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Here are the Mathematics revision sheets for class 11 and class 12 physics
Differentiation
We have two quantities x and y such that $y=f(x)$ where $f(x)$ is some function of x. We may be interested in finding the followings things
- $\frac{dy}{dx}$
- Maximum and Minimum values of y.It can be found with the method of Maxima and Minima
$\frac{dy}{dx}$ is the called the derivative of y w.r.t to x
It is defined as
$\frac{dy}{dx}=\lim_{\Delta x \to 0}\left ( \frac{\Delta y}{\Delta x} \right )$
Some commonly known functions and their derivatives are:-
| $y=x^n$ | $\frac{dy}{dx}=nx^{n-1}$ |
| $y=sinx$ | $\frac{dy}{dx}=cosx$ |
| $y=cosx$ | $\frac{dy}{dx}=-sinx$ |
| $y=tanx$ | $\frac{dy}{dx}=sec^{2}$ |
| $y=cotx$ | $\frac{dy}{dx}=-cosec^{2}$ |
| $y=secx$ | $\frac{dy}{dx}=secx tanx$ |
| $y=ln x$ | $\frac{dy}{dx}=\frac{1}{x}$ |
| $y=e^{x}$ | $\frac{dy}{dx}=e^{x}$ |
Some important and useful rules for finding derivatives of composite functions
- $\frac{d}{dx}(cy)=c\frac{dy}{dx}$ where c is constant
- $\frac{d}{dx}(a+b)=\frac{da}{dx} + \frac{da}{dx} $ where a and b are function of x
- $\frac{d}{dx}(ab)=a\frac{db}{dx}+b\frac{da}{dx}$
- $\frac{d}{dx}(\frac{a}{b})=\frac{[b\frac{da}{dx}-a\frac{db}{dx}]}{b^{2}}$
- $\frac{dy}{dx}=(\frac{dy}{da})(\frac{da}{dx})$
- $\frac{d^{2}y}{dx^{2}}=(\frac{d}{dx})(\frac{dy}{dx})$
Maximum and Minimum values of y
Step 1:
fine the derivative of y w.r.t x
$(\frac{dy}{dx})$
Step2:
Equate
$\frac{dy}{dx}=0$
Solve the equation to find out the values of x
Step3:
find the second derivative of y w.r.t x and calculate the values of
$\frac{d^{2}y}{dx^{2}}$
for the values of x from step2
if $\frac{d^{2}y}{dx^{2}}>0$ then the value of x corresponds to mimina of y then $y_{min}$ can be find out by putting this value of x
if $\frac{d^{2}y}{dx^{2}}<0$ then the value of x corresponds to maxima of y then $y_{max}$ can be find out by putting this value of x
Integration
$I=\int_{a}^{b}f(x)dx$
It reads as integration of function f(x) w.r.t. x within the limits from x=a to x=b.
