- 1. Straight from My Revision Book By the way, this is probably
repeating the other one a lot (Special thanks to Moji who is a star
)
2.
- Transformations of Graphs
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- Motion with Variable Acceleration
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- Finding Minimum or Maximum Value
3. 4.
- When we have to stretch a line along the x-axis, we have to
stretch to nwhere n is the number given
- When stretching/compressing a line along the y-axis, the
stretch is to the number given
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- Curve y=x translated to give y=f(x)
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- It is -2 because x is squared in the question and moved to the
right
- Again it is squared because the question has x
y x 0 x A (2,-4) y=x 5.
- Similarity : when the angles are the same but thelengths are
different
- Triangles are congruent if:
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- 3 sides are the same (SSS)
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- 2sides + included angle are the same (SAS)
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- 2angles + 1side are the same (ASA or AAS)
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- In a right angled triangle, the hypotenuse +1side are the same
(RHS)
- You will always be given the graph and an equation
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- 2a) x+4x-5 first draw graph in book
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- To solve the equation make a grid
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-
-
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- =0 therefore y=0 when x=-5 and 1
- Note : if this doesnt work y=0
x -6 -5 -4 -3 -2 -1 0 1 2 x 36 25 16 9 4 1 0 1 4 +4x -24 -20 -16
-12 -8 -4 0 4 8 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 y 7 0 -5 -8 -9 -8 -5
0 7 6.
- When you are told to complete the square:
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- Take the co-efficient of x at be, for example in:
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- Half this and then square it. Both add and subtract it from the
equation (so the equation doesnt change) like so:
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- This can be factorised into:
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- To check you can expand to get:
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- Remember: that if it is 4x + ... it has to be factorised into
4(x + ...) rather than (4x + ...)
7. 8.
9.
- For long equations, differentiate in parts and then
add/subtract as needed
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- 1 st(3x) multiply the power by themultiple of x=15x
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- Then minus 1 from the power=15x
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- Repeat for all parts, remembering the signs:
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- = 15x+16x - 6x [+2] ...(continued)
d dx Multiple of x Power 10.
- For a whole number i.e. The 2, it will =0
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- For things like 5x they become 5
11.
- The second derivative tells us if a point is a maximum point,
minimum point or a possible point of inflection
- This is just differentiating what you already have
differentiated
- The third derivative tells us if a point is a point of
inflection if it does not equal zero. If it does equal zero it is
not possible at this level.
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- It shouldnt equal zero but if it does just remember to putnot
possible at this levelas there may be a mistake in the
question
12.
- Integration is the inverse process of differentiation
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- Take the multiple, like 3, and divide it by the power + 1
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- In this case you would get 3/6. You then take the x, x here,
and add 1 to the power again, x
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- You place this over the multiple in the fraction to give you:
[(3x)/6]
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- Continue like this, adding and subtracting according to the
equation given
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- For a constant, like 2, simply at x, thus 2 = 2x
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- Then simplify if possible
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- The example above becomes 2x
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- Then add c where c is the constant
13.
- There are two formulae that need to be learnt:
- Note:Finding the original function given its derivative is the
same as finding the integral
1. 2. Where a is a constant 14.
- For definite integrals, numbers are given
- You must integrate as normal
- Then you take the number for the upper limit and substitute it
in for x, x=3
- Do the same for the lower limit, x=1, and subtract this from
the previous value. Giving:
- Note:As c vanishes when you work the previous expression, it is
common to replace c altogether and write:
- It can also be found using the formula:
- This can be used to find areas of curves
Upper limit Lower limit 15.
- To find the constant, you will be given co-ordinates which then
can be substituted into an expression which will be used to find
the constant
IntegrateSubstitute 16.
- DONT FORGET TO PUT IT INTO THE PREVIOUS EXPRESSION!
- You could lose marks for leaving it just like that!
17.
- Explain whyis a possible expression for the gradient of the
curve and give an alternative expression for
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-
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- So x=0 or x=2making them stationary points thus a possible
expression
-
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- Where k is a constant which is an alternative expression
- The curve passes through the point (3,2). Taking as , find the
equation of the curve
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- Substitute this into the equation from the integration:
18.
- The 2 ndderivative tells us if a point is a maximum, a minimum,
or a possible point of inflection
- The 3 rdderivative tells us if a point is a point of inflection
if it doesnt equal zero
- If it does equal zero it is not possible at this level
- However this should only occur if there was a mistake in
workings or the question
19.
- Find the first derivative
- Then find the 2 ndderivative
- Substitute one possible value for x into this equation
- You will often be asked to find the possible points
(co-ordinates) where the stationary points are. To do this, simply
substitute the possible values of x into the equation given
- It is important to say that the co-ordinate is a max/min/point
of inflection
20.
- To find an equation for v you can do the derivative of s
(ds/dt)
- To find an equation for a you can do the derivative of v
(dv/dt)
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- Find an equation for v with the following equation
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- Find an equation for a with answer above
- Similar to doing the 2 ndderivative
21.
- Area under the line which also given the velocity in a
time-distance graph
22.
23.
- Two mutually exclusive events (meaning that if something
happens, something else cannot happen)
- If a trial is conducted n times,
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- P is the probability of success in every trial
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- q is the probability of failure
- The probability of exactly r successes is:
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- Where x is number of success and r=0,1,2...
24.
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- Indices of a term = the sum of indices of expression
- However, Pascals Triangle is long to use so thebutton on the
calculator is used instead
- So, if we want to work out the co-efficient ofin the
expression, we need
- Using the co-efficients of each expansion we can get Pascals
Triangle
- Its easy to work out as you just add the previous numbers
- The triangle is symmetrical and always has 1 at the beginning
and end of each line
Power of binomial Power of x and 25. 26.
- Finding Minimum or Maximum Value
27.
- When using the equation the decides the answer you get
- If, you get anon-realnumber as it is a minus
- If , you gettworeal distinctanswers
- If, you getone real equalanswer
- It is important to know the correct terminology here
28.
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- Useto find the discriminate so it will be:
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- Then, when , you can factorise to get
- Whenorthe answers are above the x-axis therefore the answer
is:
29.
- If given an equation where the constant is negative to find the
minimum, the only possible answer is for the equation to equal
zero
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- Ascannot be any less than 0, when this is multiplied by 4 it
can be no less than zero
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- Therefore -25 has o be the minimum possible answer