Calculus - Fanyu Zhao
Key Points
1. Functions Definition
1.1 Each has only one
A function denoted of a single variable is a rule that assigns each element of a set ( written ) to exactly one element of a set Y ( ) :
1.2 Domain of
Domain of Function
Image of Function
For a given value of , there should be at most one value of .
For example,
1.4 Polynomials
2. Implicit Differentiation
For example,
Mainly two ways to take derivatives,
and plug inside
Or, simply we apply the exponential transformation, and take deriviatives later.
However, for a polynomial, we normally have to do the implicit differentiation.
3. L 'Hospital's Rule & Limitations
If there is a limitation (, which is called as the inderterminate form),
then, it could be calculated as,
For example, , at .
4. Taylor Series
Approximate a function a certain point, by a series of terms.(detailing explaination sees Blog Section 6 )
We use the 1st, 2nd, 3rd, 4th, ... n^th derivatives, etc, to approximate the function at a certain value.
For example, at .
5. Integration
5.1 Intergration by Parts
and then integrate from both sides,
as , so we would get,
For example,
We define a integral, ( is called Gamma Function)
is determined as the subscript of , and could be treated as a constant in that integral.
We integrate that formula, and would get,
If we keep doing that, we would get,
where,
so we get,
5.3 Other Tips
5.3.1
For example,
5.3.2 Decompose the Fraction - Factorisation
For example,
The further implication is that.
Any rational expression , ( with degree of < degree of ), could be rewritten as.
, where each Is,
6. Complex Number -
6.1 Definition
and could be expressed in polar co-ordinate form as,
, where
The set of all complex numbers is denoted ; and for any complex number , we could write . ( ).
6.2 Modulus
The modulus of donates is defined as,
![](/Users/mie/Desktop/Screenshot 2022-09-22 at 12.39.55.png)
6.3 Complex Conjugate
For example, if , then .
by Euler's Identity,
6.5 Euler's Fomula
The Euler's Identity is shown as, by applying Taylor's Expansion and by ,
Plug into Euler's Formula,
7.Higher Dervatives
By Fanyu Zhao