By Noel M. Morris

A pocket reference of crucial formulae overlaying: digital and electric engineering, measurements and keep an eye on, common sense, telecommunications and arithmetic. Of worth to scholars at either BTEC nationwide and better point, in addition to at undergraduate point, specially these learning digital and electric engineering.

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V ~- u d;; v2 v Leibnitz theorem Allowing u

and v

to denote the pth derivatives of u and v with respect to x, then n(n- 1) dn u (uv)=uv+nu+ - 2! dxn + n(n- 1Xn- 2) uv 3! Function of a [unction If y = f(u ), where u = cp(x ), then dy dy du -=-·dx du dx Maximum and minimum values off(x) A curve has a maximum value at a point if, at that point, d2 y dy dx =0 and dx 2 has a negative value. A curve has a minimum value at a point if, at that point ..

For every repeated factor (ax + b ) 2 of M(x) there is a · a correspond"mg par t"Ial f rae t"wn t h ere IS ax+b I -In (ax+ b) ax+ b sec 2 x ax +b 2. For every quadratic factor (ax 2 + bx + c) of M(x) corresponding partial fraction _P_ + ( In x X 1. For every linear factor (ax + b) of M(x) there is a --=----=--- + -------:: ax 2 + bx + c (ax 2 + bx + c) 2 X sin -t - (a2 - x2 )~ a I -tan -t I x2 + a2 cosh x sinh x sech 2 x a sinh x cosh x tanh x a Integration by parts fu dxdv 'Cover-up' rule If X dx = uv - fv dudx dx f(x) = p(x) (x +a)(x + b)(x +c)··· then the numerators of the separate fractions due to the factors (x +a), (x + b), etc, are determined by 'covering up' each of the factors in turn and evaluating the remainder of the expression by replacing each x term by the value of x which makes the 'covered up' factor zero.

For example, if x2 +x + 2 f(x)------(x + 3)(x + 2)(x + 1) then OPERATOR D {(-3) 2 - {(-2) 2 Dy= dy dx 3 + 2}/(-3 + 2)(-3 + 1) «x)=~~---~x-+-3-~-~ 2 + 2}/(-2 + 3)(-2 + 1) - +~~---~~-~~-~ x+2 {(-1) 2 1 + 2}/(-1 + 3)(-1 + 2) - +~~---~~-~~-~ x+1 I - f(x)= D D -1 {f(x)}= Jf(x) dx 4 4 1 =-----+-x+3 x+2 x+1 25 LAPLACE TRANSFORMS If F(t) is a function oft for values oft> 0, then the Laplace transformation of F(t) is J: e_, F(t) dt F(t) Laplace transformation 1 (unit step function) 1/s A 6 (unit impulse function) A/s 1 s+a a s(s +a) P-a (s + a)(s + (j) t (ramp function) 1 ;2 (s + a2 ) t" (n a positive integer) n!

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