Thursday, August 31, 2023

sinx^2, sinx^3, sinx^4 graph | What are the graphs of sin over 2, 3 and 4?

Below you can find the graphic drawings of 2 over sinx, 3 over sinx and 4 over sinx. We wish everyone a good work...

$(sin(x))^2$ graph:

$(sin(x))^3$ graph:

$(sin(x))^4$ graph:

Perimeter and Area Formulas of Geometric Shapes | All Geometry Formuls

You can find the perimeter and area formulas of all geometric shapes from the image below. We wish everyone a good work...

Rectangle perimeter = 2(a+b)

Rectangle area = a.b

Square perimeter = 4a

Square area = $a^2$

Triangle perimeter = a+b+c=2s

Triangle area = 1:1/2*b*h & 2: $\sqrt{s(s-a).(s-b).(s-c)}$

Right triangle perimeter = b+h+d

Right triangle area = 1/2 bh

Equilateral triangle perimeter = 3a

Equilateral triangle area = 1: 1/2 a.h & 2: $\frac{\sqrt{3}}{4}.a^2$

Isosceles right triangle perimeter = 2a+d

Isosceles right triangle area = 1/2 a^2

Parallelogram perimeter = 2(a+b)

Parallelogram area = a.h

Rhombus perimeter = 4a

Rhombus area = 1/2 d1 d2

Trapezium perimeter = Sum of its four sides

Trapezium area = 1/2 h(a+b)

Circle perimeter = 2πr

Circle area = πr^2

Semicircle perimeter = πr+2r

Semicircle area = 1/2πr^2

Ring (shaded region) perimeter = -

Ring (shaded region) area = π(R^2-r^2)

Sector of a circle perimeter = I + 2r where I = (Ɵ / 360) * 2πr

Sector of a circle area = Ɵ/360 * πr^2

Wednesday, August 30, 2023

Graph of tanhx^-1 | What is tanh^-1x graph drawing?

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The graph of tanhx to the -1 is as follows.

$tanhx^{-1}$ graph:

tanh^-1x graph

Graph of cothx^-1 | What is coth^-1x graph drawing?

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The graph of cothx to the -1 is as follows.

$cothx^{-1}$ graph:

cothx^-1 graphic

Graph of coshx^-1 | What is cosh^-1x graph drawing?

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The graph of coshx to the -1 is as follows.

$coshx^{-1}$ graph:

cosh^-1(x) graph

Graph of sinhx^-1 | What is sinh^-1x graph drawing?

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The graph of sinhx to the -1 is as follows.

$sinhx^{-1}$ graph:

sinh^-1x graph

Tuesday, August 29, 2023

Special Power Series | ex | sinx | cosx | tanx | ln(1+x) | sinhx | coshx | tanhx | sinh^-1 | tanh^-1 All Series

You can reach all special power series formulas from below. We wish you all good lessons...

$e^x$ series:

(all x)
sinx series:

(all x)

cosx series:

(all x)

tanx series:

(|x| < π/2)

$sin^{-1}x$ series:


$tan^{-1}x$ series:


ln(1+x) series:


sinhx series:

(all x)

coshx series:

(all x)

tanhx series:


$sinh^{-1}x$ series:


$tanh^{-1}x$ series:


Maclaurin Series Formula | What is the Maclaurin series?

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If the series is zero-centered (a=0), the Taylor series takes a simpler form and this special series is called the Maclaurin series after the Scottish mathematician Colin Maclaurin.

Maclaurin Series Formula:

Monday, August 28, 2023

Epicycloid Definition | Epicycloid Parametric Equations

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Parametric equations:

This is the curve described by a point P on a circle of radius b as it rolls on the outside of a circle of radius a. The cardioid is a special case of an epicycloid.

Z Transforms | All Function and Transform

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You can find all the Z transforms below. We wish you good work...

Shift Theorem

Initial value theorem

Final value theorem

provided f(∞) exists.

Inverse Formula

Sunday, August 27, 2023

Mechanics Kinematics & Centres of Mass Definitions and Formulas


Motion constant acceleration


General solution of $\frac{d^2x}{dt^2}=-w^2x$ is

x=a.coswt + b.sinwt = R.sin(wt+φ)

where $R=\sqrt{a^2+b^2}$ and cosφ = a / R, sinφ = b / R.

In polar coordinates the celocity is  and the acceleration is

Centres of mass:

The following results are for uniform bodies:

  • hemispherical shell, radius r    ⇾    $\frac{1}{2}r$                    ⇾     from centre
  • hemisphere, radius r                 ⇾    $\frac{3}{8}r$                  ⇾     from centre
  • right circular cone, height h      ⇾    $\frac{3}{4}h$                  ⇾     from vertex
  • arc, radius r and angle 2θ         ⇾    (r sinθ) / θ       ⇾     from centre
  • sector, radius r and angle 2θ    ⇾    ($\frac{2}{3}r$ sinθ) / θ    ⇾     from centre

Chebyshev Polynomials

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You can find the Chebyshev polynomials below. We wish everyone a good lesson...

Leibnitz's Theorem | Leibnitz Derivative Rule

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You can find the Leibnitz derivative theorem below. We wish you good lessons...

Product Rule & Quotient Rule & Chain Rule (Derivative)

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You can find the product rule, quotient rule and chain rule in the derivative below. We wish you good lessons.

Product Rule:


Quotient Rule:


Chain Rule:


Tuesday, August 22, 2023

Laplace Transforms | Function and Transform (Examples)

You can find Laplace transforms below. We wish everyone good work.

You can find sample solutions below.

Sunday, August 20, 2023

General Rules of Differantiation

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In the following, u, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e=2.71828... is the natural base of logarithms; In u is the natural loraithm of u [i.e. the loraithm to the base e] where it is assumed thad u>0 and all angles are in radians.

Graphs of Hyperbolic Functions (Sinhx, Coshx, Tanhx, Cothx Graphics)

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y=sinh x graph:

y=cosh x graph:

y=tanh x graph:

y=coth x graph:

y=sech x graph:

y=csch x graph:

Expression of Hyperbolic Functions In Terms of Others (Table)

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 In the following we assume x>0.