Pure Maths Flashcards
What are the steps for a proof?
- Specialise to understand eh problem
- Make a generalisation
- Calculate and simplify
- Write a conclusion using wording of phrase
What are the three types of sequence?
Convergent (to a limit)
Oscillating (Period is number of different numbers)
Divergent (to infinity)
What is the equation for the sum of the first n terms of an arithmetic progression?
Sn = n/2(2a + (n-1)d)
- N is term
- A is first term
- D is difference between the terms
What is an example of a position to term rule and a term to term rule?
- Position to term rule: tn = 6n + 5
2. Term to term rule: tn + 1 = tn + 6
For completing the square, how do you interpret it graphically?
Look at notes
How do you find the gradient of a line?
Y-Y1 / X-X1 = gradient
How do you fined the midpoint of the line?
(x1 + x2 ) / 2 = xm
(y1 + y2) / 2 = ym
How do you find the length of a line?
(deltay)^2 + (deltax)^2 = c^2
What is the trapezium rule?
- Find area under a curve, split into equal length
- ((a+b) / 2)h
What is the equation for arc length?
Theta x r
What is the equation for area of a sector?
0.5 x r^2 x theta
How do you convert from radians to degrees?
x 180/pi
How do you convert from degrees to radians?
Divide by 180/ pi
What is the equation of a circle?
(x-a) ^ 2 + (y-b)^2 = r^2
-Where (a,b) is the centre
What is the equation for a geometric sequence?
tn = a x r^n-1
- a is the first term
- r is the common ratio
What is the sum of the first n terms of a geometric progression?
Sn = a x (1-r^n) / (1-r)
What is the term to term rule of a geometric progression?
Un + 1 and state U1 =
What are the log addition rule?
logcA + logcB = logcAB
logcA + logcA = logcA^2
What is the log subtraction rule?
logcA - logcB = logc (A/B)
What is 2logcA also seen as?
logcA^2
What is the power rule for logs?
nlogcA = logcA^n
What are the steps for finding the stationary point?
- Find dy/dx
- Solve dy/dx = 0
- Then plug these x values into the original equation to get y values
- Then find the second derivative and plug in your x values to find out if the point is a maximum or minimum - (<0 it is a maximum) (>0 it is a minimum)
What are the trigometric value we need to know?
sin^2theta + cos^2theta = 1
sin / cos = tan
What is secX ?
secX = 1/cosX
What is cosecX?
cosecX = 1/sinX
What is cotX?
cotX = 1/tanX
What are the two conditions for a function?
- Completeness Criteria: Every object in the domain has an image in the codomain
- Uniqueness Criteria: each object in the domain has a unique image
Do you do g or f first in gf(x)?
f first
What do you do in differentiation?
- Bring the old power down to the front
2. Decrease the power by one
What do you do in integration? What do you always need to remember?
- Increase power by one
- Divide by new power
- c
What is dy/dx of y=e^x?
e^x
What is dy/dx of y=lnx?
1/x
What is the rule for chain rule?
nstuff^n-1 x stuff’
- (x^2 + 3x )^4
- 4(x^2 + 3x)^3 x (2x + 3)
What is e^stuff differentiated?
e^stuff x stuff’
What is ln(stuff) differentiated?
stuff’ / stuff
What is the infinite sum?
Sn = a / 1-r
What is the cosine rule?
- a squared = b squared + c squared - 2bcCosA
- CosA = b squared + c squared - a squared / 2bc
What is the sine rule?
aSinB = bSinA
What is the binomial expansion?
(1+x)^n = 1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3 …..
USE BRACKETS!
When is the binomial valid?
mod (x/number) < 1
mod(x) < 1
What does sinx differentiate to?
cosx
What does cosx differentiate to?
-sinx
What does tanx differentiate to?
sec^2x
How do you differentiate cos(stuff)?
-stuff’ x sin(Stuff)
How do you differentiate sin(stuff)?
cos(Stuff) x stuff’
What are some trigonometric identities?
sin^2x + cos^2x = 1
tan2^x + 1 = sec^2x
1 + cot^2x = cosec^2x
How do you work out R in harmonic form?
y1 = R^2sin^2alpha = 4 y2=R^2cos^2alpha = 9 y1 + y2 = 13 R^2 (sin^2alpha + cos^2alpha) = 13 R^2 (1) = 13 R = sqrt(13)
What is the maximum and minimum value of Rsin(3x+alpha)+5?
Max: R + 5
Min: -R + 5
What is the maximum and minimum value of 1/Rsin(3x+alpha)+5
max: 1/5-R
min: 1/5+R
What is an even function?
f(-x) = f(x) e.g. cos
What is an odd function?
f(-x)=-f(x) e.g. y=x or tanx
What does change of sign show?
A root (always do it to one more dp to check around but draw swell) !
What is Newton Raphson excel?
x f(x) f(x') x0
What is C4+C5/2 x a2 in excel?
0.5A$2(C4+C5)
How do you do integration by parts?
make v’ be the e (you can do it)
Check by integrating!
How do you do integration by substitution?
Change limits!
What is implicit differentiation? What does y^2 = 3x^2y + x^2 differentiate to?
2y dy/dx = 6xy = 3x^2 dy/dx +2x
