Download e-book for kindle: A Course of Mathematics for Engineers and Scientists. Volume by Brian H. Chirgwin, Charles Plumpton

By Brian H. Chirgwin, Charles Plumpton

ISBN-10: 0080063888

ISBN-13: 9780080063881

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Example text

Show that D [sin (ax -\- b)] = a sin (ax + b + j-π). Deduce that D w [sin (ax + b)] = an sin (ax -f- b + ^ηπ). Express T>n [cos (ax + 6)] in this form. 2:4 Exponentials, logarithms and hyperbolic functions The exponential function. , we define e x p (a;) as t h e function which is equal t o its own derivative a n d takes t h e value u n i t y when x = 0 . The unique function which satisfies t h e above conditions m a y be shown (see § 5:5) t o be expressible as a n infinite series in t h e form «Ρ(*) = 1 + X π + X ίΡ^ 2Γ+-3Γ + - + Χ^ 1ίΓ + ···.

X = ct, y=—. 73. x = acosnt, ί y = bsinnt, n ={= 0. τ a n d we can b y differend /djA t i a t i o n find t h e value of —— —— . This we call t h e second derivative ax \dxj à2y of y w . r . t o x a n d denote it b y A 22 (d2yldx2), f"(x) or O2y. , t h e 71 t h àny fn\x) orOny. derivative of f(x), b y - ^ , (dny/dxn), Q. X 2 n d {x ) Examples, (i) 2 = n(n — 1) xn~2. d 3 (sin*) (ii) —-z—^— = — cos x. d 1 d W h e n x = x(t), y = y(t), eqn. r. to x is equivalent t o differentiation w . r . t o t followed b y division b y x.

G{%) - l o g / O r ) . 1 dw /, * , ,, . flfia;)·/'^) y v / ö / v =g'(x)-\ogf(x) + ^ y dz * ' ' / ( s') * r . / = ax, then log y = x log a. Examples, (i) 1 dy d(a*) — —— = log a and therefore —-— = ax log a. y ax ax If y = xx, then (ii) log y = x log x ; 1 di/ — _ = l+log*, ά(«*) . - . +log*). Hyperbolic functions. There exist certain combinations of t h e exponential functions ex a n d e~x with properties which bear a close formal analogy with those of t h e trigonometric functions. We define these hyperbolic functions as follows X' 3 cosh x = -He« + t a n h a; β " " ) = sinh x cosh a: 1 X*n + * + | τ + | τ + · · · + | ^ Τ+ · · · ' e^ — e _ : r ex + e * 1 Ì cotn x = t a n h x , X5 1 sech x = cosh x ', i cosech x l sinh x Clearly cosh x is a n even function a n d cosh x > 1 whereas sinh x a n d t a n h x a r e o d d functions with j t a n h x | < 1.

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A Course of Mathematics for Engineers and Scientists. Volume 1 by Brian H. Chirgwin, Charles Plumpton


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