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Flashcards in Core A level Deck (75)
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1

What is the exponential form of a complex number?

z = re^iθ where
r = |z|
and θ = argz

2

Describe the proof for writing a complex number in exponential form

z = r(cosθ + isinθ)
Find the Maclaurin series expansion of cosθ, sinθ, and eˣ
subsitute x = iθ into the e-series, separate out the complex and real parts
you will get e^iθ = cosθ + isinθ

3

How do you go from the proof of exponential form of complex numbers to Euler's identity?

e^iθ = cosθ + isinθ
θ = π
e^iθ = -1 + 0
e^iθ + 1 = 0

4

If z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, what does z₁*z₂ =

z₁*z₂ = r₁*r₂e^i(θ₁+θ₂)

5

What is de Moivre's theorem?

zⁿ = rⁿ(cosnθ + isinnθ)

6

What can you quickly use, to prove de Moivre's theorem for all n?

Exponential form

7

z + 1/z =

2cosθ

8

z - 1/z =

2isinθ

9

zⁿ + 1/zⁿ =

2cosnθ

10

zⁿ - 1/zⁿ =

2isin(nθ)

11

If zⁿ = w, what is the general solution to z, in modulus-argument form?

z = r(cos(θ+2kπ) + isin(θ+2kπ))

12

Describe what the roots of a complex number look like on an argand diagram

The roots lie at the vertices of a regular n-gon with its centre at the origin
n = no. of roots

13

When is the Maclaurin series valid?

When all the f(0), f'(0), f''(0), ..., fʳ(0) all have finite values

14

When is an integral improper?

When:
one or both of the limits is infinite
f(x) is undefined at x = a, x = b, or any other point in the interval [a ,b]

15

If an improper integral exits, what is it described as?
If it doesn't exist?

Exists: Convergent
Doesn't exist: Divergent

16

If you have an integral, where the limits are ± ∞, how can you tell whether the integral is convergent or divergent?

Split the integral into two with limits (∞, c) and (c, -∞) where c is a number
If both integrals converge, then so does the original.
If either diverges, then the original is divergent

17

How do you calculate the mean value of a function? (In the interval [a, b])

= 1/(b-a) ∫ f(x) dx
where the integral limits are b and a

18

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then...
What is the mean value of f(x) + k?

f + k over the interval [a,b]

19

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then...
What is the mean value of -f(x)?

-f over the interval [a,b]

20

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then...
What is the mean value of kf(x)?

kf over the interval [a,b]

21

x = f(t)
y = g(t)
What is the volume of the solid that is generated when the parametric curve is rotated about the x-axis, between x = a and x = b, through 2π radians?

Volume = π ∫ y² dx (between x=b, and x = a)
= π ∫ y² dx/dt dt (between t=p and t=q)

where a = f(p) and b = f(q)

22

x = f(t)
y = g(t)
What is the volume of the solid that is generated when the parametric curve is rotated about the y-axis, between y = a and y = b, through 2π radians?

Volume = π ∫ x² dy (between y = b, and y = a)
= π ∫ x² dy/dt dt (between t = p and t = q)

where a = f(p) and b = f(q)

23

Polar co-ordinates:
Write x in terms of θ

rcosθ = x

24

Polar co-ordinates:
Write y in terms of θ

rsinθ = y

25

Polar co-ordinates:
Write r in terms of x and y

x² + y² = r

26

Polar co-ordinates:
Write θ in terms of x and y

θ = arctan (y/x)

27

What is the origin called in polar coordinates?

The pole

28

Polar co-ordinates:
What is the initial line?

Usually the positive x-axis

29

Polar co-ordinates:
What is the form of the coordinates? Eg, what are a and b in (a, b)

(r, θ)

30

Polar co-ordinates:
r = a

circle with centre O and radius a

31

Polar co-ordinates:
What shape is α = θ

Half-line through O, making an angle α with the initial line

32

Polar co-ordinates:
What shape is r = aθ

Spiral starting at O

33

Polar co-ordinates:
r = a ( p + qcosθ )
When is the curve convex (eg, egg shaped)?

p ≥ 2q

34

Polar co-ordinates:
r = a ( p + qcosθ )
When is the curve concave (at θ = π) (eg, dimple shaped)?

q ≤ p < 2q

35

Polar co-ordinates:
How do you find a tangent parallel to the initial line?

dy/dθ = 0

36

Polar co-ordinates:
How do you find a tangent perpendicular to the initial line?

dx/dθ = 0

37

sinh x =

(eˣ - e⁻ˣ)/2

38

cosh x =

(eˣ + e⁻ˣ)/2

39

tanh x =

(eˣ - e⁻ˣ)/(eˣ + e⁻ˣ)
or
( e²ˣ - 1 )/ ( e²ˣ + 1 )

40

sinh (-a) =

-sinh a

41

cosh (-a) =

cosh a

42

arsinh x =

ln( x + √(x² + 1) )

43

arcosh x =

ln( x + √(x² - 1) ), x ≥ 1

44

artanh x =

0.5 ln( (1 + x)/(1 - x), |x|<1

45

What is the hyperbolic identity that equals 1?

cosh² A - sinh² A = 1

46

What is the sine addition formula for hyperbolic functions?

sinh ( A ± B ) = sinh A cosh B ± cosh A sinh B

47

What is the cosine addition formula for hyperbolic functions?

cosh ( A ± B ) = cosh A cosh B ± sinh A sinh B

48

What is Osborne's rule?

