Math 140
Hw 6
Worked examples of selected recommended problems.
f(x)=2x1, g(x)=x23x6, h(x)=x3, k(x)=2, m(x)=x29.
Write each function as a composition
f g of simpler functions f and g.
D. f x 3x 1, gx 2x 5
F. (a) 3 3x 4
The largest inner function if gx
Math 140
Lecture 22
Homework 21 and 22 both due beginning of next week.
DOUBLEANGLE FORMULAS
sin 2 2 sin cos
cos 2 cos 2 sin 2
2 tan
tan 2 1tan 2
Proof.
sin 2 sin
sin cos cos sin 2 sin cos ,
cos 2 cos
cos cos sin sin cos 2 sin 2 ,
tan 2 tan
tan
Math 140
Lecture 21
TRIGONOMETRIC ADDITION FORMULAS
sins t sin s cos t cos s sin t
coss t cos s cos t sin s sin t
tan stan t
tans t 1tan s tan t
Replace t by t to get
sins t sin s cos t cos s sin t
coss t cos s cos t sin s sin t
tan stan t
tans t 1tan s t
Math 140
Lecture 26
CONVENTION
side a is opposite angle A,
side b is opposite angle B,
side c is opposite angle C.
SINE LAWS
sin A
a
A
c
b
B
a
C
a
sin A
sin B
b
b
sin B
sin C
c
or
c
sin C
PROOF. The area of the triangle can be written three ways:
1
2 bc s
Math 140
Lecture 25
Exam 4 & PreExam 4 Wed. 8:30am, here in Bilger.
Covers lectures 1924. Sit in your section area. Come 5
minutes early. This lecture is not on Exam 4.
Trigonometric word problems
(a) Find the exact answer.
(b) Find the decimal answer (o
Math 140
Lecture 23
RECALL
cosx y cos x cos y sin x sin y
cosx y cos x cos y sin x sin y
Adding these gives
cosx y cosx y 2 cos x cos y
Subtracting gives
cosx y cosx y 2 sin x sin y
Similarly
sinx y sin x cos y cos x sin y
sinx y sin x cos y cos x sin y
A
Math 140
Lecture 24
Exam 4, PreExam 4 Mon. 8:30am. Exam 4 here in Bilger.
Covers lectures 1923, not this lecture.
sin, cos and tan are not 11. The xaxis is a horizontal line
which crosses their graphs more than once.
They are 11 when restricted to the
Math 140 Practice Exam 4 /100
Exam 4. Lectures 1924. No calculators.
Write angles in exact radians, no decimals, no degrees.
1(10). For the graph below, find the amplitude, find the
period, and find an equation of the form y =+ Asin(Bx)
or y = +Acos(Bx).
Math 140
Lecture 29
b A circle is the set of all points such that the distance to a
center point is some constant r.
xis
minor a


v
ex
ert

m
foci 
r a
aj o
i sx

ter
cen
foci
vertex
DEFINITION. An ellipse is the set of all points such that the
su
Math 140
Lecture 28
y ax2 bx c is a vertical parabola with a vertical axis
of symmetry.
bOther parabolas have horizontal or slanted axes.
axis
parabola
focus
vertex
directrix
DEFINITION.
A parabola consists of all points equidistant between a
given focus
Math 140 Practice Exam 5
The final covers everything Lectures 130. No Gateway problems.
This practice exam covers the material since the last exam. About a
third of the exam will be on this material; the other exam problems
will be on the previous materi
Math 140
ve
rt
ex


  
int
o


 center
r
no
i
m

 
xis
a
x
rte
ve
  f o
cal
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ajo
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asy

