Pure Maths Flashcards

1
Q

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

A

sin θ = tan θ = θ

cos θ = 1- (θ^2)/2

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2
Q

what is sin kx and cos kx differentiated?

A

sin kx –> k cos kx

cos kx –> -k sin kx

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3
Q

what is e^f(x) differentiated?

A

f’(x)e^f(x)

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4
Q

what is ln[ f(x) ] differentiated?

A

f’(x) / f(x)

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5
Q

how does the chain rule work? (differentiation)

A

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

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6
Q

how does the product rule work? (differentiation)

A

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

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7
Q

what is tan kx and cot kx differentiated?

A

d/dx tan kx = k sec^2 kx

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

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8
Q

what is sec kx and cosec kx differentiated?

A

d/dx sec kx = k sec kx tan kx

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

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9
Q

how do you find dy/dx from parametric equations?

A

dy/dt / dx/dt

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10
Q

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

A

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

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11
Q

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

A

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.

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12
Q

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

A

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

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13
Q

integrate cosec x cot x

A

-cosec x +C

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14
Q

Integrate sec x tan x

A

sec x +C

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15
Q

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

A

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

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16
Q

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

A

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

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

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17
Q

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

A

put your calculator in radians, dummy

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18
Q

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

A

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

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19
Q

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

A

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

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20
Q

how could you simplify fractions with polynomials to integrate them?

A

split into partial fractions

21
Q

how does integration by substitution work?

A

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

22
Q

what is the equation for integration by parts?

A

∫u dv/dx = uv- ∫v du/dx

23
Q

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

A

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

24
Q

what is the difference between arithmetic and geometric sequences?

A

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

25
what is the formula for ∑r from 1 to n?
1/2 n(n+1)
26
when can you use the binomial expansion approximation formula on (1+x)^n ?
when |x| < 1
27
what do you have to do to (a+bx)^n to approximate it?
turn it into a^n * (1+bx/a)^n
28
what is the equation for arc length?
l= rθ
29
what is the equation for the area of a sector?
A = 1/2 r^2 θ
30
what is the equation for the area of a segment?
A = 1/2 r^2 (θ - sinθ)
31
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) ```
32
how do you prove that a complex continuous function has a root?
f(a) and f(b) have opposite signs
33
what are the 7 methods of integration?
function derivative method, trig identities, u sub, integration by parts, partial fractions, addition formulae, guess method.
34
what is a typical way of solving some equations with sin and cos?
turn it into Rsin(x+α) or Rcos(x+α)
35
what is log y ± log x ?
log (x*/y)
36
what is 1/ loga(b) ?
logb(a)
37
what is loga(b)/ loga(c) ?
logc(b)
38
what is the equation for the Nth term of a geometric sequence?
Un = ar^(n-1)
39
what is the parametric integration formula?
∫ y(t) dx/dt dt
40
what are the trig identities with cos2x?
2cos^2 x ≡ 1 + cos 2x | 2sin^2 x ≡ 1 - cos 2x
41
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
42
integrate e^kx
1/k e^kx +C
43
differentiate e^ f(x)
f'(x) e^ f(x)
44
differentiate a^f(x)
a^ f(x) * f'(x)ln(a) | if a>0
45
integrate a^kx
a^kx / kln(a) +C | if a>0
46
how do you get rid of a modulus sign in an equation?
square it
47
what is the quotient rule when | y = f(x) / g(x)?
dy/dx = [ g(x)f'(x) - f(x)g'(x) ] / g(x)^2
48
what is loga [f(x)] differentiated
1/ xlna *f'(x) Check if x needs to be f(x)
49
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