Maths Skills
Short Description
My tips for A level H2 Maths questions. Maths is all about structure, while I try to list as many possible ways to tackl...
Description
Mathematics skills Disclaimer: These notes are based on my opini on on how the questions should be tackled. Do take it with a pinch of salt, and practice to see how they are appli ed.
Miscellaneous •
When the answer you need to show is an integer multiple of a o o
•
=3 ,
To show that o o
•
∈ ℤ ℎ ℎ ∈ ℤ + ∈ ℤ+0 ∈ℤ ∪ 3 +1 −1 −2 1 3 +1 −1 −2 ⋯ 1
is a positive integer as well as 0
{0}
To show continued multiplication, division, addition or subtraction, use … o
o
…
+
+
+
Note the ‘ +’ flanking the …
•
Add the units only to your final answer
•
Don’t forget to add
to the end of a radian value
APGP •
If the question points to a trend being AP/GP but did not explicitly say so, define it o
•
Also define o o
•
____ follows an arithmetic/ geometric progression progression with first term , common ratio Sub in
,
/
or
is the
and
if known
is used
term of the arithmetic/ geometric progression
is the sum of the first
terms of the arithmetic/ geometric progression
When converting AP to GP and vice versa, convert either the common ratio or difference into the other, and solve to get the common ratio/ difference
•
When converting AP to GP and vice versa, take note of when the progression starts, so as to not double count when adding to past results. (See MSM qn 4ii)
•
When finding sum of a non AP GP sequence, split the sequence into its AP GP components, and add them up.
•
When showing the GP is convergent, need to show that | | < 1, not just
•
When dealing with percentages o
0
0.5 0.5
, you need to find a trend in the coefficient of the , usually by
referring to earlier workings to see how the coefficient is derived •
1
When asked to find in ascending powers of ,use the descending powers of method o
−1 = −1 (1 3 + −12 −2 32 + ⋯ )
−1 −1 3
(3 + )
=
(
( + 1)
)(
)
!
Mathematical induction •
When making conjecture, try to include the term number in the result obtained
2 = 4, where in this case, the nominator is the term number. 3 6 To prove by mathematical induction Let be the statement of ‘ = ’ (of given equation), ∈ ℤ+ Prove 1 is true When = 1 1 = … 1 = … = 1 + Assume is true for some k, ∈ ℤ Prove is true +1 is true o
•
For example,
4
=
o o
o o
+1 +1 +1 1 + +1 ∈ℤ ≥ When
=
+1 = … = … = Use equation
o
Since
is true
is true,
,
,
is true, by mathematical induction,
1
is true for all
Recurrence relation •
When finding the limit of ‘
+1 → ∞ +1 → → ’
=??
o
Say ‘IF the sequence converges, as
o
You get the equation ‘ = ? ? ’
,
•
Solve from here to get the exact answer
To show that they have similar roots as the graph, make it like the equation
+1 +1 → ∞ +1 → → +1 > ,if
From ‘
o
=?? Have a statement saying ‘The sequence converges to …’
When proving o
’
given, then say ‘Given that the roots of
o
,
From
(or the other way round
>
’ (the given equation)
= ??
‘As
,
= ?? > Convert
are …’
,
to
From
’
by changing both sides
>
Done by Nickolas Teo Jia Ming
•
+1 +1 +1 +1
To ? ?