Integration of some important functions are
$\int sinx dx=-cosx$
$\int cosx dx=sinx$
$\int sec^{x}dx=tanx$
$\int cosec^{x}dx=-cotx$
$\int \frac{1}{x}dx=lnx$
$\int x^{n}dx=\frac{x^{n+1}}{n+1}$
$\int e^x dx=e^x$
Useful rules for integration are
$\int cf(x)dx=c\int f(x)dx$
$\int[f(x)+h(x)]=\int f(x)dx+\int h(x)dx$
$\int f(x)g(x)dx=f(x)\int g(x)dx -\int\left ( f'(x)\int g(x)dx \right ) dx$
Trigonometry
Properties of trigonometric functions
1. Pythagorean identity
$sin^2 A +cos^2 A=1$
$1+tan^ A=sec^2 A$
$1+cot^2 A=cosec^2 A$
2. Periodic function
$sin(A+2\pi)=sinA$
$cos(A+2\pi)=cosA$
3.Even-Odd Identities
$cos(-A)=cos(A)$
$sin(-A)=-sin(A)$
$tan(-A)=-tan(A)$
$cosec(-A)=-cosec(A)$
$sec(-A)=sec(A)$
$cot(-A)=-cot(A)$
4. Quotient identities
$tan(A)=\frac {sin A}{cos A}$
$cot(A)=\frac{cos A}{sin A}$
5. Co-function identities
$sin\left ( \frac{\pi}{2}-A \right )=cos(A)$
$cos\left ( \frac{\pi}{2}-A \right )=sin(A)$
$tan \left ( \frac{\pi}{2}-A \right )=cot(A)$
$cosec \left( \frac{\pi}{2}-A \right )=sec(A)$
$sec\left ( \frac{\pi}{2}-A \right )=cosec(A)$
$cot\left ( \frac{\pi}{2}-A \right )=tan(A)$
6. Sum difference formulas
$sin(A\pm B)=sin(A)cos(B) \pm sin(B)cos(A)$
$cos(A \pm B)=cos(A)cos(B) \mp sin(A)sin(B)$
$tan(A \pm B)=\frac {tan(A) \pm tan(B)}{1 \mp tan(A) tan (B)}$
7. Double angle formulas
$cos(2A)=cos^2(A)-sin^2(A)=2cos^2 (A)-1=1-2sin^2(A)$
$sin(2A)=2sin(A)cos(A)$
$tan(2A)=\frac{2tan(A)}{1-tan^2(A)}$
8. Product to sum formulas
$sin(A)cos(B)=\frac {1}{2}[cos(A-B)-cos[A+B]$
$cos(A)cos(B)=\frac {1}{2}[cos(A-B)+cos[A+B]$
$sin(A)cos(B)=\frac {1}{2}[sin(A+B)+sin[A-B]$
$cos(A)sin(B)=\frac {1}{2}[sin(A+B)-sin[A-B]$
9. Power reducing formulas
$sin^2 A=\frac{1-cos(2A)}{2}$
$cos^2 A=\frac{1+cos(2A)}{2}$
$tan^2 A=\frac{1-cos(2A)}{1+cos(2A)}$
10. reciprocal identities
$Sin(A)=\frac{1}{cosec(A)}$
$cos(A)=\frac{1}{sec(A)}$
$tan(A)=\frac{1}{cot(A)}$
$ cosec(A)=\frac{1}{sin(A)}$
$Sec(A)=\frac{1}{cos(A)}$
$cot(A)=\frac{1}{tan(A)}$
Binomial Theorem
$(a+b)^n=C_{0}^{n}a^{n}+C_{1}^{n}a^{n-1}b+C_{2}^{n}a^{n-2}b^2+………+C_{n}^{n}b^{n}$
From the binomial formula, if we let a = 1 and b = x, we can also obtain the binomial series which is valid for any real number n if |x| < 1.
(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+………..
Geometric Series
$a,aq,aq^2,aq^3,aq^4………..aq^{n-1}$ where q is not equal to 0, q is the common ratio and a is a scale factor.The formula for the sum of the first n numbers of geometric progression
$S_{n}=\frac{a(1-q^n)}{(1-q)}$
Infinite geometric series where |q| < 1
If |q| < 1 then $a_{n} \to 0$, when n -> infinity So the sum S of such an infinite geometric progression is:
$S=\frac{1}{(1-x)}$ which is valid only for |x|
Arithmetic Progression
$a,a+d,a+2d,a+3d……….a+(n-1)d$
The sum S of the first n values of a finite sequence is given by the formula:
$S=\frac{n}{2}[(2a + d(n-1)]$
Quadratic Formula
$ax^2+bx+c=0$
then
$x=-\frac{b\pm?(b2-4ac)}{2a}$
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Mathematics revision sheet for class 11 and class 12 physics pt 1 (40.5 KiB)
Mathematics revision sheet for class 11 and class 12 physics part 2 (48.0 KiB)
