IGCSE Additional Mathematics (0606) Revision Notes
Transcript of IGCSE Additional Mathematics (0606) Revision Notes
IGCSE Additional Mathematics (0606)
Revision Notes CAIE 2020 syllabus
By: Steven Zhou, find more at ste-z.com
Table of Contents
Legend ........................................................................................................................................................................ 2
1. Functions........................................................................................................................................................... 3
2. Quadratic Function ......................................................................................................................................... 5
3. Equations, Inequalities And Graphs............................................................................................................. 8
4. Indices And Surds ......................................................................................................................................... 11
5. Factors Of Polynomials................................................................................................................................ 12
6. Simultaneous Equations ............................................................................................................................... 15
7. Logarithmic & Exponential Functions ......................................................................................................... 16
8. Straight Line Graph ..................................................................................................................................... 18
9. Circular Measures ........................................................................................................................................ 20
10. Trigonometry ........................................................................................................................................... 21
11. Permutation & Combination .................................................................................................................. 26
12. Series ........................................................................................................................................................ 32
13. Vectors In 2 Dimensions .......................................................................................................................... 34
14. Differentiation & Integration ................................................................................................................. 36
Appendix A: Formula Sheet .................................................................................................................................. 44
Appendix B: Operations On Graphs ................................................................................................................... 44
Acknowledgement................................................................................................................................................... 45
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Legend
Syllabus objective
Notation
Note and tip
Formula and Laws
Key concepts
Information and explanation
Information and explanation
1. Step 1
a) Step 1.1
b) Step 1.2
2. Step 2
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1. Functions
understand the terms: function, domain, range (image set), one-one function, inverse
function and composition of functions
use the notation ๐(๐ฅ) = ๐ ๐๐ ๐ฅ, ๐: ๐ฅ โฆ ๐๐ ๐ฅ, (๐ฅ > 0), ๐โ1(๐ฅ) and ๐2(๐ฅ) [=
๐(๐(๐ฅ))]
Function Notation
e.g. ๐(๐ฅ) = ๐ ๐๐ ๐ฅ
e.g. ๐: ๐ฅ โฆ ๐๐ ๐ฅ
One-one function
function with no 2 points have the same y-coordinate
Can be tested with the horizontal line test
draw a horizontal line and move it across
a function is one-one is the line does not touch two point on the curve at
once
explain in words why a given function does not have an inverse
find the inverse of a one-one function
use sketch graphs to show the relationship between a function and its inverse
Inverse function
๐ฅ and ๐ฆ value exchanges when ๐(๐ฅ) become ๐โ1(๐ฅ)
The domain and range exchanges
Notation - ๐โ1(๐ฅ)
A function has an inverse when it is one-one
Finding the inverse function
Graph between function and its inverse (Figure 1)
The inverse function is the reflection of the function in ๐ฆ = ๐ฅ
form composite functions
Figure 1
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Composition of Functions
Notation
When plugging ๐(๐ฅ) into ๐(๐ฅ) โ ๐๐(๐ฅ) or ๐(๐(๐ฅ))
When plugging ๐(๐ฅ) into ๐(๐ฅ) - ๐2(๐ฅ) or ๐(๐(๐ฅ))
For fg(k) and k is a given value, first calculate g(k), and then plug the result of g(k)
into f(x)
why a given function is a function โ one x value only corresponds to one y
Domain and Range
Domain
set of values of x
Notation e.g. (๐ฅ > 0)
Range
set of values of y
Notation e.g. (๐ฆ > 0)
understand the relationship between ๐ฆ = ๐(๐ฅ) and ๐ฆ = |๐(๐ฅ)|, where ๐(๐ฅ) may
be linear, quadratic or trigonometric
Modulo functions
Notation: ๐ฆ = |๐(๐ฅ)|
any part of ๐ฆ = ๐(๐ฅ) below the x axis is reflected
upwards
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2. Quadratic Function
solve quadratic equations for real roots
Solving quadratic equations for real roots
By factorizing
1. Rearrange the equation so one side is zero
2. Factorize into the form (๐๐ฅ โ ๐)(๐๐ฅ โ ๐) (a,b,c and d is constants)
3. Solve the 2 (or 1) linear equation ๐๐ฅ โ ๐ = 0 and ๐๐ฅ โ ๐ = 0
By completing the square
1. Put the constant at the right of the equal sign
2. If the coefficient of ๐ฅ2 is not 1, divide through to make it 1
3. Find and add the square number (= coefficient of ๐ฅ2
