Pure Maths Flashcards
what is the small angle approximation of sin, cos and tan θ?
sin θ = tan θ = θ
cos θ = 1- (θ^2)/2
what is sin kx and cos kx differentiated?
sin kx –> k cos kx
cos kx –> -k sin kx
what is e^f(x) differentiated?
f’(x)e^f(x)
what is ln[ f(x) ] differentiated?
f’(x) / f(x)
how does the chain rule work? (differentiation)
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
how does the product rule work? (differentiation)
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
what is tan kx and cot kx differentiated?
d/dx tan kx = k sec^2 kx
d/dx cot kx = -k cosec^2 kx
what is sec kx and cosec kx differentiated?
d/dx sec kx = k sec kx tan kx
d/dx cosec kx = -k cosec kx cot kx
how do you find dy/dx from parametric equations?
dy/dt / dx/dt
how do you differentiate functions that cannot be written as y= …? (implicit differentiation)
differentiate as normal but if differentiating a y term, add dy/dx on. then rearrange to make dy/dx the subject
how can you differentiate a function with terms with x and y?
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.
how can you tell if a point is concave, convex or a point of inflection?
concave / max (-x^2): f’‘(x) < 0
convex / min (x^2): f’‘(x) > 0
inflection: f’‘(x) = 0
integrate cosec x cot x
-cosec x +C
Integrate sec x tan x
sec x +C
how do you work out ∫(2x+3)^4 dx with the guess method?
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
what are the trig identities for (sec x)^2 and (cosec x)^2 ?
(sec x)^2 = 1 + (tan x)^2
(cosec x)^2 = 1 + (cot x)^2
what do you have to remember when integrating and differentiating trig?
put your calculator in radians, dummy
how do you integrate [f(x)]^n
add one to the power
divide by the power
divide by the differential
how do you differentiate f(x) ^n when f(x) is linear?
multiply by the power
subtract one to the power
multiply by the differential
how could you simplify fractions with polynomials to integrate them?
split into partial fractions
how does integration by substitution work?
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
what is the equation for integration by parts?
∫u dv/dx = uv- ∫v du/dx
how do you find a general solution to dy/dx = f(x)*g(y)
put the dx and Xs on one side and the dy and Ys on the other. then integrate both sides with dx or dy.
what is the difference between arithmetic and geometric sequences?
arithmetic sequences add a common difference, geometric multiply by a common ratio
what is the formula for ∑r from 1 to n?
1/2 n(n+1)
when can you use the binomial expansion approximation formula on (1+x)^n ?
when |x| < 1
what do you have to do to (a+bx)^n to approximate it?
turn it into a^n * (1+bx/a)^n
what is the equation for arc length?
l= rθ
what is the equation for the area of a sector?
A = 1/2 r^2 θ
what is the equation for the area of a segment?
A = 1/2 r^2 (θ - sinθ)
how do you express: a sinx ± b cosx
in the form Rsin(x±α) or Rcos(x∓α)?
Rsin(x+α) = Rsinxcosα ± Rcosxsinα Rcosα = a , Rsinα = b and R = √(a^2 + b^2)
how do you prove that a complex continuous function has a root?
f(a) and f(b) have opposite signs
what are the 7 methods of integration?
function derivative method, trig identities, u sub, integration by parts, partial fractions, addition formulae, guess method.
what is a typical way of solving some equations with sin and cos?
turn it into Rsin(x+α) or Rcos(x+α)
what is log y ± log x ?
log (x*/y)
what is 1/ loga(b) ?
logb(a)
what is loga(b)/ loga(c) ?
logc(b)
what is the equation for the Nth term of a geometric sequence?
Un = ar^(n-1)
what is the parametric integration formula?
∫ y(t) dx/dt dt
what are the trig identities with cos2x?
2cos^2 x ≡ 1 + cos 2x
2sin^2 x ≡ 1 - cos 2x
what do you have to remember when integrating by substitution an equation with parameters?
sub the parameters into x in the u in terms of x equation to find the new parameters
integrate e^kx
1/k e^kx +C
differentiate e^ f(x)
f’(x) e^ f(x)
differentiate a^f(x)
a^ f(x) * f’(x)ln(a)
if a>0
integrate a^kx
a^kx / kln(a) +C
if a>0
how do you get rid of a modulus sign in an equation?
square it
what is the quotient rule when
y = f(x) / g(x)?
dy/dx = [ g(x)f’(x) - f(x)g’(x) ] / g(x)^2
what is loga [f(x)] differentiated
1/ xlna *f’(x)
Check if x needs to be f(x)
what is the general formula for an exponential model?
Y = Ae^(kx)
This may be used to model the value of a house etc.
Y = Ak^x is an invalid model
