mid semester test Flashcards
(117 cards)
1
Q
NOTATION: x (E A
A
the object x is an element of the set A
2
Q
NOTATION: Ø
A
empty sets - no elements in it
3
Q
NOTATION: A <= B
A
the set A is contained in B
B is a big bag that has smaller bag A contained in it
* B can = A*
4
Q
NOTATION: A < B
A
A is contained in B but are not equal
5
Q
NOTATION: A U V
A
union of sets A and B
- in venn diagram all coloured in
6
Q
NOTATION: A n B
A
intersection of A and B
- only the bits that have both in are included
7
Q
NOTATION: A/B
A
difference of sets A and B
8
Q
NOTATION: N
A
natural numbers (1,2,3..)
9
Q
NOTATION: Z
A
pos + neg whole numbers
10
Q
NOTATION: Q
A
fractions
11
Q
NOTATION: |R
A
includes all rationals (surds and pi etc).. decimals and shiz
12
Q
NOTATION: C
A
complex numbers
13
Q
NOTATION: upside down A
A
for all
14
Q
NOTATION: sideways E
A
there exists
15
Q
domain meaning
A
set of values which the variable on the horizontial axis can take (usually x variable)
16
Q
range meaning
A
set of values which the variable can take on the vertical axis (usually y variable)
17
Q
absolute functions
A
always 2 solutions when solving equations - one + / -
- inverse graph so always + y value (unless undergone horizontial translation - you just do the same thing below the line)
18
Q
real valued functions
A
f: D -> |R is defn {meaning domain is real}
1. range is not part of notation as it is not part of the definition of the function
19
Q
[ ] include/exclude
A
include
20
Q
() include/exclude
A
exclude
21
Q
Is it a function
A
wherever you draw a line it an only cut the graph once (vertical line test)
22
Q
1 - 1
A
function can only return 1 value for each input - THE ONLY WAY INVERSE FUNCTIONS WORK
23
Q
horizontial line test
A
a function is 1-1 if the horizontial line meets the graph at most once
24
Q
inverse functions?
A
- write y = f(x) + solve for y
2. f-1(x) = and put x wherever y was
25
how to check inverse functions?
f*f-1(x) = x
f(f-1(x)) = x
--- so u find whatever that function is you put in there
26
DERIVATIVE: x^n
n*x^(n-1)
27
DERIVATIVE: e^x
e^x
28
DERIVATIVE: e^f(x)
e^f(x) *f'(x)
29
DERIVATIVE: ln x
1/x
30
DERIVATIVE: ln f(x)
f'(x)/f(x)
31
DERIVATIVE: u(x)+v(x)
u'(x)+v'(x)
32
DERIVATIVE: [f(x)]^n
n*[f(x)]^(n-1) * f'(x)
33
DERIVATIVE: f(x) *g(x)
f'(x)*g(x)+f(x)*g'(x)
34
DERIVATIVE: f(x)/g(x)
(f'(x)*g(x)+f(x)*g'(x))/[g(x)]^2
35
DERIVATIVE: sin x
cos x
36
DERIVATIVE: cos x
-sin x
37
DERIVATIVE: sin f(x)
cos f(x)*f'(x)
38
DERIVATIVE: cos f(x)
-sin f(x)*f'(x)
39
DERIVATIVE: tan x
sec^2 (x)
40
DERIVATIVE: tan f(x)
sec^2 f(x) * f'(x)
41
DERIVATIVE: arcsin x
1/(1-x^2)^(1/2)
42
DERIVATIVE: arcsin f(x)
f'(x)/(1-(f(x))^2)^(1/2)
43
DERIVATIVE: arccos x
-1/(1-x^2)^(1/2)
44
DERIVATIVE: arccos f(x)
-f'(x)/(1-(f(x))^2)^(1/2)
45
DERIVATIVE: arctan x
1/(1+x^2)
46
DERIVATIVE: arctan f(x)
f'(x)/(1+(f(x))^2)
47
INTEGRAL: x^n
(1/n+1)*x^(n+1) + c
48
INTEGRAL: sin x
-cos x +c
49
INTEGRAL: cos x
sin x +c
50
INTEGRAL: k
kx +c
51
INTEGRAL: 1/x
ln |x| +c
52
INTEGRAL: e^x
e^x +c
53
INTEGRAL: (ax+b)^n
(1/a)(1/(n+1))(ax+b)^(n+1) +c
54
INTEGRAL: 1/(ax+b)
(1/a) ln | ax+b | +c
55
INTEGRAL: e^(ax+b)
1/a e^(ax+b) +c
56
INTEGRAL: sin(ax+b)
-1/a cos (ax+b) +c
57
INTEGRAL: cos(ax+b)
1/a sin (ax+b) +c
58
INTEGRAL: a^x
1/(ln a) * a^x +c
59
INTEGRAL: sec^2 x
tan x +c
60
INTEGRAL: 1/(1-x^2)^(1/2)
arcsin x +c
61
INTEGRAL: 1/(1+x^2)
arctan x +c
62
INTEGRAL: ln x *NOT COMP*
x ln x - x +c
63
LOGS: ln m + ln n
ln (m*n)
64
LOGS: ln (m*n)
ln m + ln n
65
LOGS: ln m - ln n
ln (m/n)
66
LOGS: m ln b
ln b^m
67
LOGS: ln (m/n)
ln m - ln n
68
LOGS: ln b^m
m ln b
69
LOGS: ln e^x
x
70
LOGS: ln e
1
71
LOGS: e^0
1
72
LOGS: e^0
1
73
LOGS: ln 1
0
74
INVERSE TRIG:
| domain of sin x
x (E [-PI/2 , PI/2]
75
INVERSE TRIG:
| domain of cos x
x (E [0, PI]
76
INVERSE TRIG: which is angle and which is coordinates
| arc cos / cos
arc cos = angle
| cos = coords
77
INVERSE TRIG:
| domain of tan x
(-PI/2, PI/2)
78
what is tan = to in terms of sin and cos
tan = sin/cos
79
DIFF EQ: general solution?