To
> ?? >
When proving o
(or the other way around) < Write “Considering ”
The
on the left side is the same as the
you want to prove
on the left side of the equation
o
Replace
in terms of
o
Make it look like the graph equation (or the other equation given to you, else draw one)
It can look similar in the sense that the equation obtain is a multiple or fraction of the graph equation
Because when y is scaled, the intercept does not change
If no graph is given, and the equation of the ideal graph is not known, draw
+1 +1 +1 +1
o
(where is whatever the equation given is) = when or < The x-intercept will be the value of > Write “From the graph of = ? ? , when (not ) … (the condition given)
•
•
o
< < < > Etc. The graph is > 0 or < 0”
o
Write “Thus given
o
Write “Thus
•
…,
<
or
>
0”
To show strictly increasing or decreasing o
Show
o
Refer to above
or
>
<
When needing to eliminate a possible answer, look at the stem of the question o
•
+1 +1 +1 +1 +1 +1
Find any restrictions, like only positive numbers
Important points for a sketch o
Axial intercepts
o
Turning points
o
Asymptotes
When describing the trend of recurrence relations o
Strictly increasing/ decreasing
o
Converge to ___
Equations •
Formulate equations based on info o
•
Number of equations should equal number of vari ables
Parametric equation o
To find equation of tangent
Find gradient
= × Done by Nickolas Teo Jia Ming
Sub in the value for
Use known point to craft equation
=
(
)
is the given -coordinate,
∝
o
is the given -coordinate
To find where the tangent/ normal cuts the graph again
Sub in the parametric equation into equation of tangent/ normal
Inequalities •
When solving a fraction inequality, o
Don’t try to bring the denominator up , instead bring the other side’s si de’s equation over, and add it to the fraction
o
Use the number line method
First factorize the numerator and denominator, until you can get roots
Sketch the graph using the roots
•
•
+
has not roots,
2 ⋯
has roots
Use the method taught in graphing techniques, ∝
Graph pass though, or bounce off
∝
Graph is negative or positive when
is large
Sometimes, to change an equation into another, you need to sub in other values o
•
2 ⋯
Like
1 1
− √ −4 2 If trying to use GC to solve a polynomial, make sure the largest power is positive Convert – x 3 + x 2 … to x 3 x 2 … If cannot use number line, use
=
±
o
•
When the inequality involves a trig onometry function, account for both ‘li mbs’/ ‘sides’ of the graph (e.g. sin
=
0.253
≥
2.89
o
Use the graph to find the two points
o
Also include the lower/ upper limit if the range of y is restricted (e.g.
o
≤ ≤
0.253
≤ ≤ 2.89
)
> 1)
When an inequality involves a constant to the power of (e.g. 5 ), and a multiple of (e.g.
2 ) •
≤ ≤ →
Reject answers that are out of the amplitude of the trigonometry curve (e.g.
sin •
≤ 14 ≤
o
Solve by drawing graph on GC
When doing a compound interest question, separate t he interest rate and use sum of GP to solve o
2 ⋯ + 4000 4000(1.05) = 4000 4000(1 + 1.05 + 1 1 05 −1 ) 1.052 + ⋯ + 1.05 1.05 ) = 4000( 000( 1 05−1 4000 4000 + 4000 4000(1.05) + 4000 4000(1.05) + ( .
)
.
Graph transformation •
Statements
Done by Nickolas Teo Jia Ming
Translation of ___ (number) units in the direction of the ___ (positive or negative)
o
___ ( or )-axis
o
Scaling of factor ___ (number) parallel to the ___ ( or )-axis
o
Reflection in the ___ ( or )-axis
Graph sketching •
•
For ’( )
Turning points become -intercept
o
Keep vertical asymptotes
o
Horizontal asymptotes become
o
o
Point of inflection becomes turning point
o
For easier drawing, mark out the graph at the ends first
1 For
Ends at asymptotes
Turning point
When
is large
o
Value of horizontal asymptote become inversed
-intercept becomes vertical asymptotes
Oblique asymptotes no longer relevant Small value becomes big, big value becomes small
For | ( )| All negative values are reflected refl ected about the x-axis
For (| |)
o
All positive -values are reflected about the y-axis
o
All negative -values no longer relevant
To state the changes made to graph, o o
Step 1: , Step 2: ___ followed by ___
To find if the graph has roots when it has a turning point o
Use the determinants on the equation
•
Vertical asymptotes become x-intercept with open circle
o
•
o
o
•
( )
o
•
=0 Oblique asymptote become horizontal line, = is the gradient of the oblique asymptote
o
•
o
2 ≥ 4
0
If there are points on the graph labeled A, B etc. o
You need to show the points on the new graph
Functions, inverse and composite •
To show that a turning point exists
≥ 2 ≥
o
Find
o
Determinants of ( )
•
( )
4
0
0
To show composite exists or not
Done by Nickolas Teo Jia Ming
⊆ 13 12 14 −1 o
• •
Use the
( )=
( )=
( )
To find range of composite function gf o
Sketch graph of g with domain
o •
or the strike out version of it (i.e. not subset)
=
Use GC if equation is known
When trying to find the equation of the inverse function, and there is a quadratic function o
Use complete the square over determinants, as it is better ( for seeing which sign to reject)
o •
2 ∴ 2 ∴ ∈ℝ ≥ ℤ ℤ+ ℤ− ≠ −1 E.g.