2) to both sides
4. Complete the square
5. Square root both sides and solve
NOTE: when square rooting both sides, make sure
RHS have ยฑ
By the Quadratic formula
Arrange the equation into the form ๐๐ฅ2 +
๐๐ฅ + ๐ = 0
๐ฅ =โ๐ยฑโโ
2๐, where โ (discriminant) = ๐2 โ 4๐๐ (This will be provided
on formula sheet)
NOTE: do not divide each side by x
find the maximum or minimum value of the quadratic function ๐: ๐ฅ โฆ ๐๐ฅ2 + ๐๐ฅ + ๐
by any method
Maximum and minimum values
1. Find the vertex of the parabola
a) Method 1
i. Arrange the function into the form ๐๐ฅ2 + ๐๐ฅ + ๐ = 0
ii. The axis of symmetry is ๐ฅ = โ๐
2๐ : Therefore, the vertex is
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(โ๐
2๐, ๐(โ
๐
2๐))
b) Method 2
i. Find the two roots โ ๐ and ๐
ii. The axis of symmetry is ๐ฅ =๐+๐
2: Therefore, the vertex is
(๐ + ๐
2, ๐(๐ + ๐
2))
c) Method 3
i. Arrange into the form ๐ฆ = ๐(๐ฅ โ โ)2 + ๐ by by completing the
square
ii. The vertex is (โ, ๐)
2. Determine if the vertex is the maximum or minimum
a) Arrange the function into the form ๐๐ฅ2 + ๐๐ฅ + ๐ = 0
b) If ๐ > 0 then it is a minimum, if ๐ < 0 then it is a maximum
use the maximum or minimum value of ๐(๐ฅ) to sketch the graph or determine the
range for a given domain
Sketching the graph of quadratic function
1. For ๐ฆ = ๐(๐ฅ โ ๐)2 + ๐, ๐ฆ = ๐(๐ฅ + ๐)(๐ฅ + ๐), ๐ฆ = ๐๐ฅ2 + ๐๐ฅ + ๐:
2. Find the x-intercept (the roots) by finding ๐ฅ for ๐(๐ฅ) = 0
3. Find the y-intercept by finding the value of ๐(0)
4. Find the vertex
5. Pinpoint the important points โ intercepts and vertex
6. Draw using the fact that the graph of quadratic function is always symmetric
7. Label the axis
MARKING: 1 point for correct shape + max/min in correct quadrant; 1 mark
for labeling all the intercept
Determining the range
If the vertex is a maximum: ๐ฆ < (๐ฆ ๐ฃ๐๐๐ข๐ ๐๐ ๐กโ๐ ๐ฃ๐๐๐ก๐๐ฅ)
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If the vertex is a minimum: ๐ฆ > (๐ฆ ๐ฃ๐๐๐ข๐ ๐๐ ๐กโ๐ ๐ฃ๐๐๐ก๐๐ฅ)
Discriminant
know the conditions for ๐(๐ฅ) = 0 to have (i) two real roots, (ii) two equal roots, (iii) no
real roots
conditions of root for ๐(๐ฅ) = 0
1. Rearrange into the form ๐๐ฅ2 + ๐๐ฅ + ๐ = 0
2. Find โ
โ CONDITIONS OF ROOT FOR ๐(๐) = ๐
> ๐ two real roots
= ๐ two equal roots
< ๐ no real roots
related conditions for a given line to
(i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given
curve
related conditions for a given line to (i) intersect a given curve, (ii) be a
tangent to a given curve, (iii) not intersect a given curve
1. Substitute the function of the line into the quadratic function
2. Rearrange into the form ๐๐ฅ2 + ๐๐ฅ + ๐ = 0
3. Find โ
โ CONDITIONS
> ๐ The line intersects a given curve
= ๐ The line is a tangent to the given curve
< ๐ The line did not intersect the curve
find the solution set for quadratic inequalities
Solving quadratic inequalities
1. Solve for x
2. Sketch a graph
3. Determine the values according to the graph
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3. Equations, inequalities and graphs
solve graphically or algebraically equations of the type |๐๐ฅ + ๐| = ๐(๐ โฉพ 0)
and |๐๐ฅ + ๐| = |๐๐ฅ + ๐|
Solving modulo equations
Graphically
1. Graph both side of the equal sign (e.g. graph ๐ฆ = |๐๐ฅ + ๐| and ๐ฆ = ๐
for the equation |๐๐ฅ + ๐| = ๐ )
2. the point(s) of intersection is the solution
Algebraically
When on one side the modulo sign is removed, a ยฑ has to be added to
the other side
๐ผ๐ |๐๐ฅ + ๐| = ๐
๐โ๐๐ ๐๐ฅ + ๐ = ยฑ๐
When removing modulo on both sides, only one ยฑ has to be added
๐ผ๐ |๐๐ฅ + ๐| = |๐๐ฅ + ๐|
๐โ๐๐ ๐๐ฅ + ๐ = ยฑ(๐๐ฅ + ๐) ๐๐ ยฑ (๐๐ฅ + ๐) = ๐๐ฅ + ๐
solve graphically or algebraically inequalities of the type |๐๐ฅ + ๐| > ๐(๐ โฉพ 0),
|๐๐ฅ + ๐| โฉฝ ๐(๐ > 0) and |๐๐ฅ + ๐| โฉฝ |๐๐ฅ + ๐|
Solving modulo inequalities
Graphically
Draw each side of the equation
(e.g. for |๐๐ฅ + ๐| โฉฝ |๐๐ฅ + ๐|, draw ๐ฆ = |๐๐ฅ + ๐|, and ๐ฆ =
|๐๐ฅ + ๐| on the same axis, and parts of the ๐ฆ = |๐๐ฅ + ๐| graph
under ๐ฆ = |๐๐ฅ + ๐| is the solution)
Algebraically - Follow the same when solving modulo equations โ expect the
sign need to change direction (e.g. < ๐ก๐ >,โฅ ๐ก๐ โค) when both sides is
divided by a negative number
sketch the graphs of cubic polynomials and their moduli, when given in factorised form
๐ฆ = ๐(๐ฅ โ ๐)(๐ฅ โ ๐)(๐ฅ โ ๐)
Graphing Cubic polynomials in form ๐ฆ = ๐(๐ฅ โ ๐)(๐ฅ โ ๐)(๐ฅ โ ๐)
1. Find the x-intercept (root): x=a, x=b, x=c
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2. Find the y intercept by substituting 0 for x
3. ๐
k>0 k<0
k affects the expansion vertically โ do not affect x-intercept
4. For ๐ฆ = ๐(๐ฅ โ ๐)2(๐ฅ โ ๐)
Curve will touch the x-axis at ๐ฅ = ๐, and cut the
curve at ๐ฅ = ๐
5. For ๐ฆ = ๐(๐ฅ โ ๐)3
Curve will meet the curve at x=a:
solve cubic inequalities in the form ๐(๐ฅ โ ๐)(๐ฅ โ ๐)(๐ฅ โ ๐) โฉฝ ๐ graphically
Solving cubic inequalities in the form ๐(๐ฅ โ ๐)(๐ฅ โ ๐)(๐ฅ โ ๐) โฉฝ ๐ by graphing
1. Graph the cubic curve
2. Graph the line y=d
3. Any part of the curve that is under the line x=d is the solution of the
inequality
use substitution to form and solve a quadratic equation in order to solve a related
equation
Problem solving with Quadratics
1. Translate the words to algebraic equation, define what x is
2. Solve the equation
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3. Check if the results are practical (e.g. numbers of objects present cant be
negative)
4. Give the answers in a sentence
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4. Indices and surds
perform simple operations with indices and with surds, including rationalising the
denominator
Operations with indices
A negative base raised to odd index is always negative; a negative base