+c
80
DIFF EQ: particular solution?
+c is known e.g. +1.4
| - y(0) = 3/2 and just solve like ord eqN to give u constant value
81
DIFF EQ: how to solve
put all y on one side and all x on other (.dy / . dx) and integrate both sides
82
INVERSE FUNCTIONS: how to work out derivative for inverse USING FORMULA
1. f'(x) = [find derivative]
2. find f-1(x)
3. sub into equation
f-(x) = 1//f'(f-1(x))
!!!! put f-1(x) wherever x is in the eq !!!!
83
INVERSE FUNCTIONS: how to work out derivative for inverse DIFF DIRECTLY
- find inverse
- differentiate it
(f-1)'(x)
84
INVERSE FUNCTIONS: from table
find x val in table and go up --- e.g.
if x=3 f-1(x) =2
```
-
0
-
2
-
3
-
```
85
INTEGRALS: how do you change the numbers on the integral by subsitution
1. find the u value
2. subsitute the values on the integral into the u equation
THAT GIVES U UR NEW VALUES
86
POPN GROWTH: what is the formula
P = Ae^(k*t)
87
POPN GROWTH: what shows rate of change
dE/dt
88
EULER'S METHOD: what are the columns
x / y / hy' / y + hy'
89
EULER'S METHOD: how do you check the accuracy
integrating actual eqn
| subbing in one val given to give +c
90
INTEGRATION: subsitution steps
1. choose sub
2. calculate
du/dx = g'x
du = g'x dx
dx = 1/g'x du
3. subsitute gx = u
dx = 1/g'x
into integral
4. integrate with respect to u
5. replace u with g(x) to get final answer in terms of x
91
INTEGRATION: by parts formula
INT f(x)*g'(x) dx = f(x)*g(x) - INT f'(x)*g(x)
92
INTEGRATION: trig and log functions? (e.g. ln x or cos x)
stick a 1 in front of it and integrate to make x
93
HOW TO CHECK INTEGRATION:
differentiating
94
NEWTONS LAW OF COOLING:
T = Ta + Ce^kt
95
VECTORS: when are vectors equal
same length and direction THEY CAN BE LOCATED IN DIFF POS
96
VECTORS: add vectors (without coords)
triangle law
init point of A to endpoint of B making a triangle
b is on the terminal point of point A
97
VECTORS: multiplication
can only multiply by scalars
| negative numbers cause reversal of scalars
98
VECTORS: 0 vector
init and terminal points are the same
0 vector and 0 scalar are not the same
if k = 0 kv=0
99
VECTORS: position vector
starts at origin
100
VECTORS: addition (with coords)
v+u = (x1+x2, y1+y2)
101
VECTORS: multiplication by scalar
kv = (kx1, kx2)
102
VECTORS: if not positioned at origin
OP1 + P1P2 = OP2
P1P2 = OP2-OP1
v = (x2,y2) - (x1,y1)
= (x2-x1, y2-y1)
103
VECTORS: length of vector
|| v || = root (x^2 + y^2)
104
VECTORS: length of P1P2
||P1P2|| = root ( (x2-x1)^2 + (y2-y1)^2 )
105
VECTORS:
when is v not equal to 0
v = 0
|| v || > 0
v = 0
106
VECTORS: what is a unit vector
||v|| = 1
107
VECTORS: how to find unit vector of v formula
1/||v|| ( , ,(v vector) )
108
VECTORS: dot product
u . v = ||u|| * ||v|| cos θ
109
VECTORS: find angle between vectors
(u*v)/ ||u||*||v|| = cos θ
110
VECTORS: when r they perpendicular
u. v = 0
111
VECTORS: projection of u on v formula
b = (u.v/ ||v||^2) * v
112
VECTORS: component of u ortohonal to v
w=u-(u.v/||v||^2)*v
113
VECTORS: proof techniques - coordinates of point
-- find vector which starts at origin and terminates at that point
114
VECTORS: proof techniques - point certain distance along an interval (1/2 AB)
OA + 1/2AB = 1/2AB
115
VECTORS: proof techniques - a triangle always gives relation between 3 vectors
ABC triangle
| AB + BC = AC
116
VECTORS: proof techniques - a quadrilarial always gives relation between 4 vectors
ABCD quad
| AB + BC + CD = AD
117
VECTORS: proof techniques - a quadrilarial always gives relation between 4 vectors
ABCD quad
| AB + BC + CD = AD
