By Brian H. Chirgwin, Charles Plumpton
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Additional info for A Course of Mathematics for Engineers and Scientists. Volume 1
The final result is — x5 sin x. Exercises 2:3 n m 1. Find O (x ) considering separately the three cases (a) n < ra, (b) n = m, (c) n > m. 2. 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 + ···.
Az a(ax + by) a(xmyp) Then a; = F' (ax + by) a'x + G' (xmyP) ax where F' (ax + by) denotes the function F' (u) of the function u = ax + by. In fact we may consider F' (ax + by) as the derivative of F (ax + by), where ax + by is regarded as a (ii) If z iJzjay. = F(ax single variahIe. az ... a; Similarly Since az ay = bF'(ax :~ + by) + m xm-1yPG'(xmyP). = aF'(ax + by) + pxmyP-IG'(xmyP). r. to x aI axa(a a;I)' ,the result being denoted by ax 2 giving :y (::) 2 is denoted by ·· d erlvatlves ::~ or or f,;,;.
Sinh n a; cosh m a;. 8. log sin x. 9. log cos a;. 11. l o g i i - ^ g - j . 14. 4. eaxsinbx. * . 20. 10. log (a;3 + 1 ) Ï (a;4 - 1)τ 18. (e*-l) (e* + 1) 1 +x\ 1 — x)' 13. V» + log (l - V « ) . cosh a; + cos x sinh x + sin a; 17. t a n ( a + bx). 6. x*l°8x. § 2:5 THE T E C H N I Q U E OF D I F F E R E N T I A T I O N 47 21. Solve the equation sinh x = e~x + 1. 22. Express tanh x in terms of e2x and show directly from the definition of the hyperbolic functions that tanh x + tanh y tanh (x + v) = -ς—r~:—; :—; · v y} 1 + tanh x tanh y 23.
A Course of Mathematics for Engineers and Scientists. Volume 1 by Brian H. Chirgwin, Charles Plumpton