How to do everything in Higher maths Flashcards

be able to do maths haha

1
Q

Distance Formula

A

The Square root of

x2-x1) + (y2-y1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

m=tana

A

positive of x axis use angle given

negative of x axis take away from 180

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Equation of a straight line

A

y-b=m(x-a)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Parallel lines

A

have equal gradients

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Perpendicular

A

flip and change sign

m1 x m2 = -1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Collinearity

A

Mab = Mbc = Mac

share a common point and have the same gradient

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Point of intersection

A

Simultaneous equations

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Medians

A

Find Midpoint
Find the gradient M
Find equation of line using M and either point

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Altitudes

A

Find M then Mperp

Use third point and Mperp

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Perpendicular Bisector

A

Find Midpoint
Find Gradient M
Find Mperp
use Mperp and Midpoint

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Recurrence Relations

A

Un+1 = aun + b

or

Un-1 = aun + b

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Limits

A

a limit can only occur if - 1< a < 1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Sketching Functions

A

inside the bracket change the x coordinate

outside the bracket change the y coordinate

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Composite Functions

A

substitute and solve

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Differentiation

A

bring power down

take away one from power

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Equations of tangents

A

need one point and M
one point is ALWAYS given
M is the rate of change f’(x)
x coordinate is the equation of the gradient

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Increasing and Decreasing Functions

A

f’(x) > 0 - increasing

f’(x) < 0 - decreasing

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Stationary Points

A

when f’(x) = 0

use nature table to find nature of curve

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Curve Sketching

A

Work out x-axis intercepts / roots solve y = 0
work out y axis intercepts for x = 0
determine stationary points and their nature
sketch on a graph

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Closed Intervals

A

Max/Min values can occur at stationary points or end points of the closed interval

a range is given -a < x < b

find stationary points - use if in range

sub the interval values and stationary points into f’(x) ti find max / min

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Graphs of Derived Functions

A

All stationary points become roots
when graph is increasing - above x axis
when graph is decreasing - below x axis
plot on graph new curve

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Polynomials - factors

A

is a factor if R=0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Equations for Polynomials

A

use y=k(x-a)(x-b)

use point on graph to determine y and x and given point for a and b

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Completing the square

A

take the coefficent out of the x^2
then half everything
complete the sqaure as normal
put the coefficent back in

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

Quadratic Inequations

A

less than zero = below x axis
greater than zero = above x axis
draw a sketch and solve

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

The discriminant

A

< 0 = No real roots
= 0 , = Equal roots (one root)
> 0 = 2 Real roots

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

Conditions for Tangency

A

make line and curve equal to each other
make = 0
solve
if one value it is a tangent

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

Point of Contact

A

sub point (x) into the line to find y

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

Integration

A

Add 1 to power and divide by new power

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

Indefenite Integrals

A

must have + c !!

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

Definite Integrals

A

have limits

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
32
Q

Finding a Limit - Integration

A

integrate, then upper take away lower limit

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
33
Q

Area Under Graph

A

use graph to find limits if not given

integrate

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
34
Q

Area above and below the x axis

A

two seperate integrations

using the 3 different roots to make to seperate limits

35
Q

Differential Equations

A

integrate and have + c on the end
sub in point to x and y to solve for c
re-write equation as y = x and sub in the value of C

36
Q

Problem Solving Differential Equations

A

D = vt

-> differentiate both ways

37
Q

Circle - Intersection of a line

A

always sub line into circle and solve

38
Q

equation of circle centre

A

(0,0) x^2 + y^2 = r^2

39
Q

Circle centre (a,b) radius r

A

(x-a)^2 + (y-b)^2 = r^2 on formula sheet
match up equation with formula sheet to find radius and centre
to work out radius if not given use distance formula

40
Q

Inside Outside or on the Circumference

A

if r^2 point < r^ circle then inside
if r^2 point = r^ circle then on
if r^2 point > r^ circle then outside