raised to even index is always positive
๐๐ ร ๐๐ = ๐๐+๐
๐๐
๐๐= ๐๐โ๐ (๐ โ 0)
(๐๐)๐ = ๐๐ร๐
(๐๐)๐ = ๐๐๐๐
(๐
๐)๐ =
๐๐
๐๐ (๐ โ 0)
๐0 = 1 (๐ โ 0)
๐โ๐ =1
๐๐
๐๐
๐ = โ๐๐๐
(๐ > 0, ๐ โ โค+,
๐ โ โค)
Operation with surds
โ๐ โฅ 0, ๐ ๐๐ข๐ ๐ก โฅ 0
โ๐๐ = โ๐ ร โ๐ (for ๐, ๐ โฅ 0) โ
๐
๐=โ๐
โ๐
๐โ๐ + ๐โ๐ = (๐ + ๐)โ๐
Rationalizing the denominator
for ๐
โ๐ : multiply by
โ๐
โ๐ (since it equals to 1)
for ๐
๐+โ๐ : multiply by
๐โโ๐
๐โโ๐
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5. Factors of polynomials
BASE KNOWLEDGE
Zero for polynomial and root for equations
Adding and subtraction of polynomials can be done by collecting like
terms
Scalar can be multiplied to a polynomial by multiplying each term
Multiplying polynomials - each term of the first polynomial is multiplied
with each term of the second polynomial
Dividing polynomials
By long division
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TIP: ๐๐ can be inserted when there are no that kind of term
find factors of polynomials
Factors of a polynomial
If ๐ฅ โ ๐ is a factor of ๐(๐ฅ), then there exists a polynomial ๐(๐ฅ) such that
๐(๐ฅ) = (๐ฅ โ ๐)๐(๐ฅ)
If one factor is found, other factor could be obtained by performing division โ
the quotient is the other factor
know and use the remainder and factor theorems
Remainder theorem
When polynomial ๐(๐ฅ) is divided by ๐ฅ โ ๐ until a constant remainder ๐ is
obtained โบ ๐ = ๐(๐)
We can use the theorem to determine the value of the remainder
Factor theorem
๐ is a zero of the polynomials ๐(๐ฅ) โบ ๐ฅ โ ๐ is a factor of ๐(๐ฅ)
TIP: if ๐๐ฅ โ ๐ is a factor of ๐(๐ฅ) โบ one of the zero is ๐
๐
We can use the theorem to determine whether x-k is a factor of P(x)
solve cubic equations
Solving cubic equations
1. Identify the constant term
2. Factorize the constant term
3. Substitute the possible factor of constant term into the cubic equation until
a) the factor that allow the equation to =0 is one of the root of the cubic
equation
4. One factor is obtained by utilizing the factor theorem
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5. The other quadratic factor is obtained by performing long division of the
equation with the factor
6. Solve the quadratic equation
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6. Simultaneous equations
solve simple simultaneous equations in two unknowns by elimination or substitution
solve simple simultaneous equations in two unknowns
by elimination
๐๐ฅ๐๐๐๐๐ {4๐ฅ โ ๐ฆ = 0 ๐ฆ โ ๐ฅ = 3
๐ผ๐ ๐ค๐ ๐๐๐ 4๐ฅ โ ๐ฆ = 0 ๐ก๐ ๐ฆ โ ๐ฅ = 3, ๐กโ๐๐:
3๐ฅ = 3
๐ฅ = 1
๐ด๐๐ ๐ ๐ข๐๐ ๐ก๐๐ก๐ข๐๐๐ ๐ฅ = 1 ๐๐๐ก๐ ๐ฆ โ ๐ฅ = 3
๐ฆ โ 1 = 3
๐ฆ = 4
by substitution
๐๐ฅ๐๐๐๐๐ {2๐ฅ โ ๐ฆ = 0๐ฆ = 5
๐๐ข๐ ๐ก๐๐ก๐ข๐ก๐ ๐ฆ = 5 ๐๐๐ก๐ 2๐ฅ โ ๐ฆ = 0:
2๐ฅ โ 5 = 0
2๐ฅ = 5
๐ฅ =5
2, ๐ฆ = 5
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7. Logarithmic & exponential functions
know simple properties and graphs of the logarithmic and exponential functions
including ln ๐ฅ and ๐๐ฅ (series expansions are not required) and graphs of ๐๐๐๐ฅ +
๐ and ๐ ln(๐๐ฅ + ๐) where ๐, ๐, ๐ and ๐ are integers
graphs of the exponential functions
for ๐ฆ = ๐๐๐ฅโ๐ + ๐
๐ controls the steepness
๐ controls horizontal translation
๐ controls vertical translation, so that the horizontal asymptote of the
graph is ๐ฆ = ๐
MARKING: 1 mark for correct shape; 1 mark for intercept labeled; 1 mark for the
asymptote
Logarithms
If ๐(๐ฅ) = ๐๐ฅ ๐กโ๐๐ ๐โ1(๐ฅ) = log๐ ๐ฅ
Logarithm in base 10 is written as lg ๐
Logarithm in base ๐ is written as ln ๐
Simple properties
log๐(๐(๐ฅ)) is defined only when ๐ and ๐(๐ฅ) > 0
TIP: This can be used to determine the domain/range of a logarithmic
function
log๐ ๐๐ฅ = ๐ฅ
๐log๐ ๐ฅ = ๐ฅ (๐๐๐ ๐ฅ > 0)
๐ผ๐ ๐ = ๐๐ฅ, ๐กโ๐๐ ๐ฅ = log๐ ๐ (๐๐๐ ๐, ๐ > 0)
ln ๐ฅ = log๐ ๐ฅ
lg ๐ฅ = log10 ๐ฅ
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graphs of the logarithmic functions
find the inverse function (the exponential function) of the given logarithmic
function
๐กโ๐ ๐๐ข๐๐๐ก๐๐๐ ๐๐ ๐ฆ = ๐ ln(๐๐ฅ + ๐)
๐กโ๐๐ ๐กโ๐ ๐๐๐ฃ๐๐๐ ๐ ๐๐ข๐๐๐ก๐๐๐ ๐๐ : ๐ฅ = ๐ ln(๐๐ฆ + ๐)
ln(๐๐ฆ + ๐) =๐ฅ
๐
๐๐ฆ + ๐ = ๐๐ฅ๐
๐๐ฆ = ๐๐ฅ๐ โ ๐
๐ฆ =1
๐๐๐ฅ๐ โ
๐
๐
graph the inverse function
reflect the graph of the inverse function in the line y=x
know and use the laws of logarithms (including change of base of logarithms) solve
equations of the form ๐๐ฅ = ๐
laws of logarithms
log๐ ๐ + log๐ ๐ = log๐ ๐๐
log๐ ๐ โ log๐ ๐ = log๐๐
๐
nlog๐ ๐ = log๐( ๐)๐
log๐ ๐ =log๐ ๐
log๐ ๐ (๐๐๐ ๐, ๐, ๐ > 0, ๐๐๐ ๐, ๐ โ 1) (Change of base rule)
Nature logarithms follows the same rule
Solving equations
By equating indices
๐ผ๐ ๐๐ฅ = ๐๐ , ๐กโ๐๐ ๐ฅ = ๐
By using logarithms
Logarithms can be added to both sides of the equal sign
Then solve the logarithmic equation formed - Utilize the laws of
logarithms to solve
CAIE IGCSE Additional Mathematics (0606) Revision Notes
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8. Straight line graph
interpret the equation of a straight-line graph in the form ๐ฆ = ๐๐ฅ + ๐
Equation of a straight-line graph is ๐ฆ = ๐๐ฅ + ๐
Slope = ๐ =โ๐ฆ
โ๐ฅ=
๐ฆ2โ๐ฆ1
๐ฅ2โ๐ฅ1
y-intercept = ๐
transform given relationships, including ๐ฆ = ๐๐ฅ๐ and ๐ฆ = ๐ด๐๐ฅ , to straight line form
and hence determine unknown constants by calculating the gradient or intercept of the
transformed graph
Transforming ๐ฆ = ๐๐ฅ๐ and ๐ฆ = ๐ด๐๐ฅ to to straight line form
Add logarithms to both side of the equarion (๐๐, ๐๐ etc)
E.g. for ๐ฆ = ๐๐ฅ๐
ln ๐ฆ = ln ๐๐ฅ๐
ln ๐ฆ = ln ๐ + ln ๐ฅ๐
ln ๐ฆ = ln ๐ + ๐ ln ๐ฅ
Gradient: ๐;
y-intercept: ln ๐
Axis: ln ๐ฆ on vertical, ln ๐ฅ
on horizontal
E.g. for ๐ฆ = ๐ด๐๐ฅ
ln ๐ฆ = ln ๐ด๐๐ฅ
ln ๐ฆ = ln ๐ด + ln ๐๐ฅ
ln ๐ฆ = ln ๐ด + ๐ฅ ln ๐
Gradient: ln ๐;
y-intercept: ln ๐ด
Axis: ln ๐ฆ on vertical, ๐ฅ on
horizontal
determine unknown constants by calculating the gradient or intercept
