# Chapter 26: Curve Sketching and Oblique Asymptotes

### Curve Sketching

#### Recall

Critical points ﻿$\leftrightarrow$﻿ ﻿$f^{1}(x)=0$﻿ or undef

Increasing ﻿$\rightarrow$﻿ ﻿$f^{1}(x)>0$﻿

Decreasing ﻿$\rightarrow$﻿ ﻿$f^{1}(x)<0$﻿

Concave up ﻿$\rightarrow f^{11}(x)>0$﻿

Concave down ﻿$\rightarrow f^{11}(x)<0$﻿

Point of inflection﻿$\rightarrow$﻿ 1) Tangent line

2) Concavity change

### The Curve Sketching Algorithm

1. Identify the domain, symmetries, intercepts
2. Calculate ﻿$f^{1}(x)$﻿ and ﻿$f^{11}(x)$﻿
3. Find the critical points and their behaviour
4. Increasing/decreasing
5. Concavity and points of inflection
6. Asymptoles
7. Draw the curve

#### example

Sketch ﻿$f(x)=x^{3}-3x$﻿

1) Domain ﻿$=(-\delta ,\delta )$﻿

y-intercept

﻿$f(0)=0$﻿

x-intercept

﻿$0=x^{3}-3x$﻿

﻿$\Rightarrow x=0,-\sqrt{3}, \sqrt{3}$﻿

odd symmetry

﻿$f(-x)=(-x^{3})-3(-x)$﻿

﻿$=-x^{3}+3x=-f(x)$﻿

2) ﻿$f^{1}(x)=3x^{2}-3$﻿

﻿$f^{11}(x)=6x$﻿

3) If ﻿$f^{1}(x)=0$﻿ then ﻿$3x^{2}-3=3(x^{2}-1)=0$﻿

﻿$\rightarrow$﻿ ﻿$x=-1$﻿ or ﻿$x=+1$﻿

4) Interval ﻿$f^{1}(x)$﻿ sign behaviour 5) ﻿$f^{11}(x)=6x$﻿ ﻿$\rightarrow$﻿ if ﻿$x>0$﻿ concave up

if ﻿$x<0$﻿ concave down

6) ﻿$\lim_{x\rightarrow \delta } f(x)\lim_{x\rightarrow \delta }x(x^{2}-3)=\delta$﻿

﻿$\lim_{x\rightarrow -\delta } f(x)\lim_{x\rightarrow -\delta }x(x^{2}-3)=-\delta$﻿

7) #### Example

Sketch ﻿$f(x) = \frac{(x+1)^{2}}{1+x^{2}}$﻿

Assume ﻿$f^{1}(x) = \frac{2(1-x)^{2}}{(1+x^{2})^{2}}$﻿

﻿$f^{11}(x) = \frac{4x(x^{2}-3)}{(1+x^{2})^{3}}$﻿

1) Domain ﻿$(-\delta ,\delta )$﻿

﻿$f(-x)=\frac{(-x+1)^{2}}{1+(-x)^{2}}=\frac{(1-x)^{2}}{1+x^{2}}\neq f(x)$﻿ No symmetry

y-intercept

﻿$f(0)=\frac{1}{1}=1$﻿

x-intercept

﻿$\frac{(x+1)^{2}}{1+x^{2}}=0\Rightarrow x=-1$﻿

2) ﻿$f^{1}(x)=\frac{2(x+1)(1+x^{2})-(x+1)^{2}(2x)}{(1+x^{2})^{2}}=\frac{2(1-x^{2})}{(1+x^{2})^{2}}$﻿

﻿$f^{11}(x)=\frac{4x(x^{2}-3)}{(1+x^{2})^{3}}$﻿

3) ﻿$f^{1}(x)=0 \rightarrow x=1 or x=-1$﻿

4) Interval sign ﻿$f^{1}(x)$﻿ sign behaviour 5) ﻿$f^{11}(x)=\frac{4x(x^{2}-3)}{(1+x^{2})^{3}}$﻿ 6) ﻿$\lim_{x\rightarrow \delta }f(x)=\lim_{x\rightarrow \delta }\frac{x^{2}+2x+1}{1+x^{2}}=1$﻿

﻿$\lim_{x\rightarrow \delta }f(x)=1$﻿ ### Oblique Asymptotes

#### Define

f(x) has an OBLIQUE ASYMPTOTE

﻿$y=mx+b$﻿

﻿$\lim_{x\rightarrow \delta }\left [ f(x)-(mx+b) \right ]=0$﻿ or

﻿$\lim_{x\rightarrow -\delta }\left [ f(x)-(mx+b) \right ]=0$﻿

#### example

The function ﻿$f(x)=3x+\frac{1}{x}$﻿ has ﻿$y=3x+0$﻿ as an oblique asymptote

﻿$\lim_{x\rightarrow \delta }\left [ f(x)-(3x+0) \right ]=\lim_{x\rightarrow \delta }\left [ 3(x)+\frac{1}{x}-3x \right ]$﻿

﻿$=$﻿ ﻿$\lim_{x\rightarrow \delta }\left [ \frac{1}{x} \right ]=0$﻿

#### example

Find the oblique asymptote of

﻿$g(x)=\frac{2x^{2}+x+1}{3x+1}$﻿

﻿$g(x)-(mx+b)=\frac{2x^{2}+x+1}{3x+1}-\frac{(mx+b)(3x+1)}{3x+1}$﻿

﻿$=\frac{2x^{2}+x+1-(3mx^{2}+mx+3bx+b)}{3x+1}$﻿

﻿$=\frac{(2-3m)x^{2}+(1-m-3b)x+(1-b)}{3x+1}$﻿

We need ﻿$2-3m=0$﻿ and ﻿$1-m-3b=0$﻿

Thus, ﻿$m=\frac{2}{3}$﻿ and ﻿$b=\frac{1}{9}$﻿

﻿$y=\frac{2}{3}x+\frac{1}{9}$﻿