Calculus - Fanyu Zhao

Key Points

1. Functions Definition

1.1 Each $x$$x$ has only one $y$$y$

A function denoted $f(x)$$f (x)$ of a single variable $x$$x$ is a rule that assigns each element of a set $X$$X$ ( written $x\in X$$x \in X$ ) to exactly one element $y$$y$ of a set Y ( $y\in Y$$y\in Y$) :

1.2 Domain of $f$$f$

$Domf$$Dom f$ Domain of Function

$Imf$$Im f$ Image of Function

For a given value of $x$$x$, there should be at most one value of $y$$y$.

1.3 Implicit Form $f(x,y)=0$$f(x,y)=0$

For example,

1.4 Polynomials

2. Implicit Differentiation

For example,

Mainly two ways to take derivatives,

and plug $y=a^x$$y=a^x$ 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, $\frac{sin(x)}{x}$$\frac{sin(x)}{x}$, at $x\to 0$$x \rightarrow 0$.

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, $e^x$$e^x$ at $x=0$$x=0$.

5. Integration

5.1 Intergration by Parts

and then integrate from both sides,

as $\int y^{\prime }dx=y+C$$\int y' dx = y+C$, so we would get,

For example,

5.2 Reduction Formula

We define a integral, ($I_n$$I_n$ is called Gamma Function)

$n$$n$ is determined as the subscript of $I_n$$I_n$, 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 $ln|f(x)|$$ln|f(x)|$

For example,

5.3.2 Decompose the Fraction - Factorisation

For example,

The further implication is that.

Any rational expression $\frac{f(x)}{g(x)}$$\frac{f(x)}{g(x)}$, ( with degree of $f(x)$$f(x)$ < degree of $g(x)$$g(x)$), could be rewritten as.

, where each $F_i$$F_i$ Is,

6. Complex Number - $i$$i$

6.1 Definition

and $z$$z$ could be expressed in polar co-ordinate form as,

, where

The set of all complex numbers is denoted $\mathbb{C}$$\mathbb{C}$; and for any complex number $z$$z$, we could write $z\in \mathbb{C}$$z \in \mathbb{C}$. ( $\mathbb{R}\subset \mathbb{C}$$\mathbb{R} \subset \mathbb{C}$ ).

6.2 Modulus

The modulus of $z$$z$ donates $|z|$$|z|$ is defined as,

6.3 Complex Conjugate

For example, if $z=x+iy$$z=x+iy$, then $\overline{z}=x-iy$$\bar{z}=x-iy$.

6.4 Polar Form

by Euler's Identity,

6.5 Euler's Fomula

The Euler's Identity is shown as, by applying Taylor's Expansion and by $i^2=-1$$i^2=-1$,

Plug $\theta =\pi$$\theta = \pi$ into Euler's Formula,

7.Higher Dervatives

By Fanyu Zhao