Whenever converting between normal trig identites and hyperbolic ones:
replace cos A by cosh A
and replace sin B by sinh B
however:
replace any product of two sin terms by minus the product of the two sinh terms
eg, sin² A would go to - sinh² A

49

What does sinh x differentiate to?

cosh x

50

What does cosh x differentiate to?

sinh x

51

What does tanh x differentiate to?

sech² x

52

What does arsinh x differentiate to?

1/ √(x² + 1)

53

What does arcosh x differentiate to?

1/ √(x² - 1), x > 1

54

What does artanh x differentiate to?

1 / (1 - x²), |x| < 1

55

Differential equations:
What is separation of variables?

If dy/dx = f(x) * g(y)
then, ∫ 1/g(y) dy = ∫ f(x) dx

56

Differential equations:
How do you solve first order differential equations?

Write in the form dy/dx + P(x)y = Q(x)
then multiply by the integrating factor; e^ ∫ P(x) dx

57

Differential equations:
How do you solve second-order, homogeneous differential equations?

Find the roots of the auxiliary equation, and write in the correct form, depending on whether there is one root, two or complex

58

Differential equations:
What is the auxiliary equation?

am² + bm + c = 0
where a, b, and c are the coefficients of the derivatives (eg, a is the coefficient of the second derivative)

59

Differential equations:
Auxiliary equation, if b²- 4ac > 0, what is the general solution to the differential equation?

y = Aeᵃˣ + Beᵇˣ
where a, and b are the roots of your auxiliary equation

60

Differential equations:
Auxiliary equation, if b²- 4ac = 0, what is the general solution to the differential equation?

y = (A + Bx) eᵃˣ
where a is the root of your auxiliary equation

61

Differential equations:
Auxiliary equation, if b²- 4ac < 0, what is the general solution to the differential equation?

y = eᵖˣ ( Acos qx + Bsin qx) where p ± qi is the solution to the auxiliary equation

62

Differential equations:
What is the complementary function?

It is the general solution to the homogeneous bit of the second-order non-homogeneous equation.

63

Differential equations:
What is the particular interval?

It is a function which satisfies the original differential equation
When solving second-order non-homogeneous differential equations, it's the function of x on the other side of the equal sign to the differential equation

64

Differential equations:
What are the particular intervals for these functions?
p
p + qx
p + qx + rx²
peᵏˣ
pcosωx + qsinωx

p = λ
p + qx = λ + μx
p + qx + rx² = λ + μx + νx²
peᵏˣ = λeᵏˣ
pcosωx + qsinωx = λcosωx + μsinωx

65

Differential equations:
How do solve second-order non-homogeneous differential equations?

Solve the corresponding homo. equation to find the complementary function (C.F)
Choose a particular integral (P.I), and substitute into the original equation to the find the value of any coefficients in the P.I
The general solution = C.F + P.I

66

What is simple harmonic motion?

Motion in which the acceleration of the particle P is always towards a fixed point O on the line of motion of P
The acceleration is proportional to the displacement of P from O
Where O = the centre of oscillation

67

What is the algebraic version of the definition of simple harmonic motion?

x'' = -ω²x

68

Simple harmonic motion:
write acceleration in terms of dv/dx

x'' = v dv/dx

69

Simple harmonic motion: what is ω?

Angular velocity

70

What is the equation for a particle moving with damped harmonic motion?

d²x/dt² + k dx/dt + ω²x = 0
where k and ω² are positive constants

71

Describe (in terms of k and ω) when a particle is being heavily, critically or lightly damped

Heavily: k² > 4ω²
Critically: k² = 4ω²
Lightly: k² < 4ω²

72

What is the equation for a particle moving with forced harmonic motion?

d²x/dt² + k dx/dt + ω²x = f(t)
where k and ω² are positive constants

73

How do you solve coupled first-order linear differential equations?

By eliminating one of the dependent variables to form a second-order differential equation
Eg. dx/dt = ax + by + f(t)
dy/dt = cx + dy + g(t)

differentiate the top equation, then substitute the second equation in for dy/dt

74

What are coupled first-order linear differential equations?

dx/dt = ax + by + f(t)
dy/dt = cx + dy + g(t)

75

Coupled first-order differential equations:
dx/dt = ax + by + f(t)
dy/dt = cx + dy + g(t)

if f(t) and g(t) are both zero, what is the system said to be?

Homogeneous