Maclaurin Series Formula | What is the Maclaurin series?

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:

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Epicycloid Definition | Epicycloid Parametric Equations

 

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.

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Z Transforms | All Function and Transform

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

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Mechanics Kinematics & Centres of Mass Definitions and Formulas

Kinematics:

Motion constant acceleration

$v=u+ft$

$s=ut+\frac{1}{2}ft^2=\frac{1}{2}(u+v)t$

$v^2=u^2+2f.s$

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

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Chebyshev Polynomials

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

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Leibnitz's Theorem | Leibnitz Derivative Rule

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

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Product Rule & Quotient Rule & Chain Rule (Derivative)

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

Product Rule:

$\frac{d}{dx}(u(x).v(x))=u(x)\frac{dv}{dx}+v(x)\frac{du}{dx}$

Quotient Rule:

$\frac{d}{dx}(\frac{u(x)}{v(x)})=\frac{v(x)\frac{du}{dx}-u(x)\frac{dv}{dx}}{[v(x)]^2}$

Chain Rule:

$\frac{d}{dx}(f(g(x)))=f'(g(x))×g'(x)$

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