2 +
To show o
•
(
1) +
1=0
=1±
For answering in set notation { : } , o
Note that in certain cases,
•
=0
or
1
should be used instead.
( )
( ) Show that for the range of ( ) and ( ) is 1-1
To find ( ) from o
,
Use
( ) ( )
Permutation and Combination •
Unless the question explicitly states that the minimum number of ___ (object) each person can have is ___ (number), work on the assumption that they can have 0 ___(object)
•
•
When two elements are to be separated o
Use the slotting-in method
o
Use the 1 – together method
When any element can be repeated any number of times o
•
, taking into account restrictions
Slot in method
Take the spots next to the element elem ent away,
Do the table permutation (i.e. (n-1)!),
Slot in possible choices at the spots removed (i.e.
Remember to permutate the choices. (i.e. They swap spots)
2)
To combat double count o
•
)
When doing circular permutation, and one elements cannot have another element next to it, o
•
Use (
Divide by the permutation
When separating 10 objects into 3 groups of 3,3,4, o
Divide by 2 because the first two groups of 3 will have double count
o
For example, ABC and DEF go into identical white pouches
If they are put into unique locations, like 3 friends, permutate them as if they are unique, as the double count has been accoun ted for!
o
If there is a restriction, where you have to force choose the contents of a group, like all 3 red objects in a group, apply the double count correction, but permutate the choice you have where you exert your power to distribute the unique white pouches
Done by Nickolas Teo Jia Ming
Probability
∩ ∩
•
When showing that two event (
•
( ) or (| | |) = ( ) If not, they are not independent ) = ( ). ( ) If they are independent, ( If the event (or combined events) take up all possibilities (e.g ( ) = 1), they are o
and
) are independent or not
Check to see if (| | |) =
considered exhaustive
Vectors •
If the question says ‘produced’, the new point i s extended out of the line, l ine, and cannot be within it
•
To find area of triangle o
1 2 × The arrow above OB is removed as you multiply by length of base 1 2 sin
o •
To find point of intersection between lines o
•
Base × Height
Sub the vector equation of one line into the other
To find line of intersection between planes o
From parametric equations
Sub the parametric equation of one plane into the other
Use GC to solve the 4 unknowns using 3 equations, in terms of the 4fourth variable
o
o
•
Constant represents the position vector
Coefficient of fourth variable is the direction vector
From cartesian or scalar product equations
Convert into Cartesian equation
Use GC to solve to get the answer in terms of the third variable (
Constant represents the position vector
Coefficient of third variable is the direction vector
3
)
Manually from scalar product equation
Get direction vector of line from cross product of normal
Get position vector of line
Fix one coordinate variable (e.g. )
Get two equations by subbing the vector (e.g. ( )) into the scalar 0 product form of the two planes
Solve to find position vector
To find foot of perpendicular from point o
Let foot of perpendicular be
o
Find
(line to line)
, get
from the line equation
Done by Nickolas Teo Jia Ming
o •
Use
.
To find foot of perpendicular from point o
o
Let foot of perpendicular be
(line to plane)
, get equation of line CN
Direction vector of line is the normal of the plane
Position vector of line is
Find intersection of line and plane
Sub the vector equation of line into the scalar product equation of plane
•
= 0 to solve for
To find distance between two parallel planes o
Decide if the planes are on the same side of the origin
( . ) or
for both have same sign (+ or )means that they are on the
same side
1 2
.
( . ) or
side
+
cos
=
.
To find angle between line and plane o
Get point of intersection between line and pl ane, point
Sub the vector equation of line into the scalar product equation of plane
o
Get another point on the line, li ne, point
o
Get vector
o
Get angle between line and plane
) =
.
cos(90
To find angle between two planes o
cos
=
.