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Lecture 30

i
al ax
 f oc
s
int
ra
xis
b DEFINITION. A hyperbola is the set of all points such that
the difference of the di
PreExam 3 due 8:30am Monday, April 7
Exam 3 Mon./Tues. April 7,8 in your discussion section.
Exam covers Lectures 1318. Not this lecture.
Math 140
Lecture 20
RECALL. For C 0, the graph of
f x C is f x shifted C units to the right.
The phase shift is the
PreExam 3 due 8:30am Monday, April 7.
Exam 3 Mon./Tues. April 7,8, in your discussion section.
Exam covers Lectures 1318. Not this lecture.
Math 140
PYTHAGOREAN IDENTITIES.
sin 2 t cos 2 t 1
tan 2 t 1 sec 2 t
cot 2 t 1 csc 2 t
Lecture 19
csc t csc t cot
Math 140
Hw 7
Worked examples of selected recommended problems.
A. f x 3x 1
(a) Find f 1 x
Switch x and y in f x y to get
f y x iff
3y 1 x
3y x 1
y 1 x 1
3
Answer: f 1 x 1 x 1
3
(c) Graph, y f x, y x, y f
coordinate system.
Answer:
D. f x 2x 3 1. Find f 1
Math 140
Hw 10
Worked examples of selected recommended problems.
Graph and on the graph mark the x and yintercepts. Label the
degrees of the roots. No credit if the graph isn't smooth.
x=1 v.a.
B. y x 3x 4x 12
For large x, graph looks like x 3.
yinterce
Math 140
Hw 8
Worked examples of selected recommended problems.
v Write the function in the form: y ax h 2 k.
E.g., rewrite y x 1 2 as y x 1 2 0.
v List the roots.
v Graph. On the graph mark the vertex and give both
coordinates.
C. y x 2 2x 3
roots: 1, 3
Math 140
Hw 9
Worked examples of selected recommended problems.
Draw the picture; indicate your variables on the picture.
Write the equations which relate the variables.
Solve for the wanted quantities.
Graph and on the graph mark the x and yintercepts.
Math 140
Hw 11
Worked examples of selected recommended problems.
A. Estimate as a power of 10:
(a) 2 30
2 30 2 10 3 10 3 3 10 9
Answer: 2 30 10 9
I. Graph and on the graph mark the xintercept,
yintercept, and horizontal asymptote.
y 3 x 1
Start with 3 x
Math 140
Hw 12
Worked examples of selected recommended problems.
Rewrite each equation using logarithms rather than
exponentials (no decimal answers).
A. 9 3 2
If exponential to the base 3 sends 2 to 9, then its
inverse, log 3 , does the opposite: it send
Math 140 Lecture 15
Angles, arcs, and radians
p is approximately 3.14.
In answers, use the exact p not the decimal 3.14.
A circle of radius r has circumference 2pr.
Unit circle circumference = 2pr = 2p1 = 2p.
Greek letters: q is theta, w is omega.
DEFINIT
Math 140
RECALL.
sin2 + cos2 = 1
tan = sin /cos
cot = cos /sin
Lecture 17
sec = 1/cos
csc = 1/sin
Simplify.
`cot sec cos
`tan csc sin
RECALL.
sin2 + cos2 = 1
tan = sin /cos
cot = cos /sin
sec = 1/cos
csc = 1/sin
Simplify.
1
cos x
` tan x cot x
1
sin x
`
Math 140 Lecture 16
Friday = Last day to withdraw.
Keller 419A secretary signs withdrawals for me.
b Know the sin, cos and tan of:
0, p/6, p/4, p/3, p/2 and their associated angles in
other quadrants.
(0, 1)
p/2=90o
1
(1/ 2, l3/ 2)
p/3=60o
o (1/l2, 1/l2)
Exam 3 covers lectures 1318.
Study the recommended exercises.
Math 140
Lecture 18
DEFINITION. For any function f :
f is even iff f (x) = f (x).
f is odd iff f (x) =  f (x).
Graphically:
Left half of an even = the yaxis reflection of right half.
Left
Math 140 Lecture 14
Exam 2 covers Lectures 712. Begins Mon. March 10
PreExam 2 is due 8:30am Mon. March 10.
Last Gateway Tuesday March 11
Exponential growth
Exponential functions measure
v the size of a growing population,
v the amount of money in a comp
Math 140
Lecture 27
Lecture 28 is omitted, it is not on the final.
Next iClicker quiz is on Lecture 29.
A triangle is determined uniquely up to congruence by
(1) two sides and an included angle, and by
(2) two angles and an included side.
sideangleside