 |
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Class Notes Topics
#17: Introduction to Matrices
·
transitions in population state
·
limiting population state
·
symbolizing transitions
·
transition equations
·
representing a transition
equation as product of row of transition probabilities and columns of
categorized populations
·
symbolizing transitions by a
matrix and a population-state vector
#18: Transition Matrices
·
using transition matrix to
calculate successive population states
·
multiplying a transition matrix
by itself
·
using powers of the transition
matrix to directly calculate population after a given number of
transitions
·
geometric interpretation of
changing population states
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Objectives:
1. Determine whether a system of two
or three simultaneous linear equations in an equal number of variables
has a solution and if so find it using substitution or elimination, as
specified.
2. Write a system of simultaneous
linear equations an augmented matrix and solve by matrix reduction.
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qa_14. Determinants; Systems of Equations and their Geometrical
Interpretation
2 x 2; minors; solve system by elimination; represent as
augmented matrix and reduce; representation as |
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15 |
11.3,4 |
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Class Notes
# 20, 21, 22 |
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Class Notes Topics
#19: Matrix Representation of Equations; Stochastic Matrices; Dot
Product
·
purchasing example with three
unknowns
·
three simultaneous linear
equations represented as product of matrix and unknown vector
·
stochastic matrix and stock
market
·
dot product of two vectors
·
dot product related to
magnitudes, angle
#20: Interpreting Stochastic Matrices
·
3-variable (sane, demented, borderline) stochastic matrix,
interpretation
·
stock market interpretation
(up, down, unchanged)
#21: Identity Matrix; Matrix Reduction; Solving Matrix Equations
·
matrix multiplied by column
vector yields column vector; identity matrix
·
A x = y solved by x = A^-1 * y
·
solving a system of
simultaneous linear equations by elimination
·
matrix version of solution by
elimination
·
matrix reduction represents
solution by elimination
·
matrix reduction can be
conducted without reference to simultaneous equations
#22: Row Reduction; Inverse Matrix
·
we can solve A x = b by finding
A^-1
·
a matrix can be inverted by
reducing its matrix augmented by the identity matrix
·
when the original matrix has
been reduced to the identity, the augmented matrix will have become the
inverse matrix
·
the solution to A x = b is x =
A^-1 * b
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Objectives:
1. Apply Cramer's Rule to solve a
system of linear equations
2. Know and apply the properties of
determinants
3. Write a system of linear equations
as a matrix equation.
4. Apply the properties of matrices to
find sums of multiples and products of compatible matrices.
5. Find the inverse of a matrix.
6. Solve matrix equations by applying
the inverse matrix.
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qa_15: A X = B; inverting a matrix; cramer’s rule; solve system
by matrix reduction
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16 |
12.1,2 |
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Objectives:
1. Given an explicit or recursive rule
for a sequence, write its first several terms.
2. Use summation notation to represent
the sum of the members of a sequence.
3. Apply the additive and
multiplicative properties of sequences.
4. Recognize, write, sum and apply
arithmetic sequences.
qa_16: Pattern of sequence: representing nth term; arithmetic sequence;
partial sums; pattern of partial sums; limit of partial sums; geom seq;
sum of geom seq; harmonic sequence (note gen def reciprocals of
arithmetic seq obvious correlation with harmonics of string);
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text sections |
qa |
text |
outline |
Class Notes |
other |
query |
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17 |
12.3,5 |
qa_17 |
text_17 |
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Class Notes
# 23, 24, 25 |
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query_17 |
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Class Notes Topics
#23: Review of Analytic Geometry; Introduction to Binary
Probability
·
general equation A x^2 + B y^2
+ C x + D y + E = 0 yields conic sections.
·
summary of graphs of parabolas,
ellipses, hyperbolas
·
some parametric equations
·
histogram for coin flips
·
pascal’s
triangle
·
combinations
#24: Combinations; Binomial Formula
·
review: pendulum model
·
combinations
·
powers of a binomial
·
C(n, r) in terms of factorials
·
binomial formula
#25: Binomial Probability; Estimating Areas
·
dice probabilities: binomial
probability
·
probability of randomly
positions points lying within various regions of a rectangle
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Objectives:
1. Recognize, write, sum and apply
geometric sequences.
2. Determine convergence of infinite
geometric sequences.
3. Use the binomial theorem to expand
binomials. |
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18 |
13.2,3 |
qa_18 |
text_18 |
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Class Notes
# 26 |
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query_18 |
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Class Notes Topics
#26: Random Walks; Sierpinski
Triangle
·
random walk data
·
linearizing
data
·
agreement with formula for mean
distance after n steps of random walk
·
chaos game: 3 points.
Random point. Randomly select one of 3
points and move halfway to it. Continue.
·
result is not to fill in
original triangle; we get Sierpinski
triangle
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Objectives:
1. Apply permutations, combinations
and the properties of probability to calculate properties of specified
events. |
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qa |
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19 |
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qa_19 |
text_19 |
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query_19 |
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Objectives: |
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