To show relation between two lines o
Check if parallel
o
•
. )
To find angle between two lines
•
+ (
Direction of must be same when used for both equati ons
•
for both have different sign means that they are on different .
•
. )
1 2
•
(
1 2 =
, where
∈ℝ
If not, the two lines intersect
Check if same line
Sub the vector equation of one line into the other
If equal for all , , etc., they are the same
If not, they are skew lines
To show relation between line and plane o
Check if parallel
Done by Nickolas Teo Jia Ming
o
= 0?
.
If not, the line only intersects
Check if contained within plane
•
= ? Where p is the scalar product
.
If not, the line is parallel but not contained within the plane If yes, the line is parallel and contained within the plane
To show relation between three planes o
Find if the planes are parallel
Parallel planes have the same simplifi ed normal
If all three planes are parallel, they will not intersect
If two planes are parallel, there would be two lines of intersection
If no planes are parallel, they will form a triangle with three lines of intersection
•
•
Always try to simplify the directional vector o
Divide all by highest common fraction
o
Make the majority of the signs positive
To find possible values when 3 planes do not have a common co mmon point, consider the cases, show how they are possible/ not possible, to get the values
•
o
A triangle is formed
o
1 pair of parallel planes
When taking the absolute value (modulus) the dot or cross product, don’t try to flip (multiply by -1) the values given.
Binominal and Poisson distribution •
When asked to find the mean for a Poisson distribution, use the formula in MF 15
•
Always apply continuity correction when you’re approx imating to normal o
Always simplify first when dealing with multiples of the same distribution
3 3 12 − + 1 2 12
o
> )=
>
=
>
+
Always apply continuity correction before converting to standard normal
•
(3
(
< )=
<
+
=
<
For cases when you’re adding or subtracting two distributions,
and , after approximating
to normal, only need to do continuity correction once o
o
( ±
> )=
±
>
+
If there are multiplies of a distribution when you’re adding or subtracting two distributions, reduce to simplest form fir st (This seldom comes out)
•
3 (6 ± 3 > ) =
2 ±
>
=
2 ±
If asked for the condition for a Poisson distribution to be vali d o o o
>
+ 1 3 2
level of significant, there is insufficient evidence at ?% level of significance to reject
Usually the accept.
(0,1)
~
is approximately normally
Conclude
•
is large, by the Central Limit Theorem, Theorem,
Test with the assumption that
o
=
Define test statistic
o
=
State assumption or use Central Limit Theorem
o
∑ −11 ∑ 2 ∑ 0 1 ∑=1 −1 −µ √ 0 0
and level of significance , Define sample mean
o
Use GC if raw data is given
0
. Thus, …’
‘Since p-value < level of significant, there is sufficient evidence at ?% level of significance to reject
0
. Thus, …’
value will not equal the critical value, but if it does, can either reject or
To backtrack to find level or significance, sample mean, variance, original mean o
Determine borderline
o
or
value (critical value)
Inverse norm on the p-value
p-value of a one-tail test is the level of significance
p-value of a two-tail test is half the level of significance
p-value is near the start (like 0.05) for a lower tail test
p-value is near the end (like 0.95) for a upper tail test
Determine range of test
If
0
If
or
is rejected, test
magnitude
value
or
magnitude is greater the borderline
Test value less than borderline value when lower tail test
Test value more than borderline value when upper tai l test
0
is not rejected, test
or
o
magnitude
or
magnitude is less the borderline
Test value more than borderline value when l ower tail test
Test value less than borderline value when upper tail test
or
Find the unknown by subbing in known val ues
Done by Nickolas Teo Jia Ming
•
Usually level of significance will not equal p-value. To obtain minimum level of significance, it will differ from the obtained p-value by the smallest unit in the answer o
•
Like 0.01 for 2 d.p.