by substituting x and y value of two point (may be given by the question
or may need to be obtained from a graph)
NOTE: the two point you choose should be as far from each other as
possible (to be precise)
solve questions involving mid-point and length of a line
Midpoint of points (๐ฅ1, ๐ฆ1) and (๐ฅ2, ๐ฆ2) is
๐(๐ฅ1+๐ฅ2
2,๐ฆ1+๐ฆ2
2)
Length of a line between points (๐ฅ1, ๐ฆ1) and (๐ฅ2, ๐ฆ2) is
๐ = โ(๐ฅ2 โ ๐ฅ1)2 + (๐ฆ2 โ ๐ฆ1)2
know and use the condition for two lines to be parallel or perpendicular, including
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finding the equation of perpendicular bisectors
condition for two lines to be parallel
the slope of the two-line equals
condition for two lines to be perpendicular
if one line has slope ๐, then the second line have the slope โ1
๐ (Opposite
reciprocal)
when multiplying the slope of two perpendicular lines together, the result is -1
Finding the equation of perpendicular bisector
1. Find the gradient of the given line by using the 2 points
2. Find the gradient of the perpendicular bisector, ๐
3. Find the midpoint, (๐ฅ1, ๐ฆ1), of the given line (since it the perpendicular
bisector bisects the line)
4. Plug the midpoint, gradient into the equation ๐ฆ โ ๐ฆ1 = ๐(๐ฅ โ ๐ฅ1) or ๐ฆโ๐ฆ1
๐ฅโ๐ฅ1= ๐
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9. Circular measures
solve problems involving the arc length and sector area of a circle, including
knowledge and use of radian measure
Radian and Degree measure of angles
Degree - 1ยฐ is 1
360๐กโ of one revolution of circle
Radian - 1๐ is an angle which results the arc length to equal the radius, if the
angle is at the center of a circle
๐ ๐๐๐๐๐๐๐ = 180ยฐ
Arc length of a circle
๐ in radians: ๐ = ๐๐
๐ in degrees: ๐ =๐
3602๐๐
Sector area of a circle
๐ in radians: ๐ด =1
2๐๐2
๐ in degrees: ๐ด =๐
360๐๐2
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10. Trigonometry
know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent,
secant, cosecant, cotangent)
BASE KNOWLEDGE
Unit circle โ circle with center (0,0) and radius 1
Equation of a circle with center (0,0) and radius r - ๐๐ + ๐๐ = ๐๐
Therefore, the equation of unit circle is ๐๐ +
๐๐ = ๐
Angle measure
Positive for clockwise, negative for
anticlockwise
Six Trigonometric functions
Sine (sin)
Right triangle trigonometry: sin ๐ =๐๐๐
๐ป๐๐
In unit circle: sin ๐ is the y-coordinate
of P
โ1 โค sin ๐ โค 1 in unit circle
Cosine (cos)
Right triangle trigonometry: cos ๐ =๐ด๐ท๐ฝ
๐ป๐๐
In unit circle: cos ๐ is the x-coordinate of P
โ1 โค cos ๐ โค 1 in unit circle
Tangent (tan)
Right triangle trigonometry: ๐ก๐๐ =๐๐๐
๐ด๐ท๐ฝ
tan ๐ =sin๐
cos ๐
Secant (sec)
sec ๐ =1
sin๐
Cosecant (cosec)
cosec ๐ =1
cos๐
Cotangent (cot)
cot ๐ =1
tan๐=cos ๐
sin๐
CAIE IGCSE Additional Mathematics (0606) Revision Notes
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Special angles (use when the question said โfind exact value of sin ๐ฅ , cos ๐ฅ etc.โ)
Angle is multiple of ๐
2 (90 degrees)
Coordinates in unit circle consists of 0 and ยฑ1
Angle is multiple of ๐
4 (45 degrees) except the
๐
2s
Coordinates in unit circle consists of ยฑ1
โ2
Angle is multiple of ๐
6 (30 degrees) except the
๐
2s
Coordinates in unit circle consists of ยฑ1
2 and ยฑ
โ3
2
understand amplitude and periodicity and the relationship between graphs of related
trigonometric functions, e.g. ๐ ๐๐๐ฅ and ๐ ๐๐2๐ฅ
Periodicity
A function is periodic when ๐(๐ฅ + ๐) = ๐(๐ฅ)
Since 2๐ is one revolution, there is no change when adding 2๐ to ๐
sin ๐ = sin(๐ + ๐2๐) and cos ๐ = cos(๐ + ๐2๐)
(๐๐๐ ๐ ๐๐ ๐๐๐๐๐๐๐ ๐๐๐ ๐ โ โค)
tan ๐ = tan(๐ + ๐๐) (๐ ๐๐ ๐๐๐๐๐๐๐ ๐๐๐ ๐ โ โค)
NOTE: When performing inverse trigonometric functions, there could be
multiple answers โ need to use periodicity and check the domain
Amplitude for a graph - ๐๐๐ฅโ๐๐๐
2
draw and use the graphs of
๐ฆ = ๐ ๐ ๐๐ ๐๐ฅ + ๐
๐ฆ = ๐ ๐๐๐ ๐๐ฅ + ๐
๐ฆ = ๐ ๐ก๐๐ ๐๐ฅ + ๐
where ๐ is a positive integer, ๐ is a simple fraction or integer (fractions will have
a denominator of 2, 3, 4, 6 or 8 only), and ๐ is an integer
CAIE IGCSE Additional Mathematics (0606) Revision Notes
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Graph of sine - ๐ฆ = ๐ sin ๐๐ฅ + ๐
Amplitude = ๐
Period = 2๐
๐
Principle axis: ๐ฆ = ๐
Graph of cosine โ ๐ฆ = ๐ cos ๐๐ฅ + ๐
Amplitude = ๐
Period = 2๐
๐
Principle axis: ๐ฆ = ๐
Graph of tangent - ๐ฆ = ๐ tan ๐๐ฅ + ๐
Amplitude undefined โ do not affect the graph
Period = ๐
๐
Principle axis: ๐ฆ = ๐
use the relationships
sin2 ๐ด + cos2 ๐ด = 1
sec2 ๐ด = 1 + tan2 ๐ด, cosec2 ๐ด = 1 + cot2 ๐ด
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 24
tan ๐ด =sin๐ด
cos ๐ด, cot ๐ด =
cos ๐ด
sin๐ด
solve simple trigonometric equations involving the six trigonometric functions and the
above relationships (not including general solution of trigonometric equations)
prove simple trigonometric identities
Trigonometric relations and identities โ used to prove other identities, solve
equations
sin2 ๐ด + cos2 ๐ด = 1 (given in formula sheet)
sec2 ๐ด = 1 + tan2 ๐ด (given in formula sheet)
cosec2 ๐ด = 1 + cot2 ๐ด (given in formula sheet)
tan ๐ =sin๐
cos ๐
cot ๐ =cos๐
sin๐
Solving Trigonometric equations
1. When there is multiple trigonometric functions (e.g. having both sin ๐ and
cos ๐), simplify until only have one trigonometric functions (by e.g. dividing
each side of the equal sign by cos ๐ so the sin ๐ is turned into tan ๐, and
cos ๐ will become 1)
2. If there is reciprocal trigonometric functions (cot ๐ , sec ๐ , cosec ๐) turn them
into tan ๐ , cos ๐ , or sin ๐ (so that calculator can calculate)
3. Move the trigonometric function to LHS and all other numbers (including the
coefficient of the trigonometric function) to RHS by moving terms, dividing
both side by a number etc.
