HackMD - Collaborative Markdown Knowledge Base

--- tags: teaching --- ## Local behaviors of a function. **Concept here**: what does that mean by local or global. For example, one could say > Around zero, the function $f(x)=x^2$ seems pertty similar to $f(x)=x^3+x^2$ When I say local behavior, I just mean a very, very tiny neigborhood around that point. ## Now think about why the local behavior of $f(x)=x^2$ and $f(x)=x^2+x^3$ are similar around $x=0$ Hint: $x^2+x^3=(1+x)x^2$. Thinking by this way: $1+x$ is almost equal to 1 around $x=0$. ### What is happening around $x=0$, the term $x^2$ is very very small. BUT, at the same time, $1+x$ is not small, $1+x$ is just around $1$ Think about what is happening if we multiply a tiny number with a number that is close to 1. $$ 0.00001\times 1.000001=0.00001000001 $$ **Key point: if we multiply a number close to 1 with a number close to 0. The result would be almost determined by the number close to 0** **0 contributes more than non-zero to local behaviors** ### Now think about local behavior of $f(x)=\frac {1+x}{x^2}$ and $f(x)=\frac1{x^2}$ around $x=0$ Why the local behavior of this two function around $0$ are the same? it is beause, around $x=0$, $1+x$ is amost equal to 1, and the main contribution of the local behavior is contributed by $\frac1{x^2}$ $$ \lim_{x\rightarrow 0}\frac1{x^2}=\infty $$ ## Now let's think about how to use local behaviors to help us draw graph of functions. #### Example: Draw the graph of $$ f(x)=\frac{x}{(x+1)(x-1)} $$ **Key point: analyze its local behavior at some special point** What is special point? > It is some point where the function equals to 0 or $\infty$ Which 3 points is the point that the function is 0 or $\infty$? $x=0,x=1,x=-1$ Try to analyze the local behavior of it. When $x=-1$, we factorize the function into the following part $$ f(x)=\underbrace{\frac1{x+1}}_{\text{goes to }\infty}\cdot \underbrace{\frac x{x-1}}_{\text{around}\frac12} $$ Then the fuction $$ g_{-1}(x)=\frac12 \cdot\frac1{x+1} $$ has the same local behavior as $f(x)$ around $x=-1$ **Excercise**: Analyze local behaviros of $f$ around $x=0$ and $x=1$ When $x=0$, $$ f(x)=\underbrace{x}_{\text{goes to 0}}\cdot\underbrace{\frac1{(x+1)(x-1)}}_{\text{around}-1} $$ $$ g_0(x)=-x $$ Left the case of $x=1$ for your own excercise $$ g_1(x)=\frac12\cdot\frac{1}{x-1} $$ ### Example $$ f(x)=\frac{(x+1)^2(x+2)^3}x $$ ### The most important point is $$ x=0,x=-1,x=-2 $$ Where when we considering around $x=0$, we want to factorize this function into the following parts $$ f(x)=\frac1x\cdot(x+1)^2(x+2)^3 $$ $$ g_0(x)=\frac8x $$ When we considering around $x=-1$, we want to factorize this function into the following parts $$ f(x)=(x+1)^2\cdot \frac{(x+2)^3}x $$ $$ g_{-1}(x)=(x+1)^2\cdot(-1) $$ When we considering around $x=-2$, $$ f(x)=(x+2)^3\cdot\frac{(x+1)^2}x $$ $$ g_{-2}(x)=(x+2)^3\cdot \left(-\frac12\right) $$ How to determine the bahavior around $\infty$ For a polynomial, the behavior of the function when $x$ goes to infty, is given by the highest term $$ f(x)=x^3+3x^2+x+7 $$ $$ g_{\infty}(x)=x^3 $$ How about rational functions? The behavior of the function when $x\rightarrow \infty$ is given by the ratio of the highest term on top and bottom $$ f(x)=\frac{3x^3+2x^2+1}{2x^2+1} $$ $$ g_\infty(x)=\frac{3x^3}{2x^2}=\frac32 x $$