Here, We will jot down a Calculus formula sheet to study calculus in-depth. In this blog, we will learn about the differentials calculus with few examples

Leibnitz’s theorem

Rolle’s Theorem

Lagrange’s Mean-value theorem 

Cauchy’s Mean value theorem 

Taylor’s theorem 

Maclaurin’s theorem 

Taylor’s series 

Indeterminate forms

Tangents & Normal-Cartesian curves

Polar curves Pedal equation

Derivation of arc

Radius of Curvature

Centre of Curvature

Maxima & Minima

Asymptotes

Differentials calculus Formula Sheet

  • Leibnitz’s theorem for nth derivative of the production of two functions

If u, v be two function of x derivatives of the nth order, then

(uv)_{n} = u_n v + ^nC_1 \:u_{n-1} v_1 + ^nC_2\: u_{n-2} v_2+……..+ ^nC_r u_{n-r} v_r+….+ ^nC_n u v_n

 

Problem : Find the nth derivative of e^x(2x+1)^3.

Solution :


Take u=e^x and \: v=(2x+3)^3

So that, u=e^x    for all the integeral values of n

v_{1}=6(2x+3)^2,v_{2}=24(2x+3),v_{3}=48,v_{4},v_{5 }.... all are zeros .
By Leibnitz's theorem, 

(uv)_{n} = u_n v + ^nC_1 \:u_{n-1} v_1 + ^nC_2\: u_{n-2} v_2+ ^nC_3 u_{n-3} v_3

[e^x (2x+3)^3]_n=e^x(2x+3)^3+ne^x[6(2x+3)^2]+\frac{n(n-1)}{1\times2}e^x[24(2x+3)]+\frac{n(n-1)(n-2)}{1\times 2\times 3}e^x[48]\\
\:\:\:\:\:\:\:\:\:\:=e^x{(2x+3)^3+6n(2x+3)^2]+12n(n-1)(2x+3)+8n(n-1)(n-2)}
  • Rolle’s Theorem

If f(x) is continues function in the closest interval [a,b] ,f’(x) exists for every value of x in the open interval (a,b) and f(a)=f(b), then there is at least one value c of x in (a,b)

Such that,

f’(c)=0.

  • Lagrange’s Mean-value theorem

First form

If f(x ) is a continuous function in the closed interval [a,b] and f’x exists in an open interval (a,b), then there will be at least one of c of x in (a,b)

Such that,

\frac{f(b)-f(a)}{b-a}=f'(c)

Second form

If  b=a+h then,

a<c<b , c=a+ \theta h  where 0< \theta <1

So, the mean value theorem is as follows-

If f(x) is a continuous function in a closed interval [a,a+h] and f’(x) exists in open interval (a, a+h), then there will be at least one number \theta (0< \theta <1)

Such as,

f(a+h)=f(a)+hf'(a+\theta h)

  • Cauchy’s Mean value theorem

If f(x) and g(x) are continuous function in closed interval [a,b] and they exist in (a,b) and g’(x)≠0 for any value of x in open interval (a,b)

then, there will be a least on value of x

Such as,

\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{f'(c)}

  • Taylor’s theorem

If f(x) and its first (n-1) derivatives be continuous in [a, a+h] and fn(x) exists for every value of x in (a,a+h), then then there is at least one number \theta (0 < \theta <1)

f(a+h)=f(a)+hf'(a)+\frac {(h^2)}{2!}f"(a)+\dotso+\frac {(h^n)}{n!}f^n(a+\theta h)

  • Maclaurin’s theorem

If f(x) can be expanded as an infinite series,

f(x)=f(0)+xf'(0)+\frac {x^2}{2!}f"(0)+\frac {x^3}{3!}f'''(0)+\dotso\infty

  •  Taylor’s series

If f(x+h) can be expanded as an infinite series then

f(x+h)=f(x)+hf'(x)+\frac {h^2}{2!}f"(x)+\frac {h^3}{3!}f'''(x)+\dotso\infty

if f(x)= all derivates of all order and Rn tends to zero as n -> \infty then Taylor’s theorem become the Taylor’s series

  • Indeterminate forms

\lim_{x \to a} \frac{f(x)}{\phi(x)}=\lim_{x \to a} \frac{f'(x)}{\phi'(x)}

Form 0/0 : when \lim_{x \to a} f(x) and \lim_{x \to a} {\phi(x) }are Zero then, quotient reduces to the indeterminate form 0/0.
\lim_{x \to a} \frac{f(x)}{\phi(x)} is a finite value

  • Tangents & Normal- Cartesian curves

(Grape photo)
       1.Equation of tangent
At point (x,y) of the curve y=f(x) is

Y-y=\frac{\text{d}y}{\text{d}x}(X-x)

    2.Equation of normal

Y-y=-\frac{\text{d}x}{\text{d}y}(X-x)

  • Polar curves

    1.Angle between radius vector and tangent

    If \phi be the angle between the radius of the vector and the tangent at any point of the curve

    then,

    r=f(\theta),tan(\theta)= r\frac{\text{d}\theta}{\text{d}r}

    2.Length of the perpendicular from pole on the tangent

    (Grape)
    If p is perpendicular from the pole on the tangents

    P=r \sin\theta,
    \frac{1}{p^2}=\frac{1}{r^2}+\frac{1}{r^4}({\frac{\text{d}r}{\text{d}\theta}})^2

  • Pedal equation
    If r be the radius vector of any point on the curve and p the length of the perpendicular from the pole at that point on a tangent then, the relation between p and r is called the pedal equation.
  • Derivation of arc

if y=f(x)

then,

\frac{\text{d}s}{\text{d}x}=\sqrt{[{1}+{(\frac{\text{d}y}{\text{d}x})^2}]}

 

  • Radius of Curvature

      1.Radius of curvature for polar curve

If r=f(\theta)

then,

\rho=\frac{(r^2+r_1^2)^(\frac{3}{2})}{r^2+2r_1^2-rr_2}

2.Radius of curvature for pedal curve

\rho=r\frac{\text{d}r}{\text{d}p}

 

  • Centre of Curvature

At any point P(x,y) on the curve y=f(x) is

Given by,

\bar{x}=x-\frac{y_1(1+y_1^2)}{y_2} \bar{y}=y+\frac{1+y_1^2}{y_2}

The locus of the center of curvature for a curve is called its Evolute and the curve is called an involute of its Evolute

  • Maxima & Minima

f(x) is called maximum at x=a, if there exist a small number h, such as f(a)>both f(a-h) and f(a+h).
f(x) is called minimum at x=a, if there exist a small number h, such as f(a)<both f(a-h) and f(a+h).

  • Asymptotes

It is defined as the straight line at a finite/ infinity distance from the origin, to which a tangent to the curve to the point of contact reduces to infinity.

a_0x^n+(a_1y+b_1)x^{(n-1)}+(a_2y^2+b_2y+c_2)x^{(n-2)}+\dotso 0 (parallel to axes)

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