Pure Maths

Question 1

what is the small angle approximation of sin, cos and tan θ?

Answer 1

sin θ = tan θ = θ

cos θ = 1- (θ^2)/2

Question 2

what is sin kx and cos kx differentiated?

Answer 2

sin kx –> k cos kx

cos kx –> -k sin kx

Question 3

what is e^f(x) differentiated?

Answer 3

f’(x)e^f(x)

Question 4

what is ln[ f(x) ] differentiated?

Answer 4

f’(x) / f(x)

Question 5

how does the chain rule work? (differentiation)

Answer 5

substitute u into the equation to remove Xs. make an equation of u in terms of X. find du/dx. find dy/du from the equation of y in terms of u. find dy/dx by multiplying dy/du with du/dx

Question 6

how does the product rule work? (differentiation)

Answer 6

if y = uv,
dy/dx = u dv/dx + v du/dx.
an example of this is y = (x^2)(x+4)^4. u would be (x^2) and v would be (x+4)^4

Question 7

what is tan kx and cot kx differentiated?

Answer 7

d/dx tan kx = k sec^2 kx

d/dx cot kx = -k cosec^2 kx

Question 8

what is sec kx and cosec kx differentiated?

Answer 8

d/dx sec kx = k sec kx tan kx

d/dx cosec kx = -k cosec kx cot kx

Question 9

how do you find dy/dx from parametric equations?

Answer 9

dy/dt / dx/dt

Question 10

how do you differentiate functions that cannot be written as y= …? (implicit differentiation)

Answer 10

differentiate as normal but if differentiating a y term, add dy/dx on. then rearrange to make dy/dx the subject

Question 11

how can you differentiate a function with terms with x and y?

Answer 11

use the product rule to differentiate the x part and leave the y and add the differential of the y, adding on dy/dx, and leave the x part.

Question 12

how can you tell if a point is concave, convex or a point of inflection?

Answer 12

concave / max (-x^2): f’‘(x) < 0
convex / min (x^2): f’‘(x) > 0
inflection: f’‘(x) = 0

Question 13

integrate cosec x cot x

Answer 13

-cosec x +C

Question 14

Integrate sec x tan x

Answer 14

sec x +C

Question 15

how do you work out ∫(2x+3)^4 dx with the guess method?

Answer 15

consider y= (2x+3)^5
dy/dx = 5*(2x+3)^4 *2 = 10(2x+3)^4
so ∫(2x+3)^4 dx = 1/10 (2x+3)^5 +c

Question 16

what are the trig identities for (sec x)^2 and (cosec x)^2 ?

Answer 16

(sec x)^2 = 1 + (tan x)^2

(cosec x)^2 = 1 + (cot x)^2

Question 17

what do you have to remember when integrating and differentiating trig?

Answer 17

put your calculator in radians, dummy

Question 18

how do you integrate [f(x)]^n

Answer 18

add one to the power
divide by the power
divide by the differential

Question 19

how do you differentiate f(x) ^n when f(x) is linear?

Answer 19

multiply by the power
subtract one to the power
multiply by the differential

Question 20

how could you simplify fractions with polynomials to integrate them?

Answer 20

split into partial fractions

Question 21

how does integration by substitution work?

Answer 21

sub in u for something
find x in terms of u and sub in
find du in terms of dx by differentiating u in terms of x to sub out dx

Question 22

what is the equation for integration by parts?

Answer 22

∫u dv/dx = uv- ∫v du/dx

Question 23

how do you find a general solution to dy/dx = f(x)*g(y)

Answer 23

put the dx and Xs on one side and the dy and Ys on the other. then integrate both sides with dx or dy.

Question 24

what is the difference between arithmetic and geometric sequences?

Answer 24

arithmetic sequences add a common difference, geometric multiply by a common ratio

Question 25

what is the formula for ∑r from 1 to n?

Answer 25

1/2 n(n+1)

Question 26

when can you use the binomial expansion approximation formula on (1+x)^n ?

Answer 26

when |x| < 1

Question 27

what do you have to do to (a+bx)^n to approximate it?

Answer 27

turn it into a^n * (1+bx/a)^n

Question 28

what is the equation for arc length?

Answer 28

l= rθ

Question 29

what is the equation for the area of a sector?

Answer 29

A = 1/2 r^2 θ

Question 30

what is the equation for the area of a segment?

Answer 30

A = 1/2 r^2 (θ - sinθ)

Question 31

how do you express: a sinx ± b cosx

in the form Rsin(x±α) or Rcos(x∓α)?

Answer 31

Rsin(x+α) = Rsinxcosα ± Rcosxsinα 
Rcosα = a , Rsinα = b and R = √(a^2 + b^2)

Question 32

how do you prove that a complex continuous function has a root?

Answer 32

f(a) and f(b) have opposite signs

Question 33

what are the 7 methods of integration?

Answer 33

function derivative method, trig identities, u sub, integration by parts, partial fractions, addition formulae, guess method.

Question 34

what is a typical way of solving some equations with sin and cos?

Answer 34

turn it into Rsin(x+α) or Rcos(x+α)

Question 35

what is log y ± log x ?

Answer 35

log (x*/y)

Question 36

what is 1/ loga(b) ?

Answer 36

logb(a)

Question 37

what is loga(b)/ loga(c) ?

Answer 37

logc(b)

Question 38

what is the equation for the Nth term of a geometric sequence?

Answer 38

Un = ar^(n-1)

Question 39

what is the parametric integration formula?

Answer 39

∫ y(t) dx/dt dt

Question 40

what are the trig identities with cos2x?

Answer 40

2cos^2 x ≡ 1 + cos 2x

2sin^2 x ≡ 1 - cos 2x

Question 41

what do you have to remember when integrating by substitution an equation with parameters?

Answer 41

sub the parameters into x in the u in terms of x equation to find the new parameters

Question 42

integrate e^kx

Answer 42

1/k e^kx +C

Question 43

differentiate e^ f(x)

Answer 43

f’(x) e^ f(x)

Question 44

differentiate a^f(x)

Answer 44

a^ f(x) * f’(x)ln(a)

if a>0

Question 45

integrate a^kx

Answer 45

a^kx / kln(a) +C

if a>0

Question 46

how do you get rid of a modulus sign in an equation?

Answer 46

square it

Question 47

what is the quotient rule when

y = f(x) / g(x)?

Answer 47

dy/dx = [ g(x)f’(x) - f(x)g’(x) ] / g(x)^2

Question 48

what is loga [f(x)] differentiated

Answer 48

1/ xlna *f’(x)

Check if x needs to be f(x)

Question 49

what is the general formula for an exponential model?

Answer 49

Y = Ae^(kx)
This may be used to model the value of a house etc.
Y = Ak^x is an invalid model