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A Level Maths > Pure Year 2 > Flashcards

Pure Year 2 Flashcards

1
Q

What is important to remember about sectors and small angle approximations?

A

Angles in radians NOT degrees

Shift
Setup
2
2

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2
Q

Arc length formula

A

r{theta}

theta must be in radians

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3
Q

Sector area formula

A

0.5r²{theta}

theta must be in radians

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4
Q

Area of a segment formula

A

(0.5r²{theta} - 0.5r²sin{theta}

theta must be in radians

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5
Q

When theta is small, 9{theta}²-2{theta}+1 is what?

A

1

theta is small so approximately 0, meaning that the first 2 terms become 0 so just leaves 1

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6
Q

What is a mapping?

A

Transforms one set of numbers into a different set of numbers
It is a function if every input has a distinct output

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7
Q

Is a one-to-one mapping a function?

A

Yes

Each input has one distinct output

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8
Q

Is a many-to-one mapping a function?

A

Yes

Each input has a distinct output

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9
Q

Is a one-to-many mapping a function?

A

No

Not every input has a distinct output

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10
Q

Is y=1/x a function

A

No

No value at 0, doesn’t map anywhere

Can be made a function by restricting domain to x≠0

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11
Q

Domain…

A

The set of X values for which the function is valid

Writen as x=a

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12
Q

Range…

A

The set of Y values that the function can take

Written as a<=f(x)<=b

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13
Q

How do you deal with partial fractions?

A

Multiply up

Knock out brackets by substituting values in to make some coefficients 0

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14
Q

What is a piecewise-defined funtion?

A

A function which consists of more than one part

When drawing:
●=less/more than or equal to
○=less/more than

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15
Q

What is an inverse function?

A

The mathematical opposite of the original function

Only exists for one-to-one functions

Y becomes X, X becomes Y, reflected in line Y=X

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16
Q

How do you get an inverse function?

A

Replace f(x) with y
Swap y and x
Rearrange to make x the subject
Swap y for f(x)

Check f(x) and not any other function notation e.g g(x)

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17
Q

When do inverse functions exist

A

One-to-one functions

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18
Q

What is the domain of an inverse?

A

The range of the ordinary function

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19
Q

What is the range of an inverse function?

A

The same as the domain of the normal function

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20
Q

If f(x)=sinx, the inverse function can be written as…

A

f-¹(x)=sin-¹x=arcsinx

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21
Q

Sketch the graph of y=arcsinx

State the range and domain

A

X Intercept: 0
Y Intercept: None

Coordinates: (-1,-90) (0,0) (1,90)

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22
Q

If f(x)=cosx, the inverse function can be written as…

A

f-¹(x)=cos-¹x=arccosx

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23
Q

if f(x)=tank, the inverse function can be written as…

A

f-¹(x)=tan-¹x=arctan

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24
Q

What is the modulus of a number?

A

It is the non negative numerical value
Also known as the absolute value
Denoted as |x|

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25
Q

What is the difference between y=f(|x|) and y=|f(x)|?

A

y=f(|x|) will not have a negative value input, meaning that negative values of x will become the positive values of x. The first quadrant is reflected in the y axis into the second quadrant and the fourth into third

y=|f(x)| will not go below the x axis since y cannot be negative. All negative in the third quadrant will reflect up into the second, and all negative in the fourth reflected up into the first

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26
Q

Sketch the graph of y=arccosx

State the domain and range

A

X intercept: 1
Y intercept: 90°

Key coordinates: (-1,180) (0,90) (1,180)

Asymptotes: ?

Reflected in the y=x axis

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27
Q

Sketch the graph of y=arctanx

State the range and domain

A

Online image

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28
Q

How do you access modulus feature on calculator?

A

Shift

Abs (absolute)

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29
Q

What’s important to remember once you have solutions for a modulus function?

A

Sub back in to check no phantom solutions (signs not equal once put back in on either side of equation)

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30
Q

How do you deal with solving modulus equations?

A

Draw the graphs look for intersections

Make modulus positive, solve, check for phantom solutions
Make modulus negative, solve, check for phantom solutions

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31
Q

What are reciprocate functions called?

A

Cosecant=Cosec
Secant=Sec
Cotangent=Cot

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32
Q

What is the rule to remember reciprocal functions?

A

Look at the third letter for each, will match what is on the bottom

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33
Q

Formula for Cosecant?

A

Cosec(x)=1/sin(x)

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34
Q

What is the formula for Secant?

A

Sec(x)=1/cos(x)

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35
Q

Formula for Cotangent?

A

Cot(x)=1/tan(x) OR cos(x)/sin(x)

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36
Q

Sketch the graph of y=cosec(x) in range (range two pie to negative two pie)

A

Image online

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37
Q

Sketch the graph of y=sec(x) in range negative two pi and two pi

A

Get an image online

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38
Q

Sketch the graph of y=cot(x) in the range negative two pi to two pi

A

Get an image from online

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39
Q

Method for solving equation involving reciprocal functions?