Converting 1 tail to 2 tail, your 1 tail p-value i s half that of the 2-tail p-value (aka you time 2 your 1 tail p-value to get your 2-tail one) o
This is because your z-cal for 1 tail or 2 tail remai ns the same.
o
Thus the area of the curve great than z-cal (or less than negative z-cal) remains the same.
o
Hence your area is multiplied by 2 for conversion from 1 to 2 tail, as you ar e now taking both ends.
•
If asked to define symbols used o o o o
•
0 1 2
is the null hypothesis is the alternative hypothesis
is the population mean time/length for ____ is the population variance for ____
To join two samples together o
∑ ∑ 2 ∑ 1 ∑ 2 ∑ ∑ ∑ 2
Find the
and
of both samples
=
×
(
= (
)
)
o
Join them both to find the overall
and
o
Use the combined value to find the unbiased estimate of population mean and variance
Correlation and regression •
To explain why the o
or
in
= + is different from or , like ln The points correspond to a ___ graph, gr aph, thus the model ____ ( =
+ ) is
suitable, as by plotting ___ against ___, the graph will be linear
� �
•
To find a missing value from a regression line, sub in
•
Validity of prediction o
o
and
Linearity holds
Scatter diagram shows a linear relationship
Magnitude of r-value close to 1
Given value to predict from is within range of given data value (Interpolation)
If not, it is an extrapolation, and there is no evidence that the linear trend continues at ___ (Value given)
•
Identification of regression line to use o
Cases
One variable is i ndependent, without experimental error
Both variables have experimental error, and do no t depend on each other
•
Regression of on Regression of on
When asking to find the line of regression and make an estimation o
Working for the line of regression for the estimation is to 5 d.p (or s. f.)
Done by Nickolas Teo Jia Ming
o •
As in sub in the given vales into the equation with 5 d.p (or s.f.)
Answer for the line of regression and estimation is in 3 s.f
To explain why a liner model is not appropriate o
The magnitude of the product moment correlation coefficient is not close to 1
Which shows that there is a weak positive/negative linear correlation between ___ and ___
If it is, say ‘although the product moment correlation coefficient is close to 1, which shows a strong positive/negative linear correlation between ___ and ___’
o
The scatter diagram shows a curvilinear curvil inear relationship between
and
Because y is increasing/ decreasing at an increasing/ decreasing rate.
Complex Numbers •
Manipulating trigonometry o
Sine
o
Cosine
o
tan = cot( cot(90 90°° Conversion table 0
√ √
0
2 4
Cosine
2 0
Tangent
)
)
)
)
6
Sin
)
cos( ) = cos cos = cos (180° cos = sin(9 sin(90° 0° ) Tangent tan( ) = tan tan = tan (180°
o
√ √ √ √
sin( ) = sin sin = sin (180° sin = cos(9 cos(90° 0°
1
2 3 2 1
√ √ 4
2
2 2 2 1
3
•
√ √ √ 3
3
2 1 2 3
√ √ ∞ 2
4
2 0 2
Important relationships o
o o o o o
1 = − ∗ − + ∗ = + − = 2 c osos = 2 () 0 = − = 2 sisin = 2 () = × = ∗ ⇒ 2 = × = | | Done by Nickolas Teo Jia Ming
•
o o
•
2 2
When finding the root of ( + ) = ( +
1 +
= + = + ±
To simplify o
, take out
)
from the top and bottom 1 = −1 − + = 2 c osos 2 − = 2 sin 2 ±
±
•
•
To solve simultaneous equations of and o
Let
and
by
and
o
Compare real and imaginary coefficients to get 4 equations
o
Use GC to solve
+
1+ 3+ √ 3+ Rationalize the denominator +1+√ 3−1 3−1 1+ × √ 3− 3− = √ 3+1+ 3+ √ 3− 3− 4 √ 3+
To solve a quadratic equation of o
If you’re lucky, you may be able to identify the exact coefficient
Use completing the square method
o
Use
=
− √ −4 2 ±
When asked to show that tan
= , where
is anything, and
is a given angle
Try to use arg , where z is the compl ex number equation
To find points whereby a line and circle intersect o
Find their equations and equate them ) + ( ) = Circle : (
2 2 2 ∗ 2 ∗ ∗ 2 ∗ ∗ ∗ 2 2 ∗ ∗ 2 ∗ ∗ 2 − ∗ Line:
•
√
E.g. 1.7320 1.732050 50 … is likely to be 3
o
o •
Try using GC (set mode to imaginary)
•
respectively
When dividing two complex number, like o
•
+
=
+
To find the value of | o
+
|
Find
| | =
o
|from given values of | |, | |, |
Find
| | = Find ( + ) | + | =( + )( + | Expand | | | =( )( Sub in found values to get answer
o
o
•
To simplify 5 + 5 o
5
o
5
(
) )
(the constant and r must be the same)
+
[2 cos( cos( )] + =2
)
( )
Done by Nickolas Teo Jia Ming
•
When finding the power of a complex number,
•
( ) = For drawing on an argand diagram o
o
2 1 1
Note that |
etc. are representing fixed complex points, and are like
,
|=
arg(
|
is a circle with center , radius
)=
is a half-line, with open dot at , with angle R
o
If the inequality > or < is used, a dotted line should be drawn for all three cases
o
If the answer is an area, label it the ‘locus of ’
o
If the answer is a line, draw that line portion with a different coloured ink and label
o
Even if you’re drawing more than one complex number (e.g.
