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1
Q

algebraic division

A

divide into it and then subtract

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2
Q

partial fractions

A

type 1 - distinct linear factors
type 2 - repeated linear factors
type 3 - irreducible quadratic factors

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3
Q

differentiation - the product rule

A

f’g + g’f

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4
Q

differentiation - the quotient rule

A

(f’g - g’f) / g^2

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5
Q

differentiate sinx

A

cosx

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6
Q

differentiate cosx

A

-sinx

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7
Q

what does secx equal?

A

1/cosx

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8
Q

what does cosecx equal?

A

1/sinx

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9
Q

what does cotx equal?

A

1/tanx

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10
Q

sin^2x + cos^2x

A

= 1

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11
Q

tan^2x + 1

A

= sec^2x

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12
Q

1 + cot^2x

A

= cosec^2x

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13
Q

differentiate tax

A

sec^2x

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14
Q

differentiate secx

A

secxtanx

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15
Q

differentiate cosecx

A

-cosecxcotx

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16
Q

differentiate cotx

A

-cosec^2x

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17
Q

differentiate e^x

A

e^x

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18
Q

differentiate ln(x)

A

1/x

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19
Q

how can extrema be found?

A

Using the second derivative

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20
Q

f’‘(x) < 0

A

max turning point

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21
Q

f’‘(x) > 0

A

min turning point

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22
Q

f’‘(x) = 0

A

May be a point of inflexion - check with nature table

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23
Q

binomial theorem

A

use pascals triangle and label a and b

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24
Q

factorials

A

3! = 3 x 2 x 1

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25
Q

what does 0! =

A

1

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26
Q

^nCr - known as binomial coefficient

A

n! / r!(n-r)!

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27
Q

binomial theorem - what is the general term?

A

(n r) x^n-r y^r

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28
Q

rule 1 - what does (n r) equal?

A

(n n-r)

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29
Q

(n r-1) + (n r)

A

(n+1 r)

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30
Q

when do vertical asymptotes occur?

A

when the denominator equals zero.

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31
Q

what are the two non-vertical asymptotes?

A

horizontal and slant

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32
Q

horizontal asymptotes

A

y = 0 or y = ai

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33
Q

slant asymptotes

A

divide out: y= mx + c

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34
Q

inverse functions calculate

A

change y and x

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35
Q

when is there an inverse function

A

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.

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36
Q

how to draw the modulus function

A

reflect all negative y values in the x axis

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37
Q

what is an even function?

A

f(-x) = f(x)

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38
Q

what is an odd function?

A

f(-x) = -f(x)

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39
Q

what is a redundant equation?

A

when at least two equations are equivalent - there is an infinite number of solutions.

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40
Q

what is an inconsistent equation?

A

parallel lines - there are no solutions.

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41
Q

Gaussian elimination steps

A

express system in augmented matrix form, reduce it to upper triangular form and then perform back substitution.

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42
Q

when is a set of equations said to be ill-conditioned?

A

When small changes in the coefficients produce relatively large changes in the solution.

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43
Q

integrate sinx

A

-cosx

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44
Q

integrate cosx

A

sinx

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45
Q

what does cos^2x equal?

A

1/2 (1+cos2x)

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46
Q

what does sin^2x equal?

A

1/2 (1-sin2x)

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47
Q

what can cos2x equal?

A

2cos^2x - 1
or
1 - 2sin^2x

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48
Q

integrate e^x

A

e^x

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49
Q

integrate 1/x

A

lnx)

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50
Q

integrate sec^2x

A

tanx

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51
Q

integrate f’(x)/f(x)

A

ln(f(x))

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52
Q

how do you integrate improper fractions?

A

divide out

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53
Q

integrating by substitution steps

A

find ‘u’
differentiate ‘u’
multiply by ‘dx’

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54
Q

substitution and definite integrals

A

where the variable changes you must change the limits too.

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55
Q

how do you find the area between curve and y axis

A

it can only be done if x is expressed as a function of y.

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56
Q

what is the formula for volume of revolution?

A

integral (b a) pi y^2 dx

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57
Q

what does ‘i’ equal?

A

root -1

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58
Q

what does ‘i^2’ equal?

A

-1

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59
Q

what are complex numbers of the form?

A

z = x +iy

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60
Q

complex numbers: what do we write x as?

