Mathematical Modelling (Week 1-4) Flashcards
Define a function
A rule which operates on one or more inputs to produce a single output
E.g one to one
Many to one
Inputs of a function are also called…
The arguments of the function
Define the Domain
Set of input values for independent variable ‘x’
Define the Range:
Set of output values for dependant variable ‘y’
A function f(x) is even if:
f(-x) = f(x) for all x
The graph is symmetric at the y-axis
A function f(x) is odd if:
f(-x) = -f(x) for all x
Graph symmetric about origin, through rotating of 180⁰ clockwise / anticlockwise
Periodic function:
Graph that repeats itself every P units where P = period of function
f(x) = f(x + P)
What kind of functions are inverse functions?
One to one
The range of f(x) becomes the input/ domain.
And the domain becomes the output
How to find the inverse function
Set y = equation in terms of x
Rearrange to get x = equation in terms of y.
Then switched the y values with the x value.
Composite Functions
The composition of two function involves taking one of the functions as an input for the other to generate a new combined function.
Format: f.g(x) or f(g(x))
Can the composite function f(g(x)) = g(f(x))
Yes only when f(x) and g(x) are inverses of each other
Differentiation by first principle equation
Lim h–> 0. f(x + h) - f(x) / h
In trigonometric functions the angle is in…
Radians
What is the derivative of the function: constant
0
What is the derivative of the function: x
1
What is the derivative of the function: kx
k
What is the derivative of the function: x^n
nx^n-1
What is the derivative of the function: kx^n
knx^n-1
What is the derivative of the function: e^x
e^x
What is the derivative of the function: e^kx
ke^kx
What is the derivative of the function: ln x
1 / x
What is the derivative of the function: ln (kx)
k / x
What is the derivative of the function: sin x
cos x
What is the derivative of the function: sin (kx)
k cos(kx)
What is the derivative of the function: sin(kx+c)
k cos(kx+c)
What is the derivative of the function: cos x
-sin x
What is the derivative of the function: cos kx
-k sin(kx)
What is the derivative of the function: cos (kx +c)
-k sin(kx+c)
What is the derivative of the function: tan x
sec^2 x
What is the derivative of the function: tan kx
k sec^2 (kx)
What is the derivative of the function: tan (kx +c)
k sec^2 (kx+c)
Product Rule:
Used when you have 2 functions
If y= f(x)g(x)
dy/dx = f’(x)g(x) + g’(x)f(x)
How do you apply the product Rule to three functions?
-Rearrange the 3 function to end up with 2 only, e.g using double angle formula.
- Let y = f(x) { g(x)h(x) }
dy/dx = f’(x) { g(x)h(x) } + { g(x)h(x) }’ + f(x)
=f’gh + (g’h +
Chain Rule:
y= f(g(x))
dy/dx = df/dg x dg/dx
Differentiation by interpretation
How to find the derivative of a modulus
Rewrite the modules
E.g y= |x| y= (x^2)^1/2
why do we use implicit differentiation, and when
differentiate function, not in the form y= f(x), sometimes hard to express y as the subject.
involves differentiating both sides, and resolving to find dy/dx
Logarithmic implicit differentiation (therefore not using quotient rule)
consider y = f(x) / g(x), find dy/dx
take log of both sides
ln y = ln f - ln g
now differentiate both sides (implicit)
points on the function y=f(x) at which dy/dx = 0 are called…
stationary points
if the second derivative is greater than 0 then you have a…
local minimum
if the second derivative is less than 0 then you have a…
local maximum
how can you tell when you have a point of infliction?
dy/dx does not change sign on either side of the stationary point
if the second derivative is = 0 does that mean its a point of inflection always?
no
Partial derivatives of multivariable functions
(3D graphs)
derivative of function f(x,y) with respect to its independent variables x and y, found by differentiating f(x,y) with respect to the corresponding variable, while treating the other variable as a constant
where can you find the zero gradient, along which lines for partial derivatives of multi variable functions (3D graphs)
along the lines (0,y) and (x,0)
note you have multiple stationary points.
Maclaurin Series:
Expansion of a differentiable function f(x) about x = 0
Maclaurin Series: Equation
f(x) = f(0) + x f’(0) + x^2f’‘(0) / 2! + x^3f’’‘(0) / 3! + … + x^kf^(k)(0) / k! + …
What does the Maclaurin and Taylor Series help express?
These series help express any “well behaved” (continuous and differentiable) function as a polynomial expansion, that will converge
Taylor Series:
Expansion of a differentiable function f(x) about x =a, i e this is a generalisation of the Maclaurin series
Taylor Series: Equation
f(x) = f(a) + (x-a) f’(a) + (x-a)^2f’‘(a) / 2! + (x-a)^3f’’‘(a) / 3! + … + (x-a)^kf^(k)(a) / k! + …
What two points are noticeable when plotting a function on a graph using the Maclaurin series
Further you move from x=0, larger errors creep in, the greater the number of terms included, the better the accuracy of the approximation
What is integration?
Technique or tool used to calc area under the curve of a function (between 2 points).
It can also be viewed as an inverse / reverse process of differentiation.
Remember the constant of integration
Indefinite integration
Without limits
Always has constant of integration (“c”)
Definite Integration
Evaluated between limits
No constant of integration (“c”)
Integrate (ax + b)^n
1/a . ((ax + b) ^n+1 / n+1 ) + c
Integrate 1/x
ln |x| + c
Integrate 1 / ax + b
1/a ( ln |ax + b|) + c
Integrate e^x
e^x + c
Integrate a^x
(a^x / ln a ) + c
Integrating by parts equation:
For integrating the product of two functions, set f(x) = u and g(x) = v’ and use:
Integral uv’ = uv . Integral vu’
Use ILATE starting left to right to choose u and then v’:
I, inverse trig
L, logarithmic
A, algebraic
T, trig
E, exponential
For the Taylor series do you get a larger error by omitting a more recent or further term?
The more recent the higher the error
Integration by reduction
Integration technique which involves expressing the integral in the form of recurrence relation
How to carry out integration by substatution
Sub u as part of the complex integral
Differentiation u in terms of x
Rearrange to get dx = something in terms of du, sub instead of dx
Integrate
Change u by substituting it’s corresponding value of x
What do you do for double and triple integration?
This is when you are working out the area of 2 and 3 dimensional graphs / shapes.
Just integrate in respect to one variable each time and treat the rest as constants for the integral range of that variable, then sub in the two values and repeat process for next variable.