41
Q

General Equation of a Circle

A

x^2 + y^2 + 2gx + 2fx + c = 0 on formula sheet
expanded version of (x-a)^2 + (y-b)^2 = r^2

use square root of g^2 + f^2 - c also on formula sheet to find r

42
Q

Tangents to Circles

A

Substitute the line into the circle
expand and solve make sure to subsitute all y’s and remember to keep the ^2
take out a common factor if possible and solve to find x, should be 1 solution

43
Q

Equations of Tangents

A
Will always be PERPENDICULAR
Find M radius
Find Mperp
use y-b = m(x-a)
DO NOT substitute the centre always use the point on the line
44
Q

Trigonometry - Multiple and Compound Angles

A

Nat 5 Knowledge

use exact triangles to solve

45
Q

Addition Formula

A
sin(a±b) = sinAcosB ± CosAsinB
cos(a±b) = cosAcosB ∓ sinAsinB 

both on formula sheet

use exact triangles to find compare and match to formula sheet

if given one angle then use pythagorus to find others and solve

46
Q

Double Angle Formula

A

sin2A = 2sinAcosA
cos2A = cos^2A - sin^2A
= 2cos^2A - 1
= 1 - 2Sin^2A

All on formula sheet
match up your equation with one of the formula

47
Q

Using the formula to solve trig equations

A

Make equation = 0
must get rid of double angles
Factorise and Solve

48
Q

Vectors

A

Equal vectors = same magnitude
Zero vector = magnitude of zero
Unit vector = magnitude of 1

49
Q

Unit vector notation 3D

A

i
j
k

i (1,0,0) j(0,1,0) k(0,0,1)

50
Q

Finding the Magnitude

A

Square and add all the numbers together
then square root
Pythagoras basically

51
Q

Position Vectors

A

AB = b-a

52
Q

Parallel Vectors

A

Multiples of each other

53
Q

Collinear Points

A

Use position vectors
if they are both the same
then

they are MULTIPLES OF THE SAME VECTOR
and share a common point

create a ratio with the number outside the bracket

54
Q

Dividing a line in Given Ratio

A

find the ratio total by adding both together
find the distance between the x’s,y’s and z’s
divide by the ratio
times by the first number in the ratio
add that to the first point

55
Q

Scalar product/ dot product

A

a.b. = | a | | b | cos θ formula sheet
vectors MUST point AWAY from each other
θ is always between 0 and 180

56
Q

if dot product = 0

A

then they are perpendicular vectors

57
Q

if angle not on exact triangle

A

find the angle opposite e.g.

90 + 60 = 150
so
90 - 60 = 30
30 is opposite to 150

so if 30 is positive 150 is negative, use graph to determine positive or negative

58
Q

Scalars and Vectors in Component Form

A

a.b. = a1b1 + a2b2 + a3b3 formula sheet

remember if a.b. = 0 it is perpendicular

59
Q

Angles Between Vectors

A

cosθ = a.b. / | a | | b |

rearranged formula same method as before

60
Q

When Notation changes

A

make sure to use correct notation with dot product

61
Q

Properties of the Scalar and Dot Product

A

a. ( b + c ) = a.b. + a.c.
a. ( b - c ) = a.b. - a.c.
a. b. = b.a.

if anything is x.x or a.a. etc then θ = 0

62
Q

Derivatives of sinx and cosx

A

f(x) sinx then f’(x) = cosx
f(x) = cosx then f’(x) = -sinx

on formula sheet

a number infront DOES NOT change
remember two negatives make a positive

63
Q

Gradient at a Point

A

Find the equation of a tangent of y = 2sinx and x = π/3

find m of y=2sinx as remember f’(x) = m

then find y by subbing in x

use y-b=m(x-a)

do not mix radians and surds!

64
Q

Chain Rule

A

Bring power down as normal but put infront of bracket
Take power away as usual
Times by the DIFFERENTIAL of the bracket

all usual differential rules apply afterwards, negative powers, surds etc.