4. Apply the inverse trigonometric functions (e.g. cosโ1 ๐)
NOTE: be aware of the mode (radian/degrees)
Do NOT move the term that is previously inside the trig function at LHS to RHS
at this moment
5. Find the other possible angle in the unit circle by:
6. Apply periodicity to find all possible angles within the domain
(adding/subtracting answer by 2๐โs for sin and cos, and ๐โs for tan)
7. Move the terms (except the unknown) at LHR to RHS
8. Check to see if the answer is in the domain again
Solving quadratic trigonometric equations
CAIE IGCSE Additional Mathematics (0606) Revision Notes
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See the trig function as a whole (e.g. letting cos ๐ to be ๐) and solve
Proving simple identities (e.g. prove thatโฆ./ show that ), often needs:
The trigonometric identities and relations above
Turning sec ๐ into 1
cos ๐, cosec ๐ into
1
sin๐
Multiplying sin ๐ and cos ๐ to both the denominator and numerator (to turn
cot ๐ into cos ๐ and tan ๐ into sin ๐)
Combining fractions by making a common denominator
Splitting fraction into 2
CAIE IGCSE Additional Mathematics (0606) Revision Notes
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11. Permutation & combination
BASE KNOWLEDGE
If there are ๐ ways of performing an operation, and for each of the way there
are also ๐ ways of performing a second independent operation, then there are
๐๐ number of ways of performing the 2 operation in succession
โandโ/ โorโ
โandโ means multiplying the possibility
โorโ means adding the possibility
recognise and distinguish between a permutation case and a combination case
distinguishing between permutation and combination case
permutation โ arrangement with a definite order
combination โ selection of objects without regard to order
know and use the notation ๐! (with 0! = 1), and the expressions for permutations
and combinations of ๐ items taken ๐ at a time
Factorial notation
Notation ๐!
๐! Is the product of the first n positive integers
๐! = ๐(๐ โ 1)(๐ โ 2)(๐ โ 3)โฆร 3 ร 2 ร 1 (๐๐๐ ๐ โฅ 1)
0! = 1
Expression for permutations and combinations of ๐ items taken ๐ at a time
Permutations
๐๐ ๐ or ๐๐
๐ = ๐!
(๐โ๐)!
Combinations
๐ถ๐ ๐ or ๐ถ๐
๐ or (๐๐) =
๐!
๐!(๐โ๐)!(given in formula sheet)
answer simple problems on arrangement and selection (cases with repetition of objects,
or with objects arranged in a circle, or involving both permutations and combinations,
are excluded)
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 27
Permutation selection problems
Choosing passwords from characters (digit/numbers, letters and symbols)
A ๐ค-character password is to be chosen from the following ๐ฆ characters
Letter: โฆโฆโฆ.
Digit: โฆโฆโฆ.
Symbol: โฆโฆโฆ.
(when each character may only be used once)
No restriction: ๐ท ๐
๐
Contain ๐ letter, ๐ digit, ๐ symbol, in this order
๐ท ๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐
๐ ร ๐ท ๐๐๐๐๐๐๐ ๐๐ ๐ ๐๐๐๐ ๐๐๐๐๐๐๐๐๐
๐ ร
๐ท ๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐
๐
Start with an [letter/number/symbol] and end with an [letter/number/symbol]
๐๐๐๐๐๐๐ ๐๐ [๐ฅ๐๐ญ๐ญ๐๐ซ/๐ง๐ฎ๐ฆ๐๐๐ซ/๐ฌ๐ฒ๐ฆ๐๐จ๐ฅ] ๐๐ฏ๐๐ข๐ฅ๐ข๐๐ฅ๐ ร ๐ท ๐โ๐
๐โ๐ ร
๐๐๐๐๐๐๐ ๐๐ [๐ฅ๐๐ญ๐ญ๐๐ซ/๐ง๐ฎ๐ฆ๐๐๐ซ/๐ฌ๐ฒ๐ฆ๐๐จ๐ฅ] ๐๐ฏ๐๐ข๐ฅ๐ข๐๐ฅ๐
Contain at least ๐ [letter/number/symbol]
METHOD ONE: total number of cases minus the number of cases where
there are no [letter/number/symbol] included
๐ท ๐
๐ โ ๐ท (๐โ๐๐๐๐๐๐๐ ๐๐ ๐ฅ๐๐ญ๐ญ๐๐ซ/๐ง๐ฎ๐ฆ๐๐๐ซ/๐ฌ๐ฒ๐ฆ๐๐จ๐ฅ ๐๐๐๐๐๐๐๐๐)
๐
METHOD TWO: Adding the cases of having (1, 2, โฆ, numbers of
[letter/number/symbol] available)character together
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 28
Combination selection problems
Choosing men and women to form a group (e.g. a committee, team)
A team/committee of ๐ is to be chosen from ๐ people (with ๐ boys/men and ๐
girls/women).
No restriction: ๐ช ๐
๐
Have ๐ boys/mens and ๐ girls/womens: ๐ช ๐๐ ร ๐ช
๐๐
Have at least ๐ [boys/girls]
METHOD ONE: total number of cases minus the number of cases where
CAIE IGCSE Additional Mathematics (0606) Revision Notes
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there are no ๐ [boys/girls] included
METHOD TWO: Adding the cases of having different combination of
numbers of girls and boys that satisfies the condition (e.g. 1B2G, 2B1G, 3B
if must have at least 1 boy)
A (specific person) or B, but not both, must be included
total number of cases minus the number of cases where both of the person
is selected/not selected
When two people cannot be separated โ see them as one person
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 30
Choosing questions in exams
An exam consists of section A (containing ๐ question), and section B (containing ๐
questions)
Find the numbers of possible questions selection that can be made
Have to choose ๐ question in section A and ๐ question in section B
๐ช ๐๐ ร ๐ช
๐๐
Must answer ๐ specific questions in Section A and ๐ specific question in section
B AND Have to choose ๐ question in section A and ๐ question in section B
๐ช ๐โ๐
๐โ๐ ร ๐ช ๐โ๐
๐โ๐
Arrangements problems
Forming numbers form digits
๐ digit numbers can be formed using ๐ digit listed (no zero)
How many numbers can be formed?