A

x = Re(z)

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61
Q

complex numbers: what do we write y as?

A

y = Im(z)

62
Q

what is the complex conjugate?

A

z = x - iy

63
Q

What is the modulus of a complex number?

A

The measure of magnitude of z and is written as (z).

64
Q

What is the argument of a complex number?

A

The size of the angle, between the x axis and the line representing the complex number on the Argand diagram.

65
Q

What is the formula for the argument of z?

A

the square root of (x^2 + y^2)

66
Q

What is the formula for the argument of z?

A

tan-1(y/x)

67
Q

What is the formula for polar form of a complex number?

A

z = r(cosx + isinx)

68
Q

What is a locus?

A

A set of points.

69
Q

How do you multiply two complex numbers in polar form?

A

Add the arguments and multiply the moduli.

70
Q

How do you divide two complex numbers in polar form?

A

Take away the arguments and divide the moduli.

71
Q

use de moivre’s theorem to show z^n.

A

Raise the modulus to that power. Multiply the argument by that power.

72
Q

using de moivres theorem, how many roots would a cubic equation have?

A

three - they would be 120 degrees apart.

73
Q

what would you do if you find the fourth roots of unity?

A

find fourth roots of one, four solutions 90 degrees apart.
write 1 in polar form. z = cos0 + isin0

74
Q

what do you get when you solve an irreducible quadratic with no real roots?

A

the roots are complex conjugates of each other.

75
Q

differentiate inverse sinx

A

1/square root of (1 - x^2)

76
Q

differentiate inverse cosx

A

-1/square root of (1 - x^2)

77
Q

differentiate inverse tanx

A

1/(1 + x^2)

78
Q

differentiate inverse sin(x/a)

A

1/square root of (a^2 - x^2)

79
Q

differentiate inverse cos(x/a)

A

-1/square root of (a^2 - x^2)

80
Q

differentiate inverse tan(x/a)

A

a / (a^2 + x^2)

81
Q

when a function is complicated by powers, products and quotients of several factors, how can it be made easier?

A

By taking logarithms before differentiating.

82
Q

parametric equations: what does f’‘(x) equal?

A

d/dt (dy/dx) dt/dx

83
Q

what is an arithmetic sequence?

A

When the number separating the terms is a constant.

84
Q

What is the equation for the nth term of an arithmetic sequence?

A

Un = a + (n-1)d

85
Q

what is a in the equation for the nth term?

A

the first term

86
Q

equation for sum of the first n terms of an arithmetic sequence?

A

Sn = 1/2n (2a + (n-1)d)

87
Q

What is the equation for the nth term of a geometric sequence?

A

Un = ar^(n-1)

88
Q

equation for sum of the first n terms of a geometric series?

A

Sn = (a(1-r^n)) / (1-r)

89
Q

when r<1, what is the sum to infinity of a geometric series?

A

a/(1-r)

90
Q

what can 1/(1-x) be interpreted as?

A

The sum to infinity of a geometric series with a=1 and r=x. r must be <1.

91
Q

sigma 1

A

n

92
Q

sigma r

A

1/2n (n+1)

93
Q

sigma r^2

A

1/6n (n+1)(2n+1)

94
Q

sigma r^3

A

1/4n^2 (n+1)^2

95
Q

when trying to integrate a proper fraction, what do you do?

A

change it into partial fractions

96
Q

how can improper fractions be simplified?

A

through algebraic division

97
Q

integral of f’g dx equals?

A

fg - integral fg’ dx

98
Q

what do you use when trying to integrate something you cannot?

A

use the dummy variable, multiply it by one and then use integrating by parts.

99
Q

difference between a general solution and a particular solution?

A

general solutions include the constant c.

100
Q

when the differential equation is of the form dy/dx = f(y), what can you do?

A

you can separate the variables, integral 1/f(y) dy = integral 1dx

101
Q

proof by induction: step 1

A

prove that the statement is true for n=1

102
Q

proof by induction: step 2

A

assume it is true for n=k

103
Q

proof by induction: step 3

A

consider n=(k+1)

104
Q

proof by induction: step 4

A

prove that it is true for n=(k+1)

105
Q

proof by induction: conclusion

A

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
Q

what can the greatest common divisor of a and b be denoted by?

