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Maths > Dr Griffiths > Flashcards

Dr Griffiths Flashcards

1
Q

Define a function

A

A function is a rule which operates on an input and produces a single output

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2
Q

What do you do to the number of a;

“One-to-one” rule?

“Many to one”?

“One to many”?

A

“One-to-one” - Multiply by 1

“Many to one” - Square the numbers

“One to many” - Take the square root

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3
Q

“One to many” is not an exmaple of a ____

Why?

A

Function

Because one input produces more than one output

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4
Q

A function has a ____ and ____

A

domain** and **range

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5
Q

Define domain

A

A set of values we allow the independant variable to take

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6
Q

Define the range

A

The set of y-values in the domain

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7
Q

Functions can also be described…?

Define it

A

Parametrically : One of a set of independent variables that express the coordinates of a point

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8
Q

Real life example of continuity?

A

Can you keep your pen on the paper when drawing the curve?

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9
Q

A function is ____ if it is not continuous

A

discontinuous

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10
Q

In the unit step function,

v(t) = ? if t ≥0

v(t) = 0 if t ? 0

A

v(t) = 1 if t ≥0

v(t) = 0 if t < 0

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11
Q

A function that has a definite pattern repeated at regular intervals is said to be…?

e.g.?

A

Periodic

e.g. y=sinx

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12
Q

A function f(x) is ____ if we can find a number ‘T’ such that

f(x+t) = ? for all x

A

A function f(x) is periodic if we can find a number ‘T’ such that

f(x+t) = f(x) for all x

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13
Q

A function that is symetric about the _-axis is said to be…?

A

y-axis is said to be even

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14
Q

A even function is such that

f(-x) = ? for all of x

A

f(-x) = f(x) for all of x

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15
Q

A odd function is such that

f(-x) = ? for all of x

A

f(-x) = -f(x) for all of x

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16
Q

An exponential function has the form y = ?

A

y = ax

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17
Q

y = 2x - derivative lies ____ the curve

y = 3x - derivative lies ____ the curve

A

y = 2x - derivative lies below the curve

y = 3x - derivative lies above the curve

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18
Q

3! = ?

A

3! = 3x2x1 = 6

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19
Q

ex - exponential ____ as x → ?

e-x - expontential ____ as x → ?

A

ex - exponential growth as x → ∞

e-x - expontential decay as x → ∞

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20
Q

e always…?

A

dominates

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21
Q

If y=ex then x = ?

A

x = ln(y)

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22
Q

If it’s ex + ex then it’s ?

But if its ex - e-x then it’s ?

A

Even

Odd

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23
Q

tanh(x) = ?

coth(x) = ?

sech(x) = ?

cosech(x) = ?

A

tanh(x) = sinh(x)/cosh(x)

coth(x) = 1/tanh(x)

sech(x) = 1/cosh(x)

cosech(x) = 1/sinh(x)

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24
Q

The three different types of stationary points are…?

A
  1. Maximum point
  2. Minimum point
  3. Points of inflection
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25
Q

Maximum and minimum points can either be ____ or ____

A

local** or **global

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26
Q

Minimum point if d2f/dx2 is…?

Maximum point if d2f/dx2 is…?

A

Minimum point if d2f/dx2 is > 0

Maximum point if d2f/dx2 is <0

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27
Q

If d2f/dx2 = 0 …? so we need to?

A

we need more information.

So we need to evaluate the sign of the function f(x) near to the point x=xs

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28
Q

Any change in sign reveals a…?

A

point of inflection

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29
Q

The Maclaurin series always expands about the point…?

A

x=0

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30
Q

Maclaurin series = Taylor series about…?

A

x=0

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31
Q

e = ?

A

e = 1 + 1/1! + 1/2! + 1/3! + … 1/n!

32
Q

The limit of f(x) is _ as x approaches a, and is written as…?

A

The limit of f(x) is L as x approaches. This is written as

limx→a f(x) = L

33
Q

L’Hôpital’s Rule, if

f(x)/g(x) = 0/0

you do what?

A

Just differentiate both lines

f’(x)/g’(x)

34
Q

A sequence is simply…?

A

…a sucession of numbers

35
Q

A series is…?

A

…simply the sum of the terms of the sequence

36
Q

If the sequence of partial sums converges then we say that…??

