Vectors Summary
1. Scalar product (dot product):
a b =| a | b | cos
Laws of dot product:
(i) a b = b a
(ii) a (b + c) = a b + a c = b a + c a
(iii) a a =| a | 2 (angle between two identical vectors is 0 degrees)
(iv) a b = 0 a and b are perpendicular
App
Maclaurins Series Summary
1. Mathematical Representation
x2
x n ( n)
f ( x) = f (0) + xf ' (0) +
f " (0) + . +
f (0) + .
2!
n!
where f (n ) (0) denotes the value of the nth derivative when x = 0 .
2. Expansions of specific standard series (Available in MF
The idea to evaluate integrals involving modulus is as such:
c
For example, considering
f ( x) dx , we must first know what range of values of x for which f (x ) is
a
negative, in this case lets assume it to be f ( x ) 0 for a x b, where b < c.
c
Then
b
f
Series/Sequences and Mathematical Induction Summary
1. Summation
(a) Anatomy of the sigma notation:
5r + 2 = [5(a) + 2 ] + [5(a + 1) + 2 ] + [5(a + 2) + 2 ] + .+ [5(n) + 2 ]
n
r
a+1
a
a+2
n
r =a
r=a
r = a +1
r=n
r =a+2
r is the variable; it changes value
Integration Techniques Summary
1. List of basic formulas:
Function
Integrated Result
ax n
ax n + 1
n+ 1
(ax + b) n
(ax + b) n + 1
( a ) (n + 1)
[ f ' ( x)][ f ( x)] n
[ f ( x )] n + 1
n+ 1
[ f ' ( x )]
2
+ [ f ( x )]
1
f ( x)
tan 1
a
a
[ f ' ( x )]
[
Differential Equations Summary
1. First order differential equations
a. Variables Separable DE:
Arrange through manipulation such that the form below is achieved:
f ( x)dx = g ( y )dy
Integrate subsequently to yield the required solution.
Example: Solve
d
Applications of Integration Summary
1. Area under curve:
y = f (x)
y
a
O
b
x
b
Area of shaded region= f ( x)dx
a
y
a
O
b
x
y = f ( x)
b
Area of shaded region=| f ( x)dx |
a
y
b
x = g ( y)
a
O
x
b
Area of shaded region= g ( y )dy
a
y
b
x = g ( y)
a
x
O
b
A
For the DE of the form
dy
+ f ( x) y = g ( x),
dx
f ( x ) dx
We will multiply both sides by an integrating factor given by e
, such that the above becomes
e
f ( x ) dx
f ( x ) dx
dy
f ( x ) dx g ( x) = h( x) , where h( x) = e f ( x ) dx g ( x)
+ f ( x
Geometry Final Test Study Outline
Chapter 1: Line and Angle Relationships
a.
b.
c.
d.
e.
f.
Sets, Statements, and Reasoning (1.1)
Informal Geometry and Measurement (1.2)
Early Definitions and Postulates (1.3)
Angles and Their Relationships (1.4)
Introduct
Chapter 4 Review
1.
Name the longest line segment shown in the
quadrilateral ABCD. Explain your answer.
_
2.
The
VABC
has a perimeter of 40 cm.
The side AB = 10, BC = x + 6, and AC = 2x 3
Is the
VABC
a) isosceles, b) scalene, or d) equilateral?
Explain yo
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of rapid prehensivity by rapid forgetfulness, or slow prehensivity
and slow forgetfulness, or in the contrast between narrow, but
intense memory, and broad but approximate memory.
Certain special considerations arise with regard to the
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false flag; nobody permits the passion to show in itself; it must
receive another name, even in the mind of the woman whom it
dominates.
The first of the forms which the sexual impulse takes is false
piety, religiosity. This is somethi
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Still more thoroughly graded is the attitude toward pain, inasmuch
as barely a trace of intelligence is required, in order to know that it
is necessary to wipe away a hot liquid drop that has fallen on the
body. Every physiological tex
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and assertion of the poet as a golden truth, but only when we
have tested its correctness for the daily life. But it must be understood
that I am not saying here, that we ourselves might have been
able to make the observation, or to ab
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whom it belongs than conversely. This occurs to any one of us,
and often we can not remember the name of even a close friend for
a greater or shorter period. But we very rarely find that we do not
think of the appearance of the individ
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consequent fading in memory. Several other particular conditions
may be added.
Kant, e. g., calls attention to the power we have over our fancy:
`In memory, our will must control our imagination and our imagination must be able to dete
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attention with unconscious. An explanation of this process will
help us, perhaps, to explain many incomprehensible and improbable
things. `Even the unconscious psychic activities,-going up
and down, smoking, playing with the hands, etc
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an image lasts hardly a second. The image as a whole does not
disappear in this time, but its content endures unchanged for so
long at most. Then it fades in waves. The correctness of this
description may be tested by anybody. But I sh
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be mistaken for equality, or at least, that there should be ignorance
of the inequality. Examples, therefore, are to be used only in the
most extreme cases, and only in such wise, that the nature of the
example is made very clearly obv
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and conceit.' These properties are, however, so powerful that they
may easily lead to deception. If the judge does not understand how
to follow this prescription it does no good, but if he does understand
it he has a weapon with which
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savage and half-savage peoples, but, on the other hand, it is not
indifferent for us to know what the situation was among peoples and
times who have influenced our own culture. Let us review the
situation hastily.
A number of classical
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meet the real object I wonder about the red dress.[1]
[1] H. Gross: Korregierte Vorstellungen. In H. Gross's Archiv X, 109.
We get this situation in miniature each time we hear of a crime,
however barren the news may be,-no more than a
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remain identical, so that one may say approximately how
much may be known of any subject at the end of a fixed time, if
only one ratio is tested. To criminalists this investigation of
Ebbinghaus' is especially recommended.
The conditio
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etc. If the child is only half-interested, it will try to fill
out these lacunae by reflection and synthesis, and may conceivably
make serious blunders. The blunders and inaccuracies increase the
further back the event goes into the ch
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or a crime, and if the other, especially the above-mentioned facts,
are not against it, we are called upon to decide whether we are
considering a mental event, due to the influence of menstruation.
Icard[1] has written the best monogra
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What especially wants consideration in the real peasant is his
silence. I do not know whether the reasons for the silence of the
countrymen all the world over have ever been sought, but a gossiping
peasant is rare to find. This trait i
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of us may remember that a witness who was ready with a prompt,
and to him an indifferent reply, started thinking and gave an essentially
different answer, even contradictory to his first, when the
meaning and the effect of what he migh
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extreme tension, then the effort _in venere_, finally, perhaps also the
use of popularly well-known stimulants, etc., may easily cause
weakening, sickening, and as conclusion the death of the old man.
But the public does not draw this
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[1] L. Geiger: Der Ursprung der Sprache. Stuttgart 1869.
We also are bound to be mistaken if we presuppose the lack
of reason as a peculiarity of the uneducated only, and accept as
well thought-out the statements of people who possess