No restriction: ๐ท ๐
๐
The number formed must be [even/odd]
๐ต๐๐๐๐๐๐ ๐๐ ๐ ๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐ ๐๐๐ ๐๐๐๐ ร ๐ท ๐โ๐
๐โ๐
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 31
Arranging books on shelfs OR Arranging people in seats
๐ different books are to be arranged on a shelf. There are ๐ math books and ๐
history books. Find the number of arrangements of books
[Math/History] books need to be together
[Math/History] books cannot be together
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 32
12. Series
use the Binomial Theorem for expansion of (๐ + ๐)๐ for positive integer ๐
Binomial theorem
(๐ + ๐)๐ = ๐๐ + (๐1)๐๐โ1๐ + โฏ+ (๐
๐)๐๐โ๐๐๐ +โฏ+ ๐๐ for ๐ =
0,1,2,โฆ , ๐ (given in formula sheet)
Sigma form:
โ(๐
๐)๐๐โ๐๐๐
๐
๐=0
use the general term (๐๐)๐๐โ๐๐๐ , 0 โค ๐ โค ๐ (knowledge of the greatest term and
properties of the coefficients is not required)
General term, (๐ + 1)๐กโ term, in binomial expansion: (๐๐)๐๐โ๐๐๐
NOTE: when finding a specific term, minus the term number by one and then use
the formula
recognise arithmetic and geometric progressions
use the formulae for the ๐th term and for the sum of the first ๐ terms to solve
problems involving arithmetic or geometric progressions
use the condition for the convergence of a geometric progression, and the formula for
the sum to infinity of a convergent geometric progression
Arithmetic progressions
Have a constant difference (common difference, ๐ ) between two consecutive
number in a sequence
๐th term, ๐ข๐
๐ข๐ = ๐ + (๐ โ 1)๐
๐ is the first term, ๐ is the common difference
Sum of the first ๐ terms, ๐๐
๐๐ =1
2๐(๐ + ๐)
๐๐ =1
2๐[2๐ + (๐ โ 1)๐]
๐ is the first term, ๐ is the last term, ๐ is the common difference
Geometric progressions
Have a constant ratio (common ratio, ๐) between two consecutive number in a
sequence
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 33
๐th term, ๐ข๐
๐ข๐ = ๐๐๐โ1
๐ is the first term, ๐ is the common ratio
Sum of the first ๐ terms, ๐๐
๐๐ =๐(1โ๐๐)
1โ๐ (๐๐๐ ๐ โ 1)
๐ is the first term, ๐ is the common ratio
Sum to infinity, ๐โ , of a convergent geometric progression
convergence of a geometric progression
when the sum trends to a finite value
๐โ =๐
1โ๐ (๐๐๐ |๐| < 1)
๐ is the first term, ๐ is the common ratio
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 34
13. Vectors in 2 dimensions
use vectors in any form, e.g. (๐๐), ๐ด๐ตโโโโ โ, ๐, ๐๐ โ ๐๐
Forms of vector
๐ด๐ตโโโโ โ
๐ or ๐
Component form - (๐๐)
Unit vector form โ ๐๐ โ ๐๐
๐ = (10) and ๐ = (
01)
know and use position vectors and unit vectors
Position vector
Position vector of B relative to A is ๐ด๐ตโโโโ โ
๐ด๐ตโโโโ โ = ๐๐ตโโ โโ โ โ ๐๐ดโโ โโ โ
Point ๐(๐ฅ, ๐ฆ) has position vector ๐๐โโ โโ โ ๐๐ (๐ฅ๐ฆ) ๐๐ ๐ฅ๐ + ๐ฆ๐
Unit vector
Any vector which has a length of one unit
Base unit vectors in the x or y positive direction
๐ = (10)
๐ = (01)
Unit vector in the direction ๐ is ๐
|๐|
A vector of length ๐ in direction ๐ is ๐๐
|๐|
A vector of length ๐ parallel to ๐ is ยฑ๐๐
|๐|
find the magnitude of a vector;
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 35
Magnitude of a vector
Notation: |๐ด๐ตโโโโ โ| or |๐|
The length of a vector
If ๐ = (๐๐ฅ๐๐ฆ) = ๐ฅ๐ + ๐ฆ๐, then its magnitude is โ๐๐ฅ2 + ๐๐ฆ2
add and subtract vectors and multiply vectors by scalars
Adding vectors
If ๐ = (๐๐ฅ๐๐ฆ) and ๐ = (
๐๐ฅ๐๐ฆ), then ๐ + ๐ = (
๐๐ฅ + ๐๐ฅ๐๐ฆ + ๐๐ฆ
),
Subtracting vectors
If ๐ = (๐๐ฅ๐๐ฆ) and ๐ = (
๐๐ฅ๐๐ฆ), then ๐ โ ๐ = (
๐๐ฅ โ ๐๐ฅ๐๐ฆ โ ๐๐ฆ
),
Multiplying vectors by scalars
Scalar โ a non-vector quantity โ only has magnitude but do not have direction
If ๐ Is and scalar and ๐ = (๐ฃ๐ฅ๐ฃ๐ฆ), then ๐๐ = (
๐๐ฃ๐ฅ๐๐ฃ๐ฆ)
Parallelism โ two non-zero vector are parallel to each other if one is a scalar
multiple of another
If ๐ = ๐๐, then ๐//๐ and |๐| = |๐||๐| (for ๐ is an scalar, and ๐, ๐
is non-zero vector)
Solving equations involving component form
(๐ฅ๐ฆ) = (
๐1 โ ๐1๐2 โ ๐2
) can form the simultaneous equation {๐ฅ = ๐1 โ ๐1๐ฆ = ๐2 โ ๐2
compose and resolve velocities
Constant velocity questions
If an object initially has position vector ๐, and move with a constant velocity
๐ (speed is |๐|), its position vectorx after time t is ๐ = ๐ + ๐ก๐
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 36
14. Differentiation & Integration
Differentiation
understand the idea of a derived function, use the notations ๐โฒ(๐ฅ), ๐โฒโฒ(๐ฅ), ๐๐ฆ
๐๐ฅ, ๐2๐ฆ
๐๐ฅ2,
(= ๐
๐๐ฅ(๐๐ฆ
๐๐ฅ))
derived function โ gradient function
Notation: ๐โฒ(๐ฅ) or ๐๐ฆ
๐๐ฅ
use the derivatives of the standard functions ๐ฅ๐(for any rational ๐), sin ๐ฅ, tan ๐ฅ,
