General Stuff

Question 1

What is the exponential form of a complex number?

Answer 1

Z = re^iθ
Where:
r is modulus of z
θ is argument of z

Question 2

How do you get cos(nθ) using de Moivre’s?

Answer 2

Expanding (cosθ + isinθ)^6
Choose real terms
Change sin terms into cos using identity

Question 3

How do you get cos^n (θ) using de Moivre’s?

Answer 3

(2cosθ)^n = (z + 1/z)^n
Binomial expansion for RHS
Pair up +- powers and change into cos
Divide by 2^n

Question 4

Mean value of a function f(x) over interval [a,b]

Answer 4

1/(b-a) x integral between b and a for f(x)

Question 5

Mean value of f(x) is f bar

So what is mean value of f(x) + k

Answer 5

F bar + k

Question 6

Mean value of f(x) is f bar

So what is mean value of kf(x)

Answer 6

Kf bar

Question 7

How do you prove the derivative of arcsin(x)?

Answer 7

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

Question 8

What is the domain and range of arcsin(x)?

Answer 8

Domain: -1,1

Range -π/2,π/2

Question 9

What is the domain and range of arccos(x)?

Answer 9

Domain: -1,1
Range: 0,π

Question 10

What is the domain and range of arctan(x)?

Answer 10

Domain: -infinity,infinity
Range: -π/2,π/2

Question 11

For volumes of revolution

When rotating around the x-axis, what do you integrate?

Answer 11

πy^2

Question 12

For volumes of revolution

When rotating around the y-axis, what do you integrate?

Answer 12

πx^2

Question 13

When integrating a volume between two lines, what makes it easier?

Answer 13

Function above - function below

Question 14

How do you integrate for volume of revolution when given parametric equations?

Answer 14

Square y
Differentiate x with respect to t
Work out t values at given x values
Then change into proper format for integration

Question 15

In polar coordinates, what is x?

Answer 15

X = rcosθ

Question 16

In polar coordinates, what is y?

Answer 16

Y = rsinθ

Question 17

In polar coordinates, what is r^2?

Answer 17

r^2 = x^2 + y^2

Question 18

In polar coordinates what is θ?

Answer 18

θ=arctan(y/x)

Question 19

For sketching polar curves, what is r=a?

Answer 19

A circle with centre O and radius a

Question 20

For sketching polar curves, what is θ=α?

Answer 20

A half line through O and makes angle α with x axis

Question 21

For sketching polar curves, what is r=aθ?

Answer 21

A spiral starting at O

Question 22

What can be predicted about a polar curve in the form r=acos(nθ) or r=asin(nθ)?

Answer 22

They will have n loops symmetrically arranged around O

Question 23

How do you find the area of a sector for a polar curve, between angles α and β?

Answer 23

Integrate 1/2 r^2 dθ between α and β

Question 24

How do you find a tangent parallel to the initial line (polar)?

Answer 24

Set dy/dθ = 0

Question 25

How do you find a tangent perpendicular to initial line (polar)?

Answer 25

Set dx/dθ = 0

Question 26

What is exponential form of sinh(x)?

Answer 26

(e^x - e^-x)/2

Question 27

What is exponential form of cosh(x)?

Answer 27

(e^x + e^-x)/2

Question 28

What is exponential form of tanh(x)?

Answer 28

(e^2x - 1)/(e^2x + 1)

Question 29

What does sinh(x) look like?

Answer 29

A cubic graph

Question 30

What does cosh(x) look like?

Answer 30

A chain hanging down between two points

Question 31

First step of solving first order differential equation: dy/dx + P(x)y = Q(x)

Answer 31

Multiply every term by integrating factor
e^integral(P(x) dx)
With integral ignore +c

Question 32

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

Answer 32

y = Ae^αx + Be^βx

Where α and β are the roots

Question 33

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

Answer 33

y = (A + Bx)e^αx

Where α is the repeated root

Question 34

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

Answer 34

y = e^px(Acos(qx) + Bsin(qx))

Where roots are p+/-qi

Question 35

What is a particular integral?

Answer 35

A function which satisfies the original differential equation

Question 36

What is simple harmonic motion?

Answer 36

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

Question 37

For a particle moving with damped harmonic motion, what is the equation?

Answer 37

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

Question 38

For a particle moving with forced harmonic motion, what is the equation?

Answer 38

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

Question 39

How can you solve coupled first order linear differential equations? (Think rabbits and foxes question)

Answer 39

Eliminating one of the dependent variables to form a second order differential equation