Math 140
Hw 25
Worked examples of selected recommended problems.
2sina has 5 symbols, 2sin(a) has 7 symbols.
A. When earth E, Mercury M, and the sun S are lined up
so that VEMS is a right angle, VSEM is 21o. Given that
the distance ES from the earth to th
Math 140
Hw 27
Worked examples of selected recommended problems.
D. 4, 11/6
x, y r cos , r sin 4 cos , 4 sin
6
= 4 3 /2, 41/2 2 3 , 2
E. 4, /6
x, y r cos , r sin 4 cos , 4 sin
6
=4 3 /2, 41/2 2 3 , 2
C
b
A
a
B
c
A(a). sinB = 1/ 2 ,
What are the possible v
Math 140
Hw 28
Worked examples of selected recommended problems.
Graph the parabola. On the graph, mark and give the
coordinates for focus and vertex. Draw the directrix with
a dotted line.
First locate the focus and vertex and draw the directrix. Then ma
Math 140
Hw 26
Worked examples of selected recommended problems.
(a) Give the unsimplified exact answer obtained using the
law of sines or cosines or Straight V Thm. E.g.
a 8 2 5 2 285 cos20 0 or b
5 sin100 o
sin50 o .
An unsimplified exact answer for an
Math 140
Hw 23
Worked examples of selected recommended problems.
Write each expression as a sum or difference of
trigonometric functions. Don't calculate the answer.
A. sin 20 cos 10
1 sin20 0 10 0 sin20 0 10 0
2
1
sin10 0 sin30 0
2
1 sin10 0 1
2
2
Math 140
Hw 20
Worked examples of selected recommended problems.
(1) Graph over one period (period doesn't have to start at 0). (2) List the x-intercepts. (3) List both coordinates of the
highest and lowest points. Get the box from the amplitude, period a
Math 140
Hw 22
Worked examples of selected recommended problems.
Evaluate the expressions.
A. sin 3 .
4
2
First find cos q. The quadrant determines the sign.
2 quadrant II cos 0
9
7
cos 1 sin 2 1 16 16
(c) tan/12
tan
sin 2 2 sin cos 2 3
4
7
4
7
4
cos 2
Math 140
Hw 21
Worked examples of selected recommended problems.
(1) Simplify using an addition formula (show the step
immediately after the addition formula). First point.
(2) Complete the simplification.
Second point.
Examples:
`sinx cosy x cosx siny x
Math 140
Hw 24
Worked examples of selected recommended problems.
Evaluate exactly without a calculator: p and 2 rather than
3.14 or 1.41. Answer may by undef.
K. cosarcsin 2
7
5 symbol fraction with square root
1 2 2
7
1
A. sin 3 /2
1
Answer: /3
B. tan
Math 140
Hw 19
Worked examples of selected recommended problems.
Find the period and amplitude.
A.
period = 4 amplitude= 6
F. y 3 sin (x/2
amplitude= 3
6
-6
2
3 4
2
period= 4
3
5
1
4
x-intercepts: 0, 2, 4
0
increases on [0, 1], [3, 4]
B.
period = 4
amplit
Math 140
Hw 16
Worked examples of selected recommended problems.
For each angle, sketch the reference angle, draw an
arrow to indicate the angle's direction. A reference
angle must be positive and < /2 = 90o.
A. (a) 110o
(b) 60o
60
70
sin cos
F. Simplify
Math 140
Hw 15
Worked examples of selected recommended problems.
Remember to include any needed units, e.g., mi/hr, rad/sec. Except
for 5ab, give only exact answers. No decimals.
A. Convert 45o to radian measure.
45 o 180
4
G. Find the six trigonometric f
Math 140
Hw 18
Worked examples of selected recommended problems.
(a) Rewrite in terms of an angle in [0,p/2].
(b) Find the exact answer, no decimals.
