Monday, 16 June 2014

Integration - Definite Integral vs Indefinite Integral

Definite vs Indefinite integral

Example

Some properties of definite integrals

Integration - Formulae

Introduction
Integration is the reverse of differentiation.

If you differentiate A and get B  --> Means if you integrate B you get back A!
E.g. if you differentiate y and get dy/dx ---> if you integrate dy/dx, you get back y!


Formulae




















Sunday, 15 June 2014

Differentiation - Maximum and Minimum

The gradient at the maximum and minimum point is 0. Thus, to find maximum and minimum, these are the steps required.

Step 1: If you need to find maximum or minimum value of V, you first need an equation involving V.
Step 2: Differentiate the equation, to find e.g. dV/dx.
Step 3: Equate dV/dx to 0, and find x.
Step 4: Substitute x into equation of V (from step 1) to find the stationary value of V.
Step 5: To determine maximum or minimum, differentiate dV/dx to find d2V/dx2. Substitute in x from step 4.
If d2V/dx2 > 0, V is minimum.
If d2V/dx< 0, V is maximum.


Example question:
Find the maximum volume of the tank, if its volume, V, is given by the following equation:


Answers:

Differentiation - Rate of change

What is rate of change?
It refers to the change of a physical quantity with time. One good example is speed. 
Speed is the change in distance per unit time. If we let distance be s, and let time be t. In Mathematical terms, the rate of change of s with respect to t can be written as:

What is the units of ds/dt? Since distance, s, has a unit of metre, and time, t, has a unit of seconds, then ds/dt has units of m/s.


Chain Rule

If you know the rate of change of x with time (dx/dt),and need to find the rate of change of V with time (dV/dt), you need an equation with V and x, and subsequently differentiate it to find dV/dx. Using the following equation, dV/dt can be found. 

Differentiation: Tangent and Normal

A tangent is a line that touches the curve.

A normal is a line that is perpendicular to the normal.

Remember this formula:
(gradient of tangent) x (gradient of normal) = -1





How to find gradient of tangent at x = a?
1) Differentiate y (find dy/dx)
2) substitute x = a into dy/dx to find gradient of tangent.

How to find gradient of normal at x = a?
1) Differentiate y (find dy/dx)
2) substitute x = a into dy/dx to find gradient of tangent.
3) find gradient of normal by using the following formulae: -1 / (gradient of tangent)

How to find equation of tangent at x = a?
1) Differentiate y (find dy/dx)
2) substitute x = a into dy/dx to find gradient of tangent. Let's call it m1.
3) substitute x into the y equation, to get a y value. Let's call this y value y1.
4) equation of equation : y -y1 = m1(x-a)

How to find equation of tangent at x = a?
1) Differentiate y (find dy/dx)
2) substitute x = a into dy/dx to find gradient of tangent. Let's call it m1.
3) substitute x into the y equation, to get a y value. Let's call this y value y1.
4) equation of equation : y -y1 = m1(x-a)

How to find equation of normal at x = a?
1) Differentiate y (find dy/dx)
2) substitute x = a into dy/dx to find gradient of tangent. Let's call it m1.
3) Find gradient of normal by using the formula gradient of normal = -1/ (gradient of tangent).
Let's call this gradient of normal m2.
3) substitute x into the y equation, to get a y value. Let's call this y value y1.
4) equation of equation : y -y1 = m2(x-a)

Differentiation Formulae

Differentiation formulae for algebraic, trigonometry, exponential, and logarithm:


































Note:
f(x) and g(x) are functions.
f'(x) means differentiation of f(x).
g'(x) means differentiation of g(x).

Friday, 13 June 2014

Surds: Putting everything together in a question