cos ๐ฅ, ๐๐ฅ, ln ๐ฅ, together with constant multiples, sums and composite functions of these
differentiate products and quotients of functions
Simple Rules of differentiation
FUNCTION DERIVITIVE
๐ (a constant) 0
๐๐ (for any rational ๐) ๐๐ฅ๐โ1
๐ฌ๐ข๐ง๐ cos ๐ฅ
๐ญ๐๐ง ๐ sec2 ๐ฅ
๐๐จ๐ฌ ๐ โsin๐
๐๐ ๐๐ฅ
๐ฅ๐ง ๐ 1
๐ฅ
Deriving sums of functions: ๐ข(๐ฅ) + ๐ฃ(๐ฅ) ๐๐๐๐๐๐๐๐๐๐๐๐ก๐โ ๐ขโฒ(๐ฅ) + ๐ฃโฒ(๐ฅ)
Deriving functions with constant multiples: ๐๐ข(๐ฅ) ๐๐๐๐๐๐๐๐๐๐๐๐ก๐โ ๐๐ขโฒ(๐ฅ)
Chain rule (use when differentiating composite functions) :
๐ฆ = ๐(๐ข(๐ฅ)) ๐๐๐๐๐๐๐๐๐๐๐๐ก๐โ ๐โฒ(๐ข(๐ฅ)) ร ๐ขโฒ(๐ฅ)
Product rule: ๐ข(๐ฅ)๐ฃ(๐ฅ) ๐๐๐๐๐๐๐๐๐๐๐๐ก๐โ ๐ขโฒ(๐ฅ)๐ฃ(๐ฅ) + ๐ฃโฒ(๐ฅ)๐ข(๐ฅ)
Quotient rule: ๐ข(๐ฅ)
๐ฃ(๐ฅ) ๐๐๐๐๐๐๐๐๐๐๐๐ก๐โ
๐ขโฒ(๐ฅ)๐ฃ(๐ฅ)โ๐ฃโฒ(๐ฅ)๐ข(๐ฅ)
[๐ฃ(๐ฅ)]2
apply differentiation to gradients, tangents and normals, stationary points, connected
rates of change, small increments and approximations and practical maxima and
minima problems
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 37
use the first and second derivative tests to discriminate between maxima and minima
Second derivative
Notation: ๐โฒโฒ(๐ฅ) or ๐2๐ฆ
๐๐ฅ2
Derivative of ๐โฒ(๐ฅ) is ๐โฒโฒ(๐ฅ) (= ๐
๐๐ฅ(๐๐ฆ
๐๐ฅ))
Application of differentiation
Gradients - The value of ๐โฒ(๐) is the gradient of the tangent to ๐ฆ = ๐(๐ฅ)
at the point ๐
tangents and normal
Equation of tangent to ๐(๐ฅ) at ๐ฅ =
๐ is: ๐ฆ โ ๐(๐) = ๐โฒ(๐)(๐ฅ โ ๐)
Equation of normal to ๐(๐ฅ) at ๐ฅ = ๐ is:
๐ฆ โ ๐(๐) = โ1
๐โฒ(๐)(๐ฅ โ ๐)
stationary points (where gradient is 0) - ๐(๐ฅ) has stationary point at x-
coordinate of ๐โฒ(๐ฅ) = 0
Determining whether a stationary point is a maxima or minima
By drawing sign diagrams
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 38
By using second derivative (For a function with stationary point at ๐ฅ = ๐)
๐โฒโฒ(๐ฅ) > 0: minima
๐โฒโฒ(๐ฅ) < 0: maxima
NOTE: This method does NOT work when ๐โฒโฒ(๐ฅ) = 0
connected rates of change
TIP: ๐๐ฆ
๐๐ฅ gives the rate of change of ๐ฆ with respect to ๐ฅ
Draw a diagram that clearly shows the situation (if not given)
Label the numbers (+ distinguish between the constant and the changing
quantity)
Write an equation that connects the variables (e.g. using Pythagoreans,
trig, similar triangles)
Differentiate the equation with respect to ๐ก
NOTE: the changing quantity is a function โ need to apply chain rule
Plug into the number of the particular case given
NOTE: do NOT plug the numbers of the particular case before you have
get a generalized differential equation
(Example on next page)
Give the answer in a sentence
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 39
small increments and approximations
The small change in y: ฮดy small change in x: ฮดx
๐ฟ๐ฆ
๐ฟ๐ฅโ๐๐ฆ
๐๐ฅ (if the increment is small)
Finding the small change of y when x increases by small amount
๐ฟ๐ฆ
๐ฟ๐ฅโ๐๐ฆ
๐๐ฅ
๐ฟ๐ฆ = ๐ฟ๐ฅ ร๐๐ฆ
๐๐ฅ
Finding the small change of x when y increases by small amount
๐ฟ๐ฆ
๐ฟ๐ฅโ๐๐ฆ
๐๐ฅ
๐ฟ๐ฅ = ๐ฟ๐ฆ ร๐๐ฅ
๐๐ฆ
๐ฟ๐ฅ = ๐ฟ๐ฆ รท๐๐ฆ
๐๐ฅ
practical maxima and minima problems
1. Draw a diagram (if not given) + Label
2. Write an equation that connects the variables
3. Find the stationary points
4. Determine the nature of the stationary points and identify the asked point
according to the question (maxima/minima)
5. Give the answer in a sentence
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 40
Integration
BASE KNOWLEDGE
Fโ(x)=f(x)
understand integration as the reverse process of differentiation
Integration - the reverse process of differentiation
If Fโ(x)=f(x), then โซ๐(๐ฅ)๐๐ฅ = ๐น(๐ฅ)
integrate sums of terms in powers of ๐ฅ including 1
๐ฅ,
1
๐๐ฅ+๐
Simple rules of integration
FUNCTION INTEGRAL
๐ (a constant) ๐๐ฅ + ๐
๐๐ (for ๐ โ โ๐) ๐ฅ๐+1
๐ + 1+ ๐
๐ฌ๐ข๐ง๐ โcos ๐ฅ + ๐
๐๐จ๐ฌ ๐ sin ๐ฅ + ๐
๐๐ ๐๐ฅ + ๐
Integrating sums of functions: โซ[๐(๐ฅ) + ๐(๐ฅ)]๐๐ฅ = โซ๐(๐ฅ)๐๐ฅ + โซ๐(๐ฅ)๐๐ฅ
Integrating functions with constant multiples: โซ๐๐(๐ฅ)๐๐ฅ = ๐ โซ๐(๐ฅ)๐๐ฅ
integrate functions of the form (๐๐ฅ + ๐)๐(for any rational ๐), sin(๐๐ฅ + ๐),
tan(๐๐ฅ + ๐), cos(๐๐ฅ + ๐), ๐๐๐ฅ+๐
Integrating ๐(๐๐ฅ + ๐): โซ[๐(๐๐ฅ + ๐)]๐๐ฅ =1
๐๐น(๐๐ฅ + ๐) + ๐
evaluate definite integrals and apply integration to the evaluation of plane areas
Definite integrals
Definite integral of ๐(๐ฅ) (where ๐(๐ฅ) is continuous) in the interval ๐ โค ๐ฅ โค
๐ is โซ ๐(๐ฅ)๐
๐๐๐ฅ = [๐น(๐ฅ)]๐
๐ = ๐น(๐) โ ๐น(๐)
Rules of definite integrals
โซ ๐(๐ฅ)๐
๐๐๐ฅ = 0
โซ ๐(๐ฅ)๐
๐๐๐ฅ = โโซ ๐(๐ฅ)
๐
๐๐๐ฅ
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 41
โซ ๐๐(๐ฅ)๐
๐๐๐ฅ = ๐ โซ ๐(๐ฅ)
๐
๐๐๐ฅ
โซ ๐(๐ฅ)๐
๐๐๐ฅ + โซ ๐(๐ฅ)
๐