E. Simplify
A(a). cos11/6
cos11/6 cos12/6 /6
cos2 /6 cos/6 cos/6
1
sec 2 t
sin 2 tcos 2 t
tan 2 t1
cos
Math 140
Hw 17
Worked examples of selected recommended problems.
sin
2
Acos 2 A
A. sin Acos A
Hint: factor the top.
sin A cos Asin A cos A/sin A cos A
sin A cos A
B. sin 2 cos csc 3 sec
1
1
sin 2 cos sin 3 cos
1
sin
csc
cot A sin A cos A cot A
=
Math 140
Hw 14
Worked examples of selected recommended problems.
Write N(t) in base e form. Then solve the problem.
A. A colony of bacteria starts with 2000 bugs. Two hours
later it has 3800.
Given: N 0 2000
N2 3800
Nt N 0 e kt
tN2 2000e k2
t2000e 2k 3800
Math 414 Lecture 25
Job scheduling
A project requires completing several jobs.
Some are independent, some must precede others.
Find the minimum time required to complete the project.
We model this problem as a longest-path problem.
The nodes of the graph
Math 414 Lecture 23
Maximum Flows
`The amount of water flowing into node x increases by 5.
How much can the flow into adjacent nodes increase? The
upper number = the capacity; the lower = the flow.
4
3
10
8
5
+5
10
7
x
10
2
We are asking for potential ind
Math 414 Lecture 22
Not included in Exam 3.
ASSIGNMENT ALGORITHM
Input: an nxn cost matrix [cij ].
Output: a minimum cost assignment [xij ].
Row step. From each row, subtract the row minimum.
teach row now has a 0.
Column step. From each column, subtract
Math 414 Lecture 24
LEMMA.
v The distance from the origin to another node = the
minimum of the distances to the nodes edges.
v If the estimated distance of a cut edge is < the estimated
distances of other cut edges, then the estimated distance is
the true
Math 414 Lecture 26
LEMMA. The amount I can expect to win by consistently
playing the strategy for a given row is the minimum payoff
for that row.
THEOREM. The strategy for I with the maximum expected
win is the strategy whose row has the maximum minimum.
Math 414 Lecture 17
Integer Programming
In many problems the variables must range over integers:
the number of people you must hire, the number of chairs
you must rent, a price in cents, . .
DEFINITION. A linear programming problem is -a pure integer prog
Math 414 Lecture 21.31
Exam 3 covers Lectures 15 - 21.
In a transportation problem, suppose the total supply s
produced exceeds the total demand d, d < s.
Solution: cut back on the supply produced.
Should each plant cut back equally? No, the ones with the
Math 414 Lecture 13
FROM LAST LECTURE.
DUAL
PRIMAL PROBLEM CANONICAL+EXTRAS
max z = C:X
max z = C:X
min z = B:W
ATW > C
AX ? B
AX + . = B
X >0
X, ., > 0
W?0
Where ? is some combination of <, >, =.
The . are the slack and extra variables.
Initially arrange
Math 414 Lecture 14
Exam 2 covers Lectures 8-14
Find an optimal solution for the primal and dual.
PRIMAL
min z = x y2u
r: x u = 0
s: -x y > -3
t: x + y > 1
x, y, u > 0
Write the canonical version with extra variables.
Solve with SciLab, retaining the extr
Math 414 Lecture 15
One step of the Simplex method makes feasible
(constant-column is > 0) but nonoptimal solutions (objective row
has a negative coefficient) more optimal.
The dual method makes super-optimal (objective row
coefficients are > 0) but nonfe
CUTTING PLANE METHOD
Convert the problem to standard form with integer
coefficients. Solve.
If a primal variable is not integral, cut it off with add a new
integer-coefficient constraint.
Cutting it off, makes it nonfeasible. Use the dual method to
restor
Math 414 Lecture 16
Constant column sensitivity
Let B(brp) be the original constant column B with the
constant br replaced by a new variable p.
Let B' be the constant column (and hence solution) of the
final tableau given by the Simplex Method and let T b
Math 414 Lecture 19
All linear inequalities can be rewritten in standard form:
ax + by < c.
`y-3 < 2x
0 < x+y < 5
is equivalent to 2x y 3.
can be written as 2 linear inequalities.
In the problems below, you may use linear inequalities in
any form.
VARIABL