Straight Line (2) Composite Functions (1) Higher Past Papers Higher Past Papers by Topic 2015 by Topic 2015 Onwards Onwards www.mathsrevision.com Differentiation (2) Recurrence Relations (1) Polynomials (3) Integration (1) Trigonometry (1) The Circle (1) Vectors (2) Logs & Exponential (1) Wave Function (1) Summary of Higher Thursday 17 March 2022 Thursday 17 March 2022 Created by Mr. Lafferty Maths Dept. Created by Mr. Lafferty Maths Dept.
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Straight Line (2) Composite Functions (1) Higher Past Papers by Topic 2015 Onwards Higher Past Papers by Topic 2015 Onwards Differentiation.
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Straight Line (2)
Composite Functions (1)
Higher Past PapersHigher Past Papersby Topic 2015 by Topic 2015
Onwards Onwards
ww
w.m
ath
srevis
ion
.com
Differentiation (2)
Recurrence Relations (1)
Polynomials (3)
Integration (1)
Trigonometry (1)
The Circle (1)
Vectors (2)
Logs & Exponential (1)
Wave Function (1)
Summary of Higher
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Straight Line
Composite Functions
Higher MindmapsHigher Mindmapsby Topic by Topic
ww
w.m
ath
srevis
ion
.com
Differentiation
Recurrence Relations
Polynomials
Integration
Trigonometry
The Circle
Vectors
Logs & Exponential
Wave Function
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Main Menu
Straight Liney = mx + c
2 1
2 1
y ym
x x
m = tan θθ
Possible values for gradient
m > 0
m < 0
m = 0
m = undefined
2 22 1 2 1( ) ( )D x x y y
For Perpendicular lines the following is
true.m1.m2 = -1
Parallel lines have
same gradient
1 2 1 2,2 2
x x y y
Mid
Mindmaps
flip inx-axis
flip iny-axis
-
+
+-
Graphs & Functions
y = -f(x)
y = f(-x)
y = f(x) ± k
y = f(kx)
Move verticallyup or downs
depending on kStretch or compressvertically
depending on k
y = kf(x)
Stretch or compress
horizontally depending on k
y = f(x ± k)
Move horizontallyleft or right
depending on k
Remember we can combine
these together !!
Mindmaps
Composite Functions
Similar to composite
Area
A complex function made up of 2 or more
simpler functions
= +
f(x) = x2 - 4 g(x) = 1x
x
Domain Range
y = f(x)1y
Restriction
x2 - 4 ≠ 0
(x – 2)(x + 2) ≠ 0
x ≠ 2 x ≠ -2
But y = f(x) is x2 - 4g(f(x))
g(f(x)) =
f(x) = x2 - 4g(x) = 1x
x
Domain Range
y = g(x)
f(g(x))
y2 - 4
Restriction x2 ≠ 0
But y = g(x) is
f(g(x)) =
1x
1x
2- 4
Rearranging
Mindmaps
1 1
2 3
Basics before Differentiation/
Integration
Working with
fractions
Indices
3 x
7 4x
5 2x
6 5x
Surdsm
nx ( )m n m nx x x ( )m
m nn
xx
x
n mx
1
3x
4
7x5
6x
5
6
1
x
2
5x
2
5
1
x
1 1
2 4x x 3
4x
1
3
2
3
x
x
x
5
6
1 1
2 3
1
6
1 3
2 5 3
10
1 4
2 5
5
81 5
2 4
Format for Differentiation
/ Integration
Mindmaps
Differentiationof Polynomials
f(x) = axn
then f’x) = anxn-1
Derivative = gradient = rate of change
Graphsf’(x)=0
54
2( )
3f x
x
5
42( )
3
xf x
9
4
94
552'( )
3 6
xf x
x
1
2( ) 2 1f x x x 3 1
2 2( ) 2f x x x 1 1
2 21
'( ) 32
f x x x
1
21
'( ) 32
f x xx
f’(x)=0Stationary PtsMax. / Mini Pts
Inflection Pt
Nature Table
Gradient at a point
Equation of tangent line
Straight LineTheory
Leibniz Notation
'( )dy
f xdx
Mindmaps
Differentiations
Polynomials
Stationary PtsMini / Max PtsInflection Pts
Rate of change of a
function.
Harder functionsUse Chain Rule
Meaning
Graphs
xxn d
dn xn 1
Rules of Indices
xsin a x( )d
da cos a x( )
xcos a x( )d
da sinx a x( )
xx2 2x4 3d
d3 x2 2x4 2 2x 8x3
Gradient at a point.
Tangent
equation
Straight line Theory
Factorisation
d
dx
d
dx
(distance)
=velocity
=(velocity)
=acceleration
Real life
( ) (outside the bracket) (inside the bracket)nd d dinside