Calculus of the Hyperbolic Functions: Fresh Take

  • Differentiate and integrate hyperbolic functions and their inverse forms
  • Understand the practical situations where the catenary curve appears

Derivatives and Integrals of the Hyperbolic Functions

The Main Idea 

  • Definitions of Hyperbolic Functions:
    • sinhx=exex2sinhx=exex2
    • coshx=ex+ex2coshx=ex+ex2
  • Key Derivatives:
    • ddx(sinhx)=coshxddx(sinhx)=coshx
    • ddx(coshx)=sinhxddx(coshx)=sinhx
    • ddx(tanhx)=sech2xddx(tanhx)=sech2x
  • Key Integrals:
    • sinhx,dx=coshx+Csinhx,dx=coshx+C
    • coshx,dx=sinhx+Ccoshx,dx=sinhx+C
    • sech2x,dx=tanhx+Csech2x,dx=tanhx+C
  • Similarities with Trigonometric Functions:
    • Derivatives of sinh and sin are similar
    • But ddx(cosx)=sinxddx(cosx)=sinx, while ddx(coshx)=sinhxddx(coshx)=sinhx
  • U-substitution is often useful for integrals involving hyperbolic functions

Evaluate the following derivatives:

  1. ddx(tanh(x2+3x))ddx(tanh(x2+3x))
  2. ddx(1(sinhx)2)ddx(1(sinhx)2)

Evaluate the following integrals:

  1. sinh3xcoshxdxsinh3xcoshxdx
  2. sech2(3x)dxsech2(3x)dx

Evaluate the following integral:

xsinh(x2),dxxsinh(x2),dx

Calculus of Inverse Hyperbolic Functions

The Main Idea 

  • All hyperbolic functions have inverses with appropriate range restrictions
  • Key Derivatives:
    • ddx(sinh1x)=11+x2ddