Formulas in Pure Math 3

Formulas in Pure Math 3

Module 3

Table of contents

Limits : Slide 3

Differentiation : Slide 7

Integration : Slide 11

 Limits 

• (a) If k is a constant ,  $\lim_{x\rightarrow a}k\ =\ k$  

• (b)  $\lim_{x\rightarrow a}k\left(f\left(x\right)\right)\ =\ k\ \times\lim_{x\rightarrow a}f\left(x\right)\$  

• (c)  $\lim_{x\rightarrow a}\left[f\left(x\right)+g\left(x\right)\right]=\lim_{x\rightarrow a}f\left(x\right)+\lim_{x\rightarrow a}g\left(x\right)$  

•  $\left(d\right)\ \lim_{x\rightarrow a}\left[f\left(x\right)-g\left(x\right)\right]=\lim_{x\rightarrow a}f\left(x\right)\ -\lim_{x\rightarrow a}g\left(x\right)$  

• (e)  $\lim_{x\rightarrow a}\left[f\left(x\right)\ \times g\left(x\right)\right]=\lim_{x\rightarrow a}f\left(x\right)\ \times\lim_{x\rightarrow a}g\left(x\right)$  

• (f)  $\lim_{x\rightarrow a}\ \frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim_{x\rightarrow a}f\left(x\right)}{\lim_{x\rightarrow a}g\left(x\right)}\$  

• (g)  $\lim_{x\rightarrow a}\left[f\left(x\right)\right]^n=\left[\lim_{x\rightarrow a}f\left(x\right)\right]^n$  (assuming that the nth power is defined.

continued

•  $\lim_{x\rightarrow a}\ ^n\sqrt{f\left(x\right)}=^n\sqrt{\lim_{x\rightarrow a}f\left(x\right)}$  (Assuming that ythe nth root is defined. )

•  $Note\ :$  

•  $\lim_{x\rightarrow a}x\ =a$  

•  $\lim_{x\rightarrow0}\ \frac{\sin x}{x}=1$   ( $\lim_{x\rightarrow0}\frac{\sin ax}{ax}$   Also counts )

•  $\lim_{x\rightarrow0}\ \frac{1-\cos x}{x}=0$  

•  $\lim_{x\rightarrow+\infty}\left(1+\ \frac{1}{x}\right)^x=e$  

 

L'Hopital's  Rule 

Limits of the form   $\lim\ \frac{f\left(x\right)}{g\left(x\right)}$   can be evaluated by this rule (if they exist) in the indeterminate cases where f(x) and g(x) both approach 0 or both approach +infinity or - infinity . The rule states :

 $\lim\ \frac{f\left(x\right)}{g\left(x\right)}=\lim\ \frac{f^1\left(x\right)}{g^1\left(x\right)},$  
where   $f^1\left(x\right)$   and  $g^1\left(x\right)$   are the first derivatives of f(x) and g(x) respectively, with respect to x. 
 

 

Continuity

A function f(x) is said to be continuous at x = a if 

1.  $\lim_{x\rightarrow a}\ f\left(x\right)\ =\ L\ must\ exist$  
2. f (a) must exist 
3. f (a) = L 

 

 Differentiation 

 $9.\ Differentiation\ \cdot\ y=x^n\rightarrow\ \frac{dy}{dx}=nx^{n-1}$  
 $\cdot\ y=ax^n\ \rightarrow\ \frac{dy}{dx}=anx^{n-1}$  
 $\cdot\ y=+\ or\ -\ \left(whole\ number\right)x\ \rightarrow\ \frac{dy}{dx}=whole\ number$   $\cdot\ y=+\ or\ -\ whole\ number\ \ \rightarrow\ \frac{dy}{dx}=0$  
 $\cdot\ y=\sin x\ \rightarrow\ \frac{dy}{dx}=\cos\ x$  
 $y=\cos\ x\ \rightarrow\ \frac{dy}{dx}=-\ \sin x\$  

 $\cdot y=\sin\ \left(ax+b\right)\rightarrow\ \frac{dy}{dx}=a\cos\ \left(ax+b\right)$   $\cdot\ y=\cos\ \left(ax+b\right)=-\sin\ \left(ax+b\right)\times a$   $\cdot y=\cos\ \left(f\left(x\right)\right)\rightarrow\ \frac{dy}{dx}=-\sin\left(f\left(x\right)\right)\times f^1\left(x\right)$   $\cdot\ y=\sin\ \left(f\left(x\right)\right)\rightarrow\ \frac{dy}{dx}=\cos\ \left(f\left(x\right)\right)\ \times f^1\left(x\right)$  

*   $\left(ax+b\right)^n\rightarrow n\left(ax+b\right)^{n-1}\times a$  

*  $\tan x\rightarrow\sec^2x$  
*  $\tan\left[f\left(x\right)\right]=\sec^2\left[f\left(x\right)\right]\times f'\left(x\right)$  
*  $\tan\left(ax+b\right)\rightarrow\sec^2\left(ax+b\right)\times a$  
*  $a\cdot f\left(x\right)\rightarrow a\cdot f'\left(x\right)$ 

 

• Product Rule :  $\frac{dy}{dx}=U\ \frac{dv}{dx}+V\ \frac{du}{dx}$   

• Quotient Rule :  $\frac{dy}{dx}=\frac{V\ \frac{du}{dx}-U\ \frac{dv}{dx}}{V^2}$  

• Chain Rule :  $y=f\ \left(g\left(x\right)\right)\ \rightarrow\ \frac{dy}{dx}=f^{-1}\left(g\left(x\right)\right)\ g^{-1}\left(x\right)$  where  $f^{-1}\&\ g^{-1}$  are the differentiated functions / the derivatives.

•  $\left(ax+\ b\right)^n=n\left(ax+b\right)^{n-1}\left(a\right)$  

•  $At\ a\ S.p\ \frac{dy}{dx}=0\ \ \ \frac{d^2y}{dx^2}:2nd\ derivative\ test$  (Maximum, Minimum, Point of inflection)    

•  Increasing function  $\frac{dy}{dx}>0\ \ \ \ \ Decrea\sin g\ function\ \frac{dy}{dx}<0$  

*  $x=f\left(t\right)\ and\ y\ =\ g\left(t\right)$  
 $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{g'\left(t\right)}{f'\left(t\right)}$  

* First Principles

 $\lim_{h\rightarrow0}\ \frac{f\left(x+h\right)-f\left(x\right)}{h}$  

Integration

•  $\int_{ }^{ }x^ndx\ =\ \frac{x^{n+1}}{n+1}+c$  

•  $\int_{ }^{ }ax^ndx=a\int_{ }^{ }x^ndx=a\left(\frac{x^{n+1}}{n+1}\right)+c$  : where a is a constant and placed outside the integral sign to be multiplied by the integrated function later.

•  $\int_{ }^{ }\left(ax+b\right)^n\ dx=\frac{\left(ax+b\right)^{n+1}}{n+1}\times\left(\frac{1}{a}\right)+c$  

•  $\int_{ }^{ }\cos\ x\ dx\ =\sin x\ +c$  

•   $\int_{ }^{ }a\rightarrow ax+c$  

continued..

•  $\int_{ }^{ }-\sin x\ dx=\cos\ x+c$  

•  $\int_{ }^{ }\sin x\ dx\ =-\cos x+c$  

•  $\int_{ }^{ }\sin\ \left(ax+b\right)dx=\frac{-\cos\left(ax+b\right)}{a}+c$  

•  $\int\ \cos\left(ax+b\right)dx=\frac{\sin\left(ax+b\right)}{a}+c$  

•  $\int_{ }^{ }\sec^2\left(x\right)\rightarrow\tan x+c$  

•  $\int_{ }^{ }\sec^2\left(ax+b\right)\rightarrow\ \frac{\tan\left(ax+b\right)}{a}+c$  

*  $\int_{ }^{ }\left(an+b\right)^n\rightarrow\ \frac{1}{a}\ \frac{\left(ax+b\right)^{n+1}}{n+1}+c\ ,\ n\ne1$  

*  $\int_{ }^{ }a\cdot f\left(x\right)dx\equiv a\int_{ }^{ }f\left(x\right)dx$  
*  $\int_x^y\left[f\left(x\right)\ \frac{+}{ }g\left(x\right)\right]dx=\int_x^yf\left(x\right)dx\ \frac{+}{ }g\left(x\right)dx$  
* $\int_{ }^{ }\ \frac{1}{x}dx=\ln x+c$  
*  $\int_{ }^{ }\cos\ ax\ \rightarrow\ \frac{\sin ax}{a}+c$  
* $\int_{ }^{ }\sin\ ax=\ \frac{-\cos ax}{a}+c$  
*  $To\ integrate\ \int_{ }^{ }\sin^2x\ change\ to\ \int_{ }^{ }\ \frac{1}{2}\left(1-\cos2x\right)dx\ \ \ \left[Identity\ used\ \cos2x=1-2\sin^2x\right]$  
*  $To\ integrate\ \int\cos^2x\ dx\ change\ to\ \int\ \frac{1}{2}\left(1+\cos2x\right)dx\ \left[Identity\ used\ \cos2x=2\cos^2x-1\right]$  

* To integrate  $\int\tan^2x\ dx\ change\ to\ \int\left(\sec^2x-1\right)dx\ \ \ \left[Identity\ used\ 1+\tan^2x=\sec^2x\right]$  

* To integrate  $\int\sin x\cos\ change\ to\ \int\ \left(\frac{1}{2}\sin\ 2x\right)\ dx\ \ \ \left[Identity\ used\ \sin\ 2x=2\sin\ x\ \cos\ x\right]$  

Definite Integrals

 $\int f\left(x\right)dx=\lceil g\left(x\right)\rceil_a^b\ =g\left(b\right)-g\left(a\right)$  
where f (x) dx  is the derivative to be integrated , g (x) is the integrated f (x) & a and b are the limits.
 * If a < b < c,  $\int_a^bf\left(x\right)+\int_b^cf\left(x\right)\ dx=\int_a^cf\left(x\right)\ dx$  

Area Under A curve 
 $Region\ bounded\ by\ \left(above\right)the\ x\ axis\ :\ A=\int_a^by\ dx$    $Region\ is\ under\ the\ x\ axis:\ A\ =-\int_{x_1}^{x_2}y\ dx\$    $\ Region\ is\ on\ the\ left\ side\ of\ the\ y\ axis\ \left(bounded\ by\ the\ y\ axis\ not\ x\ axis\right)\ :\ A=-\int_{y_1}^{y_2}xdy$   $Region\ on\ the\ right\ of\ the\ y\ axis\ \left(bounded\ by\ the\ y\ axis\ \right):\ A\ =\int_{y_1}^{y_2}\ x\ dy$  
Same process as definite integrals except the numerical answer is  $^2\ units\$  because the "area" of the region was found. 

 

* Region rotated about the x axis :  $V=\pi\int_{x_1}^{x_2}y^2dx$  
*Region rotated about the y axis :  $V=\pi\int_{y_1}^{y_2}x^2dy$  

*  $\int_x^af\left(x\right)d=\int_x^af\left(a-x\right)dx,\ a>0$