maths Flashcards
(152 cards)
1
Q
algebraic division
A
divide into it and then subtract
2
Q
partial fractions
A
type 1 - distinct linear factors
type 2 - repeated linear factors
type 3 - irreducible quadratic factors
3
Q
differentiation - the product rule
A
f’g + g’f
4
Q
differentiation - the quotient rule
A
(f’g - g’f) / g^2
5
Q
differentiate sinx
A
cosx
6
Q
differentiate cosx
A
-sinx
7
Q
what does secx equal?
A
1/cosx
8
Q
what does cosecx equal?
A
1/sinx
9
Q
what does cotx equal?
A
1/tanx
10
Q
sin^2x + cos^2x
A
= 1
11
Q
tan^2x + 1
A
= sec^2x
12
Q
1 + cot^2x
A
= cosec^2x
13
Q
differentiate tax
A
sec^2x
14
Q
differentiate secx
A
secxtanx
15
Q
differentiate cosecx
A
-cosecxcotx
16
Q
differentiate cotx
A
-cosec^2x
17
Q
differentiate e^x
A
e^x
18
Q
differentiate ln(x)
A
1/x
19
Q
how can extrema be found?
A
Using the second derivative
20
Q
f’‘(x) < 0
A
max turning point
21
Q
f’‘(x) > 0
A
min turning point
22
Q
f’‘(x) = 0
A
May be a point of inflexion - check with nature table
23
Q
binomial theorem
A
use pascals triangle and label a and b
24
Q
factorials
A
3! = 3 x 2 x 1
25
what does 0! =
1
26
^nCr - known as binomial coefficient
n! / r!(n-r)!
27
binomial theorem - what is the general term?
(n r) x^n-r y^r
28
rule 1 - what does (n r) equal?
(n n-r)
29
(n r-1) + (n r)
(n+1 r)
30
when do vertical asymptotes occur?
when the denominator equals zero.
31
what are the two non-vertical asymptotes?
horizontal and slant
32
horizontal asymptotes
y = 0 or y = ai
33
slant asymptotes
divide out: y= mx + c
34
inverse functions calculate
change y and x
35
when is there an inverse function
when there is a one-to-one correspondence between x and y. If there is two x values for on y value, restrict the domain.
36
how to draw the modulus function
reflect all negative y values in the x axis
37
what is an even function?
f(-x) = f(x)
38
what is an odd function?
f(-x) = -f(x)
39
what is a redundant equation?
when at least two equations are equivalent - there is an infinite number of solutions.
40
what is an inconsistent equation?
parallel lines - there are no solutions.
41
Gaussian elimination steps
express system in augmented matrix form, reduce it to upper triangular form and then perform back substitution.
42
when is a set of equations said to be ill-conditioned?
When small changes in the coefficients produce relatively large changes in the solution.
43
integrate sinx
-cosx
44
integrate cosx
sinx
45
what does cos^2x equal?
1/2 (1+cos2x)
46
what does sin^2x equal?
1/2 (1-sin2x)
47
what can cos2x equal?
2cos^2x - 1
or
1 - 2sin^2x
48
integrate e^x
e^x
49
integrate 1/x
lnx)
50
integrate sec^2x
tanx
51
integrate f'(x)/f(x)
ln(f(x))
52
how do you integrate improper fractions?
divide out
53
integrating by substitution steps
find 'u'
differentiate 'u'
multiply by 'dx'
54
substitution and definite integrals
where the variable changes you must change the limits too.
55
how do you find the area between curve and y axis
it can only be done if x is expressed as a function of y.
56
what is the formula for volume of revolution?
integral (b a) pi y^2 dx
57
what does 'i' equal?
root -1
58
what does 'i^2' equal?
-1
59
what are complex numbers of the form?
z = x +iy
60
complex numbers: what do we write x as?
x = Re(z)
61
complex numbers: what do we write y as?
y = Im(z)
62
what is the complex conjugate?
z = x - iy
63
What is the modulus of a complex number?
The measure of magnitude of z and is written as (z).
64
What is the argument of a complex number?
The size of the angle, between the x axis and the line representing the complex number on the Argand diagram.
65
What is the formula for the argument of z?
the square root of (x^2 + y^2)
66
What is the formula for the argument of z?
tan-1(y/x)
67
What is the formula for polar form of a complex number?
z = r(cosx + isinx)
68
What is a locus?
A set of points.
69
How do you multiply two complex numbers in polar form?
Add the arguments and multiply the moduli.
70
How do you divide two complex numbers in polar form?
Take away the arguments and divide the moduli.
71
use de moivre's theorem to show z^n.
Raise the modulus to that power. Multiply the argument by that power.
72
using de moivres theorem, how many roots would a cubic equation have?
three - they would be 120 degrees apart.
73
what would you do if you find the fourth roots of unity?
find fourth roots of one, four solutions 90 degrees apart.
write 1 in polar form. z = cos0 + isin0
74
what do you get when you solve an irreducible quadratic with no real roots?
the roots are complex conjugates of each other.
75
differentiate inverse sinx
1/square root of (1 - x^2)
76
differentiate inverse cosx
-1/square root of (1 - x^2)
77
differentiate inverse tanx
1/(1 + x^2)
78
differentiate inverse sin(x/a)
1/square root of (a^2 - x^2)
79
differentiate inverse cos(x/a)
-1/square root of (a^2 - x^2)
80
differentiate inverse tan(x/a)
a / (a^2 + x^2)
81
when a function is complicated by powers, products and quotients of several factors, how can it be made easier?
By taking logarithms before differentiating.
82
parametric equations: what does f''(x) equal?
d/dt (dy/dx) dt/dx
83
what is an arithmetic sequence?
When the number separating the terms is a constant.
