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