A

Change
Flip
Solve

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40
Q

Solve sec(x)=0

A

1/cos(x)=0

cos(x)=1/0

1/0 = no solutions

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41
Q

Solve cosec(x)=0

A

1/sin(x)=0

sin(x)=1/0

1/0 = no solutions

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42
Q

Solve cot(x)=0 in range 0° to 360°

A

1/tan(x)=0

tan(x)=1/0

1/0 = asymptotes, meaning 90°, 270°

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43
Q

2 trig functions from year 12

A

tan(x)=sin(x)/cos(x)

sin²(x)+cos²(x)=1

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44
Q

2 trig identities from year 13, derived from sin²(x)+cos²(x)=1

A

1+tan²(x)=sec²(x)

1 with a tan is sexy

1+cot²(x)=cosec²(x)

1 in a cot is cosy

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45
Q

How can you derive the two trig identities from sin²(x)+cos²(x)

A

Divide by cos

Divide by sin

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46
Q

Where do the addition formula appear?

A

Page 6 of formula booklet

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47
Q

Double angle formula for sin2x

A

2sinxcosx

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48
Q

Double angle formulas for cos2x

A

cos²x-sin²x

2cos²x-1

1-2sin²x

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49
Q

Double angle forumla for tan2x

A

2tanx
______
1-tan²x

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50
Q

When is dy/dx used

A

Gradient or rate of change
Tangent or normal
Increasing or decreasing
Stationary point or turning point

51
Q

When is d²y/dx² used

A

Nature of a stationary point
Checking if minimum or maximum
Finding minimum or maximum

52
Q

Differentiate tan(x)

A

sec²(x)

53
Q

Differentiate x^n

A

nx^(x-1)

54
Q

Differentiate e^x

A

e^x

55
Q

Differentiate 4e^2x

A

8e^2x

56
Q

Differentiate e^sin(x)

A

cos(x)e^sin(x)

57
Q

Differentiate ln(x)

A

1/x

58
Q

Differentiate ln(x²+1)

A

2x/(x^2+1)

59
Q

Differentiate a^x

A

a^xln(a)

60
Q

Differentiate 2^3x

A

3(2^3xln(2))

61
Q

Differentiate sin, cos, -sin, -cos

A

cos
-sin
-cos
sin

62
Q

Bracket rule for differentiating (f(x))^n

A

n(f(x))^n-1

63
Q

Steps for product and quotient rule

A

Identify u and v
Differentiate u and v
Substitute into formula

64
Q

Product rule formula

A

u’v+uv’

65
Q

Quotient rule formula

A

(u’v-uv’)/v^2

66
Q

dx/dy …

A

reciprocal of dy/dx

67
Q

When is a function concave

A

If and only if f’‘(x)<0 for every x-value in given interval

68
Q

When is a function convex

A

If and only if f’‘(x)>0 for every x-value in given interval

69
Q

What is a point of inflection

A

Point at which a curve changes from being concave to convex or vice versa
f’‘(x) changes sign
Does not have to be a stationary point

70
Q

How do you prove a point of inflection

A

d^y/dx^2
Equal to 0
Opposite signs either side very close to point

71
Q

If y=u/v

Then dy/dx=…

A

(u’v-uv’)/v²

72
Q

How do you put something in harmonic form

A 
Rewrite the RHS using addition formula
Compare coefficients
Find tan a
Use pythagoras to find R
Write out your function in full
73
Q

How do you find exact values

A

Draw a triangle
Lable OAH
Check signs

74
Q

Signs for sin(x) from 0 to 360

A

P
P
N
N

75
Q

Signs for cos(x) going from 0 to 360

A

P
N
N
P

76
Q

Signs for tan(x) going from 0 to 360

A

P
N
P
N

77
Q

Explain a cast diagram

A

C bottom right (cos)
A top right (all)
S top left (sin)
T bottom left (tan)

Starting at A, draw arrow anticlockwise

Letter in each quadrant is what is positive for that range

78
Q

Harmonic motion

asin(x)+bcos(x)

A

R sin(x+a)

79
Q

acos(x)+bsin(x)

A

R cos(x-a)

80
Q

asin(x)-bcos(x)

A

R sin(x-a)

81
Q

acos(x) - bsin(x)

A

R cos(x+a)

82
Q

How do you integrate e^x

A

e^x + c

83
Q

How do you integrate e^4x

A

0.25 e^4x + c

84
Q

How do you integrate e^0.25x

A

4e^0.25x + c

85
Q

How do you integrate

Sin
Cos
-Sin
-Cos

A

-Cos
Sin
Cos
-Sin

86
Q

How do you integrate sin(2x)