label the axis as
and
o
Label the radius of the circle Label the angle of the roots
o
Label the roots as
To find To find
To find o
To find o
•
=
To find maximum and minimum value of | |, after a circle locus is drawn Take the distance to the center, plus radius for maximum, minus radius for minimum
∴ ℎ ℎ ∈ ℤ+ ∴ 2−12 ℎ ℎ ∈ ℤ+ ∴ ℎ ℎ ∈ ℤ+ ∴ ℎ ℎ ∈ ℤ+ − when is real sin( ) = 0 when
cos(
To find o
=
,
is imaginary
)=0
Because
•
), it’s okay to
The magnitude/ modulus portion of the answer
o
o
•
and
alone works as well
Draw a dotted circle, which all the roots lie on
o •
( ).
o
o •
( ) and
When drawing an argand diagram after finding the complex roots of an equation,
•
If the angle is negative, label it as negative on the argand diagram |=| | is the perpendicular bisector of the line joining and
it ‘locus of ’
•
=
(
)
,
is odd
when is positive cos( ) < 0 =2 Because is even, and is an integer when is negative cos( ) < 0 = (2 1) Because is odd, and is an integer when
is real/ imaginary and positive/ negative
Sub the
obtained from evaluating the real/ imaginary step, into the positive/
negative step, so as to relate the two restrictions.
•
(
When solving o
o
) = 0, and they give that the roots include cotangent
=1
=
It can also be equal to stuff other than 1, just find the roots of those
The simpler complex is the numerator
1−
Done by Nickolas Teo Jia Ming
The modulus-argument form, denominator and numerator will change
depending on the question
= −
o
Use the simplify method given earlier
Convert from polar to rectangular form to solve (as per the method given earlier)
o
•
1+ 1 1 =
o
Let
o
•
For questions that first asks you to find ( + ) =
and apply the answer found earlier to solve
=
Find
=
+
When asked to transform one point to another, e.g. o
and then hence
1+
in terms of in the polar form
to
= State the transformation The point representing
o
is transformed from the point representing
anti-clockwise rotation about the origin by enlargement about the origin by factor
, followed by an
by an
Sampling •
Applying quota sampling to scenario Issue
o
Population
o
•
Teenagers at a shopping mall
Sub-population
o
•
How often teenagers aged 12-18 go shopping
Male and female teenagers
Applying stratified sampling to scenario o
Define the non-overlapping strata that the population is going to be grouped in
o
Calculate and state the sample size from each strata
o
State that the selection of sampling sampli ng units within the strata is random
Limitations Quota
o
Sample not representative of population
Not everyone in the desired population has an equal chance of being selected
Maclaurin’s series 1. When a.
is small, for trigonometry
∓
Multiples like 2 , 3 can be subbed to replace
in the formula directly
b. Additions or subtractions, like i. cos
±
± , needs to be broken up using MF15 = cos cos± sin sin±
Done by Nickolas Teo Jia Ming
2. When asked to find Maclaurin’s series for , find a.
Then sub
b. Then sub
= 0 into the equation for = 0 and
, , …
, , … into
i. Use the -value found earlier
when solving for and so on When you have a Maclaurin’s series to a power, use binomial to expand 6 6 6 a. (cos ) = 1 + ⋯ = 1 + + ⋯ = 1 + 6 + + 2 4 2 4 2 4 6 5 + 2 + ⋯ 2 2 4 ii. Use the value of
3.