84
What is the equation for the nth term of an arithmetic sequence?
Un = a + (n-1)d
85
what is a in the equation for the nth term?
the first term
86
equation for sum of the first n terms of an arithmetic sequence?
Sn = 1/2n (2a + (n-1)d)
87
What is the equation for the nth term of a geometric sequence?
Un = ar^(n-1)
88
equation for sum of the first n terms of a geometric series?
Sn = (a(1-r^n)) / (1-r)
89
when r<1, what is the sum to infinity of a geometric series?
a/(1-r)
90
what can 1/(1-x) be interpreted as?
The sum to infinity of a geometric series with a=1 and r=x. r must be <1.
91
sigma 1
n
92
sigma r
1/2n (n+1)
93
sigma r^2
1/6n (n+1)(2n+1)
94
sigma r^3
1/4n^2 (n+1)^2
95
when trying to integrate a proper fraction, what do you do?
change it into partial fractions
96
how can improper fractions be simplified?
through algebraic division
97
integral of f'g dx equals?
fg - integral fg' dx
98
what do you use when trying to integrate something you cannot?
use the dummy variable, multiply it by one and then use integrating by parts.
99
difference between a general solution and a particular solution?
general solutions include the constant c.
100
when the differential equation is of the form dy/dx = f(y), what can you do?
you can separate the variables, integral 1/f(y) dy = integral 1dx
101
proof by induction: step 1
prove that the statement is true for n=1
102
proof by induction: step 2
assume it is true for n=k
103
proof by induction: step 3
consider n=(k+1)
104
proof by induction: step 4
prove that it is true for n=(k+1)
105
proof by induction: conclusion
thus if it is true for n=k then it is also true for n=(k+1) but since it is true for n=1, then by induction it is true for all n in the set of natural numbers.
106
what can the greatest common divisor of a and b be denoted by?
(a,b)
107
euclidean algorithm: if a = qb + r then...
(a,b) = (b,r)
108
If the gcd of two numbers is 1, what are the numbers said to be?
Co-prime or relatively prime.
109
What is Maclaurin series?
f(0) + f'(0)x/1! + f''(0)xx/2! + f'''(0)xxx/3!...
110
Maclaurin series for e^x, when is it valid?
For all x in the set of real numbers.
111
Maclaurin series for sinx/cosx, when is it valid?
For all x in the set of real numbers.
112
Maclaurin series for ln(1+x), when is it valid?
-1
113
Maclaurin series for (1+x)^n, when is it valid?
-1
114
Maclaurin series for tan-1x, when is it valid?
-1
115
What is the transpose of a matrix? (A' or A^T)
interchanging rows and columns.
116
What does (AB') equal?
B'A'
117
AI = IA =...
A
118
A'A = I if ...
orthogonal
119
How do you find detA of a 2x2 matrix?
ad - bc
120
How do you find the inverse of a 2x2 matrix?
1/ad-bc (adjA)
121
What happens in detA is 0?
The matrix has no inverse and is said to be singular.
122
What does A^-1A equal?
I
123
How do you find the inverse of a 3x3 matrix?
Use Gaussian elimination for A and I and make all ones in A and then resultant is inverse.
124
1st order linear differential equation form:
dy/dx + P(x)y = Q(x)
125
1st order linear differential equation IF:
e^integral of p(x)dx
126
2nd order linear differential equation form:
ad2y/dx2 + bdy/dx + cy = Q(x)
127
2nd order linear differential equation, what is it called whenn q(x) = 0?
The equations are homogeneous.
128
2nd order linear differential equation, what is it called whenn q(x) doesn't equal 0?
The equations are non-homogeneous.
129
Find complimentary function (solution to corresponding homogeneous equation)
Use the auxiliary equation.
130
How do you disprove a statement?
Give one counter example to disprove a conjective.
131
Direct proof: What do you do if n is even?
Let n = 2k , n in the set of natural numbers.
132
Direct proof: What do you do if n is odd?
Let n = 2k-1 , n in the set of natural numbers.
133
Proof by contradiction: What do you do initially?
Assume the opposite, eg., if n^2 +1 is odd, then n is even -> if n^2 +1 is odd, then n is odd.
134
Proof by contrapositive: what is the contrapositive of p->q?
not q implies not p.
135
What does rational mean?
It can be written as a fraction.
136
What would you do to prove something is rational?
Let it = m/n where m and n share no common factor, m, n in the set of integers.
137
What is the scalar product/dot product formula?
a.b. = detAdetBcos(angle)
138
How do you find the angle between two vectors?
cos(angle) = a.b./ detAdetB
139
What is the vector product/cross product?
a x b = ndetAdetBsin(angle)
140
What does it tell you if det(a x b) = 0?
The two vectors are parallel.
141
What does a x b =?
-(bxa)
142
What is the cartesian equation of a plane?
ax + by + cz = k
143
What is the angle between two planes the same as?
The angle between the two normals.
144
What is the vector equation of a plane?
r = a + tb + uc
145
What are parametric equations of a plane?
x = a + tb + uc
y = ...
z = ...
146
To find the equation of a plane, what do you need?
A point on the plane and two vectors parallel with the plane.
147
For the equation of a line in space, what do you need?
A point on the line and a direction vector (a vector parallel to the line)
148
The angle between a line and a plane: Two steps:
Find the acute angle between the normal and the line, subtract this value from 90.
149
Intersection between line and plane: Steps
Sub x, y and z of line into plane equation to find t. Sub t to get coordinates of intersection.
150
Intersection of two lines: Steps
Show they are not parallel, express parametric equations, find x and y, sub these into 3rd equation to find intersection point or if they are skew.
151
Intersection of two planes: steps
Cross the two normals, z = 0, find x and y,
152
Intersection of three planes: steps
use Gaussian elimination
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