How to do everything in Higher maths Flashcards
be able to do maths haha
Distance Formula
The Square root of
x2-x1) + (y2-y1
m=tana
positive of x axis use angle given
negative of x axis take away from 180
Equation of a straight line
y-b=m(x-a)
Parallel lines
have equal gradients
Perpendicular
flip and change sign
m1 x m2 = -1
Collinearity
Mab = Mbc = Mac
share a common point and have the same gradient
Point of intersection
Simultaneous equations
Medians
Find Midpoint
Find the gradient M
Find equation of line using M and either point
Altitudes
Find M then Mperp
Use third point and Mperp
Perpendicular Bisector
Find Midpoint
Find Gradient M
Find Mperp
use Mperp and Midpoint
Recurrence Relations
Un+1 = aun + b
or
Un-1 = aun + b
Limits
a limit can only occur if - 1< a < 1
Sketching Functions
inside the bracket change the x coordinate
outside the bracket change the y coordinate
Composite Functions
substitute and solve
Differentiation
bring power down
take away one from power
Equations of tangents
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
Increasing and Decreasing Functions
f’(x) > 0 - increasing
f’(x) < 0 - decreasing
Stationary Points
when f’(x) = 0
use nature table to find nature of curve
Curve Sketching
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
Closed Intervals
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
Graphs of Derived Functions
All stationary points become roots
when graph is increasing - above x axis
when graph is decreasing - below x axis
plot on graph new curve
Polynomials - factors
is a factor if R=0
Equations for Polynomials
use y=k(x-a)(x-b)
use point on graph to determine y and x and given point for a and b
Completing the square
take the coefficent out of the x^2
then half everything
complete the sqaure as normal
put the coefficent back in
Quadratic Inequations
less than zero = below x axis
greater than zero = above x axis
draw a sketch and solve
The discriminant
< 0 = No real roots
= 0 , = Equal roots (one root)
> 0 = 2 Real roots
Conditions for Tangency
make line and curve equal to each other
make = 0
solve
if one value it is a tangent
Point of Contact
sub point (x) into the line to find y
Integration
Add 1 to power and divide by new power
Indefenite Integrals
must have + c !!
Definite Integrals
have limits
Finding a Limit - Integration
integrate, then upper take away lower limit
Area Under Graph
use graph to find limits if not given
integrate
Area above and below the x axis
two seperate integrations
using the 3 different roots to make to seperate limits
Differential Equations
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
Problem Solving Differential Equations
D = vt
-> differentiate both ways
Circle - Intersection of a line
always sub line into circle and solve
equation of circle centre
(0,0) x^2 + y^2 = r^2
Circle centre (a,b) radius r
(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
Inside Outside or on the Circumference
if r^2 point < r^ circle then inside
if r^2 point = r^ circle then on
if r^2 point > r^ circle then outside
General Equation of a Circle
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
Tangents to Circles
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
Equations of Tangents
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
Trigonometry - Multiple and Compound Angles
Nat 5 Knowledge
use exact triangles to solve
Addition Formula
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
Double Angle Formula
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
Using the formula to solve trig equations
Make equation = 0
must get rid of double angles
Factorise and Solve
Vectors
Equal vectors = same magnitude
Zero vector = magnitude of zero
Unit vector = magnitude of 1
Unit vector notation 3D
i
j
k
i (1,0,0) j(0,1,0) k(0,0,1)
Finding the Magnitude
Square and add all the numbers together
then square root
Pythagoras basically
Position Vectors
AB = b-a
Parallel Vectors
Multiples of each other
Collinear Points
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
Dividing a line in Given Ratio
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
Scalar product/ dot product
a.b. = | a | | b | cos θ formula sheet
vectors MUST point AWAY from each other
θ is always between 0 and 180
if dot product = 0
then they are perpendicular vectors
if angle not on exact triangle
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
Scalars and Vectors in Component Form
a.b. = a1b1 + a2b2 + a3b3 formula sheet
remember if a.b. = 0 it is perpendicular
Angles Between Vectors
cosθ = a.b. / | a | | b |
rearranged formula same method as before
When Notation changes
make sure to use correct notation with dot product
Properties of the Scalar and Dot Product
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
Derivatives of sinx and cosx
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
Gradient at a Point
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!
Chain Rule
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.
Integrals of sinx and cosx
∫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
Definite Integrals
same rules as before this time with limits.
Finding the area under the curve y=sinx between x=0 and x= 2π
Find the area of one using normal rules, then double as as areas are equal and the range is 2π
Harder Chain Rule Examples
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
Chain Rule - Trig Functions
Keep the BRACKET the SAME!
differentiate outside the bracket
times by differentiated bracket
all usual rules apply, bringing power down etc if necessary.
Integrating Chain Rule
Add 1 to power
Divide by new power
also divide by the differential of the bracket, NOT the INTEGRAL
Definite Integrals
Same method, integrate chain rule
use limits solve
Integrating sin(ax+b) and cos (ax+b)
∫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
Wave Function
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
Graphs of Exponential functions
a>0 and a≠1 for y=a^x
you will get two coordinates of (0,1) and (1,0)
for 0<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
Transformation of exponential function graphs
work the SAME as normal transformations of graphs
e
approximately = 2.718
logs!
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
e^x and lnx
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…
Rules for logarithms
log10xy = log10x + log10y (multiply)
log10 x/y = logx - log10y (divide)
nlog10x = log10x^n (make the power come infront the log)
Formula of Experimental data
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
whether it is y = kn^n or y=ab^x
if logy and logx on graph then y=kx^n
if logy and x then y = ab^x
For both use y = mx+c to solve as it is much easier
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