๐๐๐ฅ = โซ ๐(๐ฅ)
๐
๐๐๐ฅ
โซ [๐(๐ฅ) + ๐(๐ฅ)]๐
๐๐๐ฅ = โซ ๐(๐ฅ)
๐
๐๐๐ฅ + โซ ๐(๐ฅ)
๐
๐๐๐ฅ
Evaluation of plane areas using definite integrals
If 2 functions ๐(๐ฅ) and ๐(๐ฅ), intersecting at ๐ฅ = ๐ and ๐ฅ = ๐, and
๐(๐ฅ) > ๐(๐ฅ) for all ๐ โค ๐ฅ โค ๐, then the area of the region between th
intersection is ๐ด = โซ [๐(๐ฅ) โ ๐(๐ฅ)]๐๐ฅ๐
๐
TIP: x-axisโ equation is ๐ฆ = 0
apply differentiation and integration to kinematics problems that involve displacement,
velocity and acceleration of a particle moving in a straight line with variable or
constant acceleration, and the use of ๐ฅโ ๐ก and ๐ฃโ ๐ก graphs
BASE KNOWLEDGE
Displacement & distance
Displacement โ the straight-line distance relative to the initial position
Interpreting displacement of a Particle P
moving in a straight line
Velocity and speed
Velocity โ vector quantity (both magnitude and direction)
Average ๐ฝ๐๐๐๐๐๐๐ =๐ ๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐
๐๐๐๐ ๐๐๐๐๐
Interpreting velocity of a Particle P moving
in a straight line
Speed โ scalar quantity (only magnitude)
Has unit ๐๐โ๐
Acceleration
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 42
Average ๐จ๐๐๐๐๐๐๐๐๐๐๐ =๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐
๐๐๐๐ ๐๐๐๐๐
Interpreting acceleration of a Particle P moving in a straight line
Has unit ๐๐โ๐
Application of differentiation and integration to kinematics problems (involving
displacement, velocity and acceleration of a particle moving in a straight line with
variable or constant acceleration)
Obtaining displacement function from velocity function and velocity function
from acceleration function
Integrate
๐ can be found by plugging the value given
Finding the displacement in a time interval ๐ก1 โค ๐ก โค ๐ก2
๐ท๐๐ ๐๐๐๐๐๐๐๐๐ก = ๐ (๐ก2) โ ๐ (๐ก1) = โซ ๐ฃ(๐ก)๐ก2๐ก1
Finding the distance traveled in a time interval ๐ก1 โค ๐ก โค ๐ก2
1. Find ๐ฃ(๐ก)
2. Find the values when ๐ฃ(๐ก) = 0 and draw the sign diagram to find, if
present, the time when the particle changes direction
3. Find ๐ (๐ก) (the constant of integration, c, may be included)
4. Find ๐ (๐ก2), ๐ (๐ก1) and ๐ (๐ก) for each time the direction changes
5. Draw a motion diagram
6. Identify + calculate the total distance travled
Finding the specific value of velocity/acceleration
1. Find the function (๐ฃ(๐ก) or ๐(๐ก))
2. Plug in the specific time value
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 43
3. Solve for ๐ฃ(๐ก) or ๐(๐ก)
Finding the time when displacement relative to initial
position/velocity/acceleration equals to a specific value
1. Find the function (๐ฃ(๐ก), ๐(๐ก) or ๐ (๐ก))
2. Plug in the specific initial position/velocity/acceleration value
3. Solve for ๐ก
TIP: โStationaryโ means ๐ฃ(๐ก) = 0; โAt the origin"/โinitial positionโ means
๐ (๐ก) = 0
Finding the time when the particle reverses/changes direction
1. Find ๐ฃ(๐ก)
2. Find value of ๐ก when ๐ฃ(๐ก) = 0
3. Draw a sign diagram โ maxima/minima point is where the particle
reverses direction
Finding the time when the particle is at its maxima/minima displacement
1. Find ๐ฃ(๐ก)
2. Find value of ๐ก when ๐ฃ(๐ก) = 0 and draw a sign diagram to find the
maxima/minima
Finding the time when the particle is at its maxima/minima velocity
1. Find ๐(๐ก)
2. Find value of ๐ก when ๐(๐ก) = 0 and draw a sign diagram to find the
maxima/minima
๐ฅโ ๐ก (๐ -๐ก) and ๐ฃโ ๐ก graphs
displacement-time graph (๐ฅโ ๐ก or ๐ -๐ก graph)
slope=velocity
velocity-time graph (๐ฃโ ๐ก graph)
area under the line=distance traveled
slope=acceleration
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 44
Appendix A: Formula Sheet
Appendix B: Operations on graphs
๐(โ๐ฅ): reflection in the ๐ฆ-axis
โ๐(๐ฅ): reflection in the ๐ฅ-axis
๐(๐ฅ)+๐: translation of ๐ units
parallel to ๐ฆ-axis
๐(๐ฅ + ๐): translation of โ ๐ units
parallel to ๐ฅ-axis
๐(๐๐ฅ): stretch, scale factor 1/๐
parallel to ๐ฅ-axis
๐๐(๐ฅ): stretch, scale factor ๐ parallel
to ๐ฆ-axis
1/ ๐(๐ฅ): every point become (1/x,
1/y)
CAIE IGCSE Additional Mathematics (0606) Revision Notes
By Steven Zhou 45
Acknowledgement
This work is licensed under a Creative Commons Attribution-
NonCommercial 4.0 International License.
Some Information and diagrams come from:
YK Pao schoolโs Teacherโs resources (worksheets, PowerPoints etc.)
Cambridge Assessment International Education (CAIE) International General
Certificate of Secondary Education (IGCSE) Additional Mathematics (0606)
Syllabus for examinations from 2020
CAIE IGCSE Additional Mathematics Past papers & corresponding mark schemes
(2013-2016)
Haese Mathematicsโ Textbook