Mathematical Modelling (Week 1-4)

Question 1

Define a function

Answer 1

A rule which operates on one or more inputs to produce a single output

E.g one to one
Many to one

Question 2

Inputs of a function are also called…

Answer 2

The arguments of the function

Question 3

Define the Domain

Answer 3

Set of input values for independent variable ‘x’

Question 4

Define the Range:

Answer 4

Set of output values for dependant variable ‘y’

Question 5

A function f(x) is even if:

Answer 5

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

The graph is symmetric at the y-axis

Question 6

A function f(x) is odd if:

Answer 6

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

Graph symmetric about origin, through rotating of 180⁰ clockwise / anticlockwise

Question 7

Periodic function:

Answer 7

Graph that repeats itself every P units where P = period of function

f(x) = f(x + P)

Question 8

What kind of functions are inverse functions?

Answer 8

One to one
The range of f(x) becomes the input/ domain.

And the domain becomes the output

Question 9

How to find the inverse function

Answer 9

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.

Question 10

Composite Functions

Answer 10

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))

Question 11

Can the composite function f(g(x)) = g(f(x))

Answer 11

Yes only when f(x) and g(x) are inverses of each other

Question 12

Differentiation by first principle equation

Answer 12

Lim h–> 0. f(x + h) - f(x) / h

Question 13

In trigonometric functions the angle is in…

Answer 13

Radians

Question 14

What is the derivative of the function: constant

Answer 14

0

Question 15

What is the derivative of the function: x

Answer 15

1

Question 16

What is the derivative of the function: kx

Answer 16

k

Question 17

What is the derivative of the function: x^n

Answer 17

nx^n-1

Question 18

What is the derivative of the function: kx^n

Answer 18

knx^n-1

Question 19

What is the derivative of the function: e^x

Answer 19

e^x

Question 20

What is the derivative of the function: e^kx

Answer 20

ke^kx

Question 21

What is the derivative of the function: ln x

Answer 21

1 / x

Question 22

What is the derivative of the function: ln (kx)

Answer 22

k / x

Question 23

What is the derivative of the function: sin x

Answer 23

cos x

Question 24

What is the derivative of the function: sin (kx)

Answer 24

k cos(kx)

Question 25

What is the derivative of the function: sin(kx+c)

Answer 25

k cos(kx+c)

Question 26

What is the derivative of the function: cos x

Answer 26

-sin x

Question 27

What is the derivative of the function: cos kx

Answer 27

-k sin(kx)

Question 28

What is the derivative of the function: cos (kx +c)

Answer 28

-k sin(kx+c)

Question 29

What is the derivative of the function: tan x

Answer 29

sec^2 x

Question 30

What is the derivative of the function: tan kx

Answer 30

k sec^2 (kx)

Question 31

What is the derivative of the function: tan (kx +c)

Answer 31

k sec^2 (kx+c)

Question 32

Product Rule:

Answer 32

Used when you have 2 functions If y= f(x)g(x) dy/dx = f'(x)g(x) + g'(x)f(x)

Question 33

How do you apply the product Rule to three functions?

Answer 33

-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 +

Question 34

Chain Rule:

Answer 34

y= f(g(x)) dy/dx = df/dg x dg/dx Differentiation by interpretation

Question 35

How to find the derivative of a modulus

Answer 35

Rewrite the modules E.g y= |x| y= (x^2)^1/2

Question 36

why do we use implicit differentiation, and when

Answer 36

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

Question 37

Logarithmic implicit differentiation (therefore not using quotient rule)

Answer 37

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)

Question 38

points on the function y=f(x) at which dy/dx = 0 are called...

Answer 38

stationary points

Question 39

if the second derivative is greater than 0 then you have a...

Answer 39

local minimum

Question 40

if the second derivative is less than 0 then you have a...

Answer 40

local maximum

Question 41

how can you tell when you have a point of infliction?

Answer 41

dy/dx does not change sign on either side of the stationary point

Question 42

if the second derivative is = 0 does that mean its a point of inflection always?

Answer 42

no

Question 43

Partial derivatives of multivariable functions (3D graphs)

Answer 43

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

Question 44

where can you find the zero gradient, along which lines for partial derivatives of multi variable functions (3D graphs)

Answer 44

along the lines (0,y) and (x,0) note you have multiple stationary points.

Question 45

Maclaurin Series:

Answer 45

Expansion of a differentiable function f(x) about x = 0

Question 46

Maclaurin Series: Equation

Answer 46

f(x) = f(0) + x f'(0) + x^2f''(0) / 2! + x^3f'''(0) / 3! + ... + x^kf^(k)(0) / k! + ...

Question 47

What does the Maclaurin and Taylor Series help express?

Answer 47

These series help express any "well behaved" (continuous and differentiable) function as a polynomial expansion, that will converge

Question 48

Taylor Series:

Answer 48

Expansion of a differentiable function f(x) about x =a, i e this is a generalisation of the Maclaurin series

Question 49

Taylor Series: Equation

Answer 49

f(x) = f(a) + (x-a) f'(a) + (x-a)^2f''(a) / 2! + (x-a)^3f'''(a) / 3! + ... + (x-a)^kf^(k)(a) / k! + ...

Question 50

What two points are noticeable when plotting a function on a graph using the Maclaurin series

Answer 50

Further you move from x=0, larger errors creep in, the greater the number of terms included, the better the accuracy of the approximation

Question 51

What is integration?

Answer 51

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

Question 52

Indefinite integration

Answer 52

Without limits Always has constant of integration ("c")

Question 53

Definite Integration

Answer 53

Evaluated between limits No constant of integration ("c")

Question 54

Integrate (ax + b)^n

Answer 54

1/a . ((ax + b) ^n+1 / n+1 ) + c

Question 55

Integrate 1/x

Answer 55

ln |x| + c

Question 56

Integrate 1 / ax + b

Answer 56

1/a ( ln |ax + b|) + c

Question 57

Integrate e^x

Answer 57

e^x + c

Question 58

Integrate a^x

Answer 58

(a^x / ln a ) + c

Question 59

Integrating by parts equation:

Answer 59

For integrating the product of two functions, set f(x) = u and g(x) = v' and use: Integral uv' = uv . Integral vu'

Question 60

Use ILATE starting left to right to choose u and then v':

Answer 60

I, inverse trig L, logarithmic A, algebraic T, trig E, exponential

Question 61

For the Taylor series do you get a larger error by omitting a more recent or further term?

Answer 61

The more recent the higher the error

Question 62

Integration by reduction

Answer 62

Integration technique which involves expressing the integral in the form of recurrence relation

Question 63

How to carry out integration by substatution

Answer 63

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

Question 64

What do you do for double and triple integration?

Answer 64

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.