Core A level Flashcards
What is the exponential form of a complex number?
z = re^iθ where
r = |z|
and θ = argz
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
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
If z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, what does z₁*z₂ =
z₁z₂ = r₁r₂e^i(θ₁+θ₂)
What is de Moivre’s theorem?
zⁿ = rⁿ(cosnθ + isinnθ)
What can you quickly use, to prove de Moivre’s theorem for all n?
Exponential form
z + 1/z =
2cosθ
z - 1/z =
2isinθ
zⁿ + 1/zⁿ =
2cosnθ
zⁿ - 1/zⁿ =
2isin(nθ)
If zⁿ = w, what is the general solution to z, in modulus-argument form?
z = r(cos(θ+2kπ) + isin(θ+2kπ))
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
When is the Maclaurin series valid?
When all the f(0), f’(0), f’‘(0), …, fʳ(0) all have finite values
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]
If an improper integral exits, what is it described as?
If it doesn’t exist?
Exists: Convergent
Doesn’t exist: Divergent
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
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
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]
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]
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]
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)
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)
Polar co-ordinates:
Write x in terms of θ
rcosθ = x
Polar co-ordinates:
Write y in terms of θ
rsinθ = y
Polar co-ordinates:
Write r in terms of x and y
x² + y² = r
Polar co-ordinates:
Write θ in terms of x and y
θ = arctan (y/x)
What is the origin called in polar coordinates?
The pole
Polar co-ordinates:
What is the initial line?
Usually the positive x-axis
Polar co-ordinates:
What is the form of the coordinates? Eg, what are a and b in (a, b)
(r, θ)
Polar co-ordinates:
r = a
circle with centre O and radius a
Polar co-ordinates:
What shape is α = θ
Half-line through O, making an angle α with the initial line
Polar co-ordinates:
What shape is r = aθ
Spiral starting at O
Polar co-ordinates:
r = a ( p + qcosθ )
When is the curve convex (eg, egg shaped)?
p ≥ 2q
Polar co-ordinates:
r = a ( p + qcosθ )
When is the curve concave (at θ = π) (eg, dimple shaped)?
q ≤ p < 2q
Polar co-ordinates:
How do you find a tangent parallel to the initial line?
dy/dθ = 0
Polar co-ordinates:
How do you find a tangent perpendicular to the initial line?
dx/dθ = 0
sinh x =
(eˣ - e⁻ˣ)/2
cosh x =
(eˣ + e⁻ˣ)/2
tanh x =
(eˣ - e⁻ˣ)/(eˣ + e⁻ˣ)
or
( e²ˣ - 1 )/ ( e²ˣ + 1 )
sinh (-a) =
-sinh a
cosh (-a) =
cosh a
arsinh x =
ln( x + √(x² + 1) )
arcosh x =
ln( x + √(x² - 1) ), x ≥ 1
artanh x =
0.5 ln( (1 + x)/(1 - x), |x|<1
What is the hyperbolic identity that equals 1?
cosh² A - sinh² A = 1
What is the sine addition formula for hyperbolic functions?
sinh ( A ± B ) = sinh A cosh B ± cosh A sinh B
What is the cosine addition formula for hyperbolic functions?
cosh ( A ± B ) = cosh A cosh B ± sinh A sinh B
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
What does sinh x differentiate to?
cosh x
What does cosh x differentiate to?
sinh x
What does tanh x differentiate to?
sech² x
What does arsinh x differentiate to?
1/ √(x² + 1)
What does arcosh x differentiate to?
1/ √(x² - 1), x > 1
What does artanh x differentiate to?
1 / (1 - x²), |x| < 1
Differential equations:
What is separation of variables?
If dy/dx = f(x) * g(y)
then, ∫ 1/g(y) dy = ∫ f(x) dx
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
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
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)
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
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
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
Differential equations:
What is the complementary function?
It is the general solution to the homogeneous bit of the second-order non-homogeneous equation.
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
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
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
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
What is the algebraic version of the definition of simple harmonic motion?
x’’ = -ω²x
Simple harmonic motion:
write acceleration in terms of dv/dx
x’’ = v dv/dx
Simple harmonic motion: what is ω?
Angular velocity
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
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
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
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
What are coupled first-order linear differential equations?
dx/dt = ax + by + f(t) dy/dt = cx + dy + g(t)
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