A

-0.5cos(2x) + c

87
Q

How do you integrate cos(0.5x)

A

2sin(0.5x) + c

88
Q

How do you integrate a^x

A

a^x/ln(a)

89
Q

How do you integrate 2^x

A

2^x/ln(2) + c

90
Q

Integrate 2^3x

A

2^3x/3ln(2) + c

91
Q

Reverse Bracket Rule

A

(f(x))^n = (f(x))^n+1/(n+1)(f’(x))

92
Q

Integrate (2x+1)⁵

A

(2x+1)⁶/12 + c

93
Q

How can you integrate a complex fraction using Y12 methods

A

Split the fraction into a polynomial by multiplying up by the denominator ^-1

94
Q

When do you use ln to integrate a fraction

A

If the numerator is a multiple of the derivative of the denominator

95
Q

Integrate 3/(1+3x)

A

3ln(1+3x) ÷ 3 = ln(x) + c

96
Q

Integrate 3/x

A

3ln(x) + c

97
Q

Integrate x/(3x²+1)

A

xln(3x²+1)/6x + c = ln(3x²+1)/6 + c

98
Q

When do you use algebraic division in integration

A

If you have an improper fraction with a numerator of same power or higher than the denominator

99
Q

When do you use partial fractions for integration

A

If the denominator has a higher power than the numerator

AND the denominator can be factories I multiple brackets

100
Q

How do you simplify integrating trig

A

Use identities
Use double angles
Check if then in the book

101
Q

Relationship between product rule and integration by parts

A

Opposites

102
Q

Integration by parts formula

A

Page 7

uv - ∫ vu’ dx

103
Q

Acronym for labelling u then v in integration by parts

A

Logs
Algebra
Trig
Exponentials

104
Q

Integrate ln(x)

A

1/x + c

105
Q

Integrate (lnx)^2

A

u=(lnx)^2
v=x

xln(x)^2 - 2xln(x) + 2x + c

106
Q

Integration by substitution method

A 
Chose the substitution u
Differentiate du/dx
Rearrange for dx
Substitute, replacing dx and cancelling
Sort limits if applicable
Simplify
Integrate
Substitute back out if necessary
107
Q

What is the Reimann Sum

A

Adding together rectangles to estimate the area under a curve
More rectangles means more accurate the estimate
ALWAYS DO RECTANGLES UNDER CURVE, DO NOT CROSS LINE

108
Q

What happens during the Riemann Sum as the number of rectangles increases

A

The value for area approaches a limit which is the actual value of the integral

109
Q

How do you prove that increasing number of triangles is more accurate for Riemann Sum

A

Draw a curve in double positive axis (1st quadrant)
Draw one rectangle
Repeat with 2 and 3
Conclude with; more triangles means more of the area under the graph is filled by rectangles

110
Q

Definition of a cartesian equation

A

Give a direct relationship between x and y eg y=(x+1)^2

111
Q

Definition of a parametric equation

A

x and y are defined in terms of a third variable known as the parameter
eg x=t-1 and y=t^2

112
Q

How do you go from a parametric equation to a cartesian equation without trig

A

Rearrange one to get t

Substitute t into the other equation

113
Q

How do you go from a parametric equation to a cartesian equation with trig

A

Chose an appropriate identity
Rearrange the parametric equation
Substitute into identity

114
Q

For a parametric equation x=p(t) and y=q(t) with cartesian equation y=f(x)

Domain of f(x)
Range of f(x)

A 
Domain = Range of p(t)
Range = Range of q(t)
115
Q

Rule for differentiating parametric equations

A

(dy/dt)/(dx/dt) = (dy/dt) x (dt/dx) = dy/dx = y’/x’

116
Q

Formula to integrate parametric equations

A

∫ y dx/dt dt

117
Q

How do you integrate parametric equations

A

Swap limits from x to t
Get dx/dt
Plug into formula

118
Q

Segment

A

Area created by joining two ends of an arc with a straight line

119
Q

Where are small angle formulas for trig found

A

Page 6 formula booklet

120
Q

Integrating trig

A

Swap for an identity so easier

Cant have extra x on bottom

121
Q

Integrating trig with sin or cos squared

A

Use double angle

122
Q

Integrate lnx

A

by parts
u=lnx
v=x

xlnx-x+c

123
Q

Continous function

A

No vertical asymptote

Drawn without taking a pen off the page

124
Q

When will Newton raphson fail

A

Stationary point chosen to start with
Derivative will be 0
Can’t divide by zero since its invalid
Tangent to the x axis means it won’t intersect the x axis

Start point close to stationary point
Gradient of tangent is so small it intercepts the x axis a long way from x0

A Level Maths (6 decks)

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