A

(a,b)

107
Q

euclidean algorithm: if a = qb + r then…

A

(a,b) = (b,r)

108
Q

If the gcd of two numbers is 1, what are the numbers said to be?

A

Co-prime or relatively prime.

109
Q

What is Maclaurin series?

A

f(0) + f’(0)x/1! + f’‘(0)xx/2! + f’’‘(0)xxx/3!…

110
Q

Maclaurin series for e^x, when is it valid?

A

For all x in the set of real numbers.

111
Q

Maclaurin series for sinx/cosx, when is it valid?

A

For all x in the set of real numbers.

112
Q

Maclaurin series for ln(1+x), when is it valid?

A

-1<x<1

113
Q

Maclaurin series for (1+x)^n, when is it valid?

A

-1<x<1

114
Q

Maclaurin series for tan-1x, when is it valid?

A

-1<x<1

115
Q

What is the transpose of a matrix? (A’ or A^T)

A

interchanging rows and columns.

116
Q

What does (AB’) equal?

A

B’A’

117
Q

AI = IA =…

A

A

118
Q

A’A = I if …

A

orthogonal

119
Q

How do you find detA of a 2x2 matrix?

A

ad - bc

120
Q

How do you find the inverse of a 2x2 matrix?

A

1/ad-bc (adjA)

121
Q

What happens in detA is 0?

A

The matrix has no inverse and is said to be singular.

122
Q

What does A^-1A equal?

A

I

123
Q

How do you find the inverse of a 3x3 matrix?

A

Use Gaussian elimination for A and I and make all ones in A and then resultant is inverse.

124
Q

1st order linear differential equation form:

A

dy/dx + P(x)y = Q(x)

125
Q

1st order linear differential equation IF:

A

e^integral of p(x)dx

126
Q

2nd order linear differential equation form:

A

ad2y/dx2 + bdy/dx + cy = Q(x)

127
Q

2nd order linear differential equation, what is it called whenn q(x) = 0?

A

The equations are homogeneous.

128
Q

2nd order linear differential equation, what is it called whenn q(x) doesn’t equal 0?

A

The equations are non-homogeneous.

129
Q

Find complimentary function (solution to corresponding homogeneous equation)

A

Use the auxiliary equation.

130
Q

How do you disprove a statement?

A

Give one counter example to disprove a conjective.

131
Q

Direct proof: What do you do if n is even?

A

Let n = 2k , n in the set of natural numbers.

132
Q

Direct proof: What do you do if n is odd?

A

Let n = 2k-1 , n in the set of natural numbers.

133
Q

Proof by contradiction: What do you do initially?

A

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
Q

Proof by contrapositive: what is the contrapositive of p->q?

A

not q implies not p.

135
Q

What does rational mean?

A

It can be written as a fraction.

136
Q

What would you do to prove something is rational?

A

Let it = m/n where m and n share no common factor, m, n in the set of integers.

137
Q

What is the scalar product/dot product formula?

A

a.b. = detAdetBcos(angle)

138
Q

How do you find the angle between two vectors?

A

cos(angle) = a.b./ detAdetB

139
Q

What is the vector product/cross product?

A

a x b = ndetAdetBsin(angle)

140
Q

What does it tell you if det(a x b) = 0?

A

The two vectors are parallel.

141
Q

What does a x b =?

A

-(bxa)

142
Q

What is the cartesian equation of a plane?

A

ax + by + cz = k

143
Q

What is the angle between two planes the same as?

A

The angle between the two normals.

144
Q

What is the vector equation of a plane?

A

r = a + tb + uc

145
Q

What are parametric equations of a plane?

A

x = a + tb + uc
y = …
z = …

146
Q

To find the equation of a plane, what do you need?

A

A point on the plane and two vectors parallel with the plane.

147
Q

For the equation of a line in space, what do you need?

A

A point on the line and a direction vector (a vector parallel to the line)

148
Q

The angle between a line and a plane: Two steps:

A

Find the acute angle between the normal and the line, subtract this value from 90.

149
Q

Intersection between line and plane: Steps

A

Sub x, y and z of line into plane equation to find t. Sub t to get coordinates of intersection.

150
Q

Intersection of two lines: Steps

A

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
Q

Intersection of two planes: steps

A

Cross the two normals, z = 0, find x and y,

152
Q

Intersection of three planes: steps

A

use Gaussian elimination

advanced higher maths (2 decks)

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