General Stuff Flashcards
What is the exponential form of a complex number?
Z = re^iθ
Where:
r is modulus of z
θ is argument of z
How do you get cos(nθ) using de Moivre’s?
Expanding (cosθ + isinθ)^6
Choose real terms
Change sin terms into cos using identity
How do you get cos^n (θ) using de Moivre’s?
(2cosθ)^n = (z + 1/z)^n
Binomial expansion for RHS
Pair up +- powers and change into cos
Divide by 2^n
Mean value of a function f(x) over interval [a,b]
1/(b-a) x integral between b and a for f(x)
Mean value of f(x) is f bar
So what is mean value of f(x) + k
F bar + k
Mean value of f(x) is f bar
So what is mean value of kf(x)
Kf bar
How do you prove the derivative of arcsin(x)?
Y = arcsin(x) Sin(y) = x Differentiate for dx/dy Flip to get dy/dx Use trig identity to show cos is sqrt(1 - sin^2) Sin(y)=x So sub that in
What is the domain and range of arcsin(x)?
Domain: -1,1
Range -π/2,π/2
What is the domain and range of arccos(x)?
Domain: -1,1
Range: 0,π
What is the domain and range of arctan(x)?
Domain: -infinity,infinity
Range: -π/2,π/2
For volumes of revolution
When rotating around the x-axis, what do you integrate?
πy^2
For volumes of revolution
When rotating around the y-axis, what do you integrate?
πx^2
When integrating a volume between two lines, what makes it easier?
Function above - function below
How do you integrate for volume of revolution when given parametric equations?
Square y
Differentiate x with respect to t
Work out t values at given x values
Then change into proper format for integration
In polar coordinates, what is x?
X = rcosθ
In polar coordinates, what is y?
Y = rsinθ
In polar coordinates, what is r^2?
r^2 = x^2 + y^2
In polar coordinates what is θ?
θ=arctan(y/x)
For sketching polar curves, what is r=a?
A circle with centre O and radius a
For sketching polar curves, what is θ=α?
A half line through O and makes angle α with x axis
For sketching polar curves, what is r=aθ?
A spiral starting at O
What can be predicted about a polar curve in the form r=acos(nθ) or r=asin(nθ)?
They will have n loops symmetrically arranged around O
How do you find the area of a sector for a polar curve, between angles α and β?
Integrate 1/2 r^2 dθ between α and β
How do you find a tangent parallel to the initial line (polar)?
Set dy/dθ = 0
How do you find a tangent perpendicular to initial line (polar)?
Set dx/dθ = 0
What is exponential form of sinh(x)?
(e^x - e^-x)/2
What is exponential form of cosh(x)?
(e^x + e^-x)/2
What is exponential form of tanh(x)?
(e^2x - 1)/(e^2x + 1)
What does sinh(x) look like?
A cubic graph
What does cosh(x) look like?
A chain hanging down between two points
First step of solving first order differential equation: dy/dx + P(x)y = Q(x)
Multiply every term by integrating factor
e^integral(P(x) dx)
With integral ignore +c
For the general solution to the second order differential equation:
a d^2 y/dx^2 + b dx/dy + c = 0
What is the general solution form when:
b^2 > 4ac
y = Ae^αx + Be^βx
Where α and β are the roots
For the general solution to the second order differential equation:
a d^2 y/dx^2 + b dx/dy + c = 0
What is the general solution form when:
b^2 = 4ac
y = (A + Bx)e^αx
Where α is the repeated root
For the general solution to the second order differential equation:
a d^2 y/dx^2 + b dx/dy + c = 0
What is the general solution form when:
b^2 < 4ac
y = e^px(Acos(qx) + Bsin(qx))
Where roots are p+/-qi
What is a particular integral?
A function which satisfies the original differential equation
What is simple harmonic motion?
Motion in which the acceleration of a particle P is always towards a fixed point O on the line of motion of P. The acceleration is proportional to the displacement from O
For a particle moving with damped harmonic motion, what is the equation?
d^2 x/dt^2 + k dx/dt + ω^2 x = 0
x is displacement from a fixed point at time t
k and ω^2 are positive constants
For a particle moving with forced harmonic motion, what is the equation?
d^2 x/dt^2 + k dx/dt + ω^2 x = f(t)
x is displacement from a fixed point at time t
k and ω^2 are positive constants
How can you solve coupled first order linear differential equations? (Think rabbits and foxes question)
Eliminating one of the dependent variables to form a second order differential equation
