Mathematics revision sheet 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 followings things
1. \frac{dy}{dx}
2. Maximum and Minimum values of y.It can be find 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
1. \frac{d}{dx}(cy)=c\frac{dy}{dx} where c is constant
2. \frac{d}{dx}(a+b)=\frac{da}{dx} + \frac{da}{dx} where a and b are function of x
3. \frac{d}{dx}(ab)=a\frac{db}{dx}+b\frac{da}{dx}
4. \frac{d}{dx}(\frac{a}{b})=\frac{[b\frac{da}{dx}-a\frac{db}{dx}]}{b^{2}}
5. \frac{dy}{dx}=(\frac{dy}{da})(\frac{da}{dx})
6.\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.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 a 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}

Download this post as pdf

Popular Posts