HYPERBOLIC FUNCTIONS

HYPERBOLIC FUNCTIONS

1. SIX HYPERBOLIC FUNCTION

   • sinh(spell:shine) =1/2(e^x -e^-x) cosh(cosh)=1/2(e^x-e^-x) tanh(tanch)=(e^2x -1)/(e^2x +1) csch(cosheck)=2/(e^x -e^-x) sech(sheck)=2/(e^x +e^-x) coth(coth)=(e^2x +1)/(e^2x -1)
   • GRAPHS
     • deduced by looking at e^x and sketch
   • Identities
     • i apperciate my exsitence from Moon and back = =
     • Oborn's rule
       • 1. firstly, if we assume that sinx corresponds to sinhx, cosx to coshx ... 2. Oborn's rule states that when you meet a product of sinx in application of their corresponding identities, change the sign. e.g. (sinx)^2 + (cosx)^2 = 1→ -(sinhx)^2 + (cosh)^2 =1
       • ABOUT 1
         • tanh^2 +sech^2 =1 coth^2 -csch^2 =1 cosh^2 -sinh^2 =1
     • 2倍角
       • e.g. csch2x= 1/(sinh2x)= 1/(2sinxcosx)

2. INVERSE HYPERBOLIC FUNCTION

   • sin^-1 OR arcsinx =y -->siny =x sin(arcsinx) =x
   • GRAPHS
     • 1.the graph is the reflection of hyperbolic functions about y =x 2. it must be 1-1 function(mapping), so domain limited
   • EXPRESSED IN Log(Ln)
     • IHF -->HF -->(By definition) expressed in e^x form --> quadratic function of e^x  ---> mapping decided by x=ln(...) ---> domain decided by graph eg. arcsinx =y, siny =x =1/2(e^x -e^-x) -->...  

3. DIFFERENTIATION

   • rate of change(看图说话) dy/dx>0 -->positive derivate dy/dx<0 -->negative derivate
   • 工具
     • 2nd-order derivative (higer-order derivatives)
     • FS: ???
     • FOR Composite functions
       • Quotient rule
       • Product rule
       • Chain rule
         • nx OR x^n please be aware of the constant change.
         • e.g. d/dx(arctanhx) --> x^2--(tanhx)^2, (tanhx)'=(sechx)^2 -->...
   • d/dx( six HF )
     • expressed in e^x form then deriviate it...
   • d/dx( six arcHF )
     • 1.arcsinx=y--> siny=x (use sub) 2.dy/dx=1/(dx/dy) 3.indentities: about 1--> to convert y-->x
     • e.g. d/dx(arcsinhx)=√(1+x^2)--> arcsinhx=y, sinhy=x, dx/dy(sinhy)=dx/dy, coshy=dx/dy--> dy/dx=1/coshy= 1/√(1+sinhy^2) = 1/√(x^2+1)

4. INTEGRATION

   • if y=∫f(x)dx, dy/dx=f(x)
   • 工具 (泛)
     • TWO DERIVED FORMULAS
       • ∫f'(x) /f(x) =ln f(x) +c ∫f(x)^n *f'(x) =f(x)^(n+1)/(n+1) +c --> reverse of product rule
       • e.g. ∫tanhx dx & ∫cothx dx --> derived formulas
         • trick for ∫tanhx dx *
     • BY PARTS *
       • when ∫(u)(dv/dx) dx OR ∫(u)(1) dx is harder than ∫(du/dx)(v) dx -->it change GUESS WORK from u(dv/dx) to v(du/dx) -->(现象) 处理 ∫(高次) dx 
       • to integrate HF & arcHF & lnx, we use integration by parts with v'=1.
       • REDUCTION FORMULAS for ∫(f(x))^g(n) * (linear function)^power dx
         • find a reduction formula= generate a recurrance sequence. i.e. an intergral In is expressed recurrisively. So as long as we know how to calculate Im for some m, this will work.
         • recurrance relation *
           • make your life easier= = when calculate I3 OR I4...  注意 domain of n changes.(recurrance relation & In=... differes.) founded by use intergarte by parts several times.> similar to the recurrance relation of number(xn+1=xn + A...)
           • 易错点: [...]limits 和 变形后应第一反应带入I(n) other than guess relation
             • 在inte 公式的第一步就做变形.就是变形成 ∫u v' dx, (v: linear function) 对于[类似的v]为quadratic function, 变形组合linear function 相乘-->既可以一步integer 的式子*
             • many reduction formula need a fiddle... while excusefucking me?
           • integrate by parts *2 OR change integrating agent then integrate by part *1 to figure out
             • then many reduction formulas need a fiddle like this: '加减改变项目' e.g. (1-x^2)^3/2=(1-x^2)*(1-x^2)^1/2
       • u: reducing agent for which has high power
       • v': integrating agent for which can be integrate easily
         • e.g. ∫arctanhx ... *
     • FS + DERIVED FORMULAS
       • ∫1/(x^2+C)^n dx= arcHF +c... *
         • for every x--> f(x) =Ax +B(拿出A) 每一个x的位置,都能换成f(x). * 需要注意, A 需要 reverse. * 需要注意, 我们关注的是 [被替换的x] 这一项, 而非改变该式子的其他 [未被替换项].
       • ∫(分子)/(分母) dx: 分子是分母的崎次变化的那种 ??
         • e.g. ∫2e^x/(9+e^2x) dx= 2/3 arctan (e^x/3) *
     • PARTIAL FRACTIONS *
       • e.g. ∫1/(1-x^2) dx= 1/2∫ 1/(1-x) + 1/(1+x) dx= 1/2ln[(1+x)/(1-x)] + c--> arctanhx
     • RECOGNISE A STANDARD FUNCTION : REVERSE WORK
     • USING FORMULAS TO CHANGE INTEGRAND
       • EXPRESSED IN e^x FORM
         • when ∫ e^x dx is easier than ∫...
         • e.g. ∫sinhx dx, ∫coshx dx
       • Identities
     • STANDARD SUBSTITUTIONS ??
       • use FS for inspiration. every x--> f(x)=Ax+B, 拿出A??
       • 2. change integrand, by identities ...
       • 1. change respect
       • 3. change limit boundaries
   • APPLICATION
     • Area of a surface of revolution, A=∫ 2πy ds--> (ds/dx) dx while ds/dx=√(1+(dy/dx)^2) OR (ds/dt) dt while ds/dt=√[(dy/dt)^2+(dx/dt)^2]
     • arc length
       • by Pythagorean theorem
         • s=∫√1+(dy/dx)^2 dx   =∫√1+(dx/dy)^2 dy --> introducing parameters s=∫√[(dx/dt)^2 + (dy/dt)^2] dt
   • ∫six arcHF dx *
     • integration by parts as we already can integrate u'(x)-- but still can not integrate u(1).
   • ∫HF dx
     • ∫sechx dx OR ∫cschx dx
       • 工具: 1. e^x-->e^2x          2. FS          3. do reverse work
       • 工具: 1. e^x-->e^2x          2. FS          3. do reverse work
       • e.g. ∫sechx dx= ∫ 2/(e^x+e^-1) dx= ∫ 2e^x/(e^2x+1) dx=> reverse work <= FS: d/dx(2arctan(e^x))= 2e^x/(e^2x+1) -->∫sechx dx= 2artan(e^x) *
     • ∫sinhx dx OR ∫coshx dx
       • expressed in e^x form. then see.
     • ∫tanhx dx OR ∫cothx dx
       • derived formulas

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