工程数学(一).pdf

工程數學 一 補充資料 1 呂學育 September 2004 Definition The derivative of a function f is denoted as f and defined as following ax afxf af ax lim h xfhxf xfxfD h x lim 0 Example If 2 xxf find xfand 2 f h xfhxf xf h lim 0 h xhx h 22 0 lim h hhx h 2 0 2 lim 2 lim 0 hx h x2 422 2 f xfxgxgxfxgxfDx h xgxfhxghxf xgxfD h x lim 0 h xgxfxghxfxghxfhxghxf h lim 0 h xfhxf xg h xghxg hxf hh lim lim 00 xfxgxgxf Example Find 33 xxxxDx xxxf 3 xxxg 3 13 13 232333 xxxxxxxxxxDx xx26 5 hxg xf D xg xf hxg hxf h x 0 lim lim 0 xghxhg xfhxgxghxf h lim 0 xghxhg xfxghxgxgxfhxf h lim 0 xghxg xfxgxgxf h lim 2 0 xg xfxgxgxf h Example Find 4 2 2 x x Dx 2 xxf 4 2 xxg 22 2222 2 2 4 4 4 4 x xxDxxD x x D xx x 22 22 4 2 4 2 x xxxx 22 4 8 x x The Chain Rule xgxgfxgfDx Set xgfxF h xgfhxgf xF h lim 0 h xghxg xghxg xgfhxgf h lim 0 xgxgf Dff first derivative of f fDff 2 second derivative of f fDff 3 third derivative of f fDff 4 4 fourth derivative of f x read delta x a change of x in x xfy xfxxfy x y xf dx dy x lim 0 dx dy xf 2 2 dx yd xf 3 3 dx yd xf n n n dx yd xf dx dv dx du vu dx d dx du v dx dv uuv dx d zfy and xgz so that xgfy dx dz dz dy dx dy xgf dz dy xg dx dz differential dx dy xfy dx dx dy dxxfdy xfdxxfDx CxfdxxfDx Definition A function is called 1 1 one to one function if and only if 21 xfxf for any two differential elements 1 x and 2 x of the domain of f If xfyyx is a 1 1 function then its inverse designated by 1 f is xfyxy xfy 1 yfx If f and g are inverse functions which are differentiable then 1 xgf xg whenever 0 xgf xxgf By implicit differentiation xDxgfD xx by the chain rule 1 xgxgf and 1 xgf xg Definition For any positive number a 1 a the exponential function with base a is defined by x axf domain f yxyx aaa yx y x a a a xyyx aa set x axf By definition for any number x h xfhxf xf h lim 0 h aa xhx h 0 lim h aaa xhx h 0 lim h a a h x h 1 lim 0 h a axf h h x 1 lim 0 xx x akaD h a k h h 1 lim 0 The inverse of the exponential function x axf is called the logarithm to the base a abbreviated a log domain 0 log a Thus yx a log if and only if xa y kx xD ax 1 log for all 0 x 1loglog log 1 1 1 aa x a x xtdt kx dt tk x x a 1 11 log h a k h h 1 lim 0 2 a 693 0 k 3 a 099 1 k The Natural Logarithm dt t x x 1 1 ln domain 0 ln xx e logln 71828 2 e n n n e 1 1lim x xDx 1 ln 2 2 1 ln x xDx baablnln ln ba b a lnlnln ara r lnln The natural exponential function x ex exp domain exp Definition The hyperbolic sine designated by sinh 2 sinh xx ee domain sinh The hyperbolic cosine designated by cosh 2 cosh xx ee domain cosh xxe x coshsinh Periodic function xxsin p2sin xxcos p2cos sinhcoscoshsin sin xxhx h xhx xD h x sin sin lim sin 0 h xxx h sinsinhcoscoshsin lim 0 h xx h sinhcos 1 coshsin lim 0 h x h x hh sinh lim cos 1cosh lim sin 00 0 1cosh lim 0 h h 1 sinh lim 0 h h xcos h xhx xD h x cos cos lim cos 0 h x h x