A

…the series converges

37
Q

What would be the first thing to spot in something like this?

I = ∫ sin(x) cos(x) dx

A

That if you dy/dx sin(x) then you get cos(x) so

d(sin(x))/dx = cos (x)

so sub in the bold part into the previous equation for cos(x)

38
Q

Tricky intergrates can often be simplified by…?

A

…using a suitable substitution

39
Q

The mean value of a function f(x) between points A and B is…?

A
40
Q

The root means square value of a function f(x) between points A and B is…?

A
41
Q

The RMS (____ ____ ____) value is used when…?

A

The RMS (Root mean square) value is used when… determining the strength of voltage supplies

42
Q

What is the area of a circle?

A

πr2

43
Q

A solid rotating about the x-axis,

r = ?

Equation is v =

A

r=y

v =

44
Q

Rotating about the y-axis,

r = ?

Equation v = ?

A

r = x

v =

45
Q

How do you measure the length of a curve? (Eqn)

A

Simply apply pythagoras’ theorem to determine the length of a hypotenuse (kinda)

46
Q

An explicit function has the form y = ?

Whereas an implicit function has the form…

A

Explicit y = f(x)

Implicit f(x,y) = 0

47
Q

y = arcsin(x)

What does x = ?

A

x = sin(y)

48
Q

We can use the idea of implicit differentiation to determine the…

A

…derivative of logarithmic type functions

49
Q

Define Partial differentiation

A

Partitial differentiation is define as the rate of change of a function of multiple variables, with respect to one (or more) independant variables

50
Q

∂ = ?

d = ?

A

∂ = Many

D = One

51
Q

When taking the partial derivative with respect to one independant variable the others are viewed as…?

A

…constants

52
Q

fxx means?

A

Differentiate with respect to x twice

53
Q

Stationary points occur when…?

A

df/dx = 0

54
Q

What would functions of two variables look like?

A

= fxxfyy - f(xy)2

55
Q

If D < 0 = ?

If D > 0 and fxx > 0 = ?

If D > 0 and fxx < 0 = ?

If D = 0

A

If D < 0 = Saddle point

If D > 0 and fxx > 0 = Minimum point

If D > 0 and fxx < 0 = Maximum point

If D = 0 - We need some more info

56
Q

D = ?

f(xy) = ?

A

D = fxxfyy - f(xy)2

f(xy) = 0 (for whatever reason)

57
Q

j = ?

A

√-1

58
Q

j is the…

A

…‘Imaginary number’

59
Q

In general, we write complex numbers in the form;

A

z = 1 +- j

60
Q

The modulus of a complex number is…?

A

|z| = √(a2+b2)

61
Q

Give an example of a complex conjugate pair

A

z = a + bj

z* = a - bj

62
Q

What do you plot complex numbers on?

A

An Argand diagram

63
Q

On an Argand diagram, what is the x and y-axis?

A

x-axis = real part of z

y-axis = imaginary part of z

64
Q

For complex numbers,

x = ?

y = ?

A

x = rcos(ø)

y = rsin(ø)

65
Q

In an argand diagram when plotting complex numbers, what parts of the diagram do you add or minus π?

A

3rd = -π

2nd = +π

ø > π other way round

66
Q

What is Elver’s identity?

A

ejπ + 1 = 0

67
Q

rejø = ?

A

rejø = r(cos(ø) + jsin(ø))

68
Q

What is De Moivre’s theorem?

A

cos(nø) + jsin(nø) = [cos(ø) + jsin(ø)]n

69
Q

In De Moivre’s theorem,

cos(nø) + jsin(nø) = ?

A

cos(nø) + jsin(nø) = [cos(ø) + jsin(ø)]n

70
Q

De Moivre’s theorem is particulary useful for…?

A

…finding roots

71
Q

Non linear homogeneous equations = ?

Linear non-homegneous = ?

A

NLH = 0

LNH = x2

72
Q

The function I(x) is known as…?

A

The intergrating factor

73
Q

For first ODE’s, what do we want it to look like?

A

dy/dx + yp(x) = q(x)

74
Q

I(x) = ?

A
75
Q

In first ODE’s, v(x) = ?

A
76
Q
A

Maths (4 decks)

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