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4. When asked why the approximation of the Maclaurin’s series is not good/reliable a.
Calculate the percentage error
− × 100% 100%
i.
b. Say the error is significant
Differentiation 5. When dealing with a.
Let
ln
ln be
.
( )
b. Differentiate c.
Replace
.
( )
with
6. When differentiating inverse trigonometry a.
Let
be the equation
b. Throw the inverse trigonometry over c.
Differentiate with respect to x
d. Find
e. Use original
= to change all into i. Use right angle triang le to convert between trigonometry •
Draw the triangle, and you have two sides known from the ration in the
•
equation
Find the other side using Pythagoras P ythagoras theorem
7. For trigonometry with powers greater than 1 a.
Differentiate as per normal with chain rule
8. When dealing with more than one (i nverse) trigonometry function a. Do product rule b. Apply ‘differentiating inverse trigonometry’ if needed to di fferentiate one term 9. To show that there is not turning point a.
Let
= 0
b. Show that the resulting equation is invalid i. Possibly due to restrictions stated in the questions, like
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•
must be positive
10. To show that tangent does (not) cut the graph again a.
Sub the equation of tangent into the equation of graph i. One solution will be the original point ii. If there’s another, that other will wil l be the other point
11. When they say that the tangent is parallel to the -axis a.
= ∞
i. Denominator of the equation of
Solve using the denominator equal to 0
•
12. When trying to join two different variable, like a.
is 0
and
Differentiate first with respect to a common variable like
b. Then
= ×
13. Whenever they say
is small, find a way to use Maclaurin’s series
Integration 1. When using MF 15 a.
2 ∫ −1 1
When the term has ( i. ii.
b.
(
) instead of
= sin
)
is any constant
−1
2
1
, multiply to the result
2. When dealing with fractions a.
See if can use inverse trigonometry in MF15
b. Check for
( )
( )
i. If both sides of the post-integration equation has a ln(|? ? |), let the constant be ln and combine it with the other ln(|? ? |) to make it easier to handle.
ln(| |) = ln(| |) + l n = ln(| |) If numerator is a constant i. Bring up the denominator using inverse power •
c.
d. If numerator is a variable i. Do partial fractions if the denominator has roots ii. Use completing the square if possible
2
2
3. When dealing with sin ( ) or cos ( ) a.
Convert using double angle formula
4. In integration by parts a.
Follow LIATE i. L = best u, as easiest to differentiate
b. For when you have more than one trigonometry function c.
If you get back the original function, just throw it over.
5. In integration by substitution a.
Find
b. Change all
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c.
Change limits
6. When dealing with trigonometry a.
′
If the
are different (e.g.
2 and
2
and ), use formula to convert to the same X
b. When dealing with tangent, try to convert to cosine and sine and look for patterns
and double angle formula Integrate tan2( ) by converting it to sec 2 ( ) 1 i. Integrate sec 2 ( ) = tan( ) To integrate sin2( ) or cos2 ( ) like
c. d.
( )
( )
i. Convert to cos(2 ) using the MF15
7. When finding area under a curve, if it crosses the axis a.
You can translate the curve by replacing i. For example,
=
+1
or
with the translated function
8. When dealing with parametric a. When finding area i. Use
∫
and sub in the parametric forms
•
Change the limits
•
Change the
•
Change all
9. To get rid of modulus from ln a.
into
Throw the modulus to the other side and incorporate it into the constant i. | + | = .
∴
, where + = = ± b. Get rid of the modulus first before you key in values to find c, if possible
Rate of change 1. Radius and height are related a.
Convert height to radius first
b. Then differentiate to find rate of change
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