65
Q

Integrals of sinx and cosx

A
∫cosdx = sinx + c
∫sinxdx = -cosx + c

both on formula sheet

number in front doesn’t change

usual steps of add one to power and divide by new power of things without sin and cos

66
Q

Definite Integrals

A

same rules as before this time with limits.

67
Q

Finding the area under the curve y=sinx between x=0 and x= 2π

A

Find the area of one using normal rules, then double as as areas are equal and the range is 2π

68
Q

Harder Chain Rule Examples

A

bring power down multiple by bracket differentiated

when there are more than just one number being multiplied i.e. it contains x’s and x^2’s then put in a bracket

69
Q

Chain Rule - Trig Functions

A

Keep the BRACKET the SAME!
differentiate outside the bracket
times by differentiated bracket

all usual rules apply, bringing power down etc if necessary.

70
Q

Integrating Chain Rule

A

Add 1 to power
Divide by new power
also divide by the differential of the bracket, NOT the INTEGRAL

71
Q

Definite Integrals

A

Same method, integrate chain rule

use limits solve

72
Q

Integrating sin(ax+b) and cos (ax+b)

A
∫cos(ax+b) = 1/a sin(ax + b) + c
∫sin(ax+b) = - 1/acos(ax + b) + c

both on formula sheet

usual rules, integrate OUTSIDE the bracket as that stays the same
divide by the differential of the bracket and new power if there is one.

if there are 2 things to integrate make sure to keep them seperate

73
Q

Wave Function

A

acosx + bsinx = kcos(x-a)

if it asks you in the form of kcos(x-a) use formula sheet to expand what is after the k may vary so you must compare with the formula sheet

then pull out kcosa= and ksina=

use these numbers to find k with pythagorus

then use trig identity sun over the sea (c) gives you a tan, ksina/kcosa = tana

use K and a and sub into the original kcos(x-a)

if you have a negative value for sin/cos then use CAST diagram and tick corresponding

i.e. +sin tick where sin is positive and -cos tick where cos is negative

74
Q

Graphs of Exponential functions

A

a>0 and a≠1 for y=a^x

you will get two coordinates of (0,1) and (1,0)

75
Q

for 0<a></a>

A

you will get two coordinates of (0,1) ans (1,a)

as a^0 = 1 and a^1 = a for all a>0

76
Q

Transformation of exponential function graphs

A

work the SAME as normal transformations of graphs

77
Q

e

A

approximately = 2.718

78
Q

logs!

A

if f(x) = a^x , then f^-1 (x) = logax then f^-1 (x) = a^x

: logex = base e
ln = log to base e of x
log = log10x = base 10

79
Q

e^x and lnx

A

e^x and ln^x are inverse functions

lnx=logex

this works for other bases i.e.

10^x is the inverse of log10x

and 2^x if log2x etc…

80
Q

Rules for logarithms

A

log10xy = log10x + log10y (multiply)

log10 x/y = logx - log10y (divide)

nlog10x = log10x^n (make the power come infront the log)

81
Q

Formula of Experimental data

A

y = kx^n

log10y = log10kx^n
log10y = log10k + log10x^n
log10y = nlog10x + log10k

normal rules plus our k

to find n find the gradient

to find k subsitute a point into the equation rearrange to solve for k and take exponential of both sides to find k

82
Q

whether it is y = kn^n or y=ab^x

A

if logy and logx on graph then y=kx^n

if logy and x then y = ab^x

83
Q

For both use y = mx+c to solve as it is much easier

A

so y=ab^x then y=mx+c

log10y= 2x+1 if m=2 and c = 1 (my example heheh) i’m using log10y as that is the other part of the graph

take exponential and well solve then match up your answer to y=ab^x to see what a and b are

do the same for y=kx^n but just take both log10x and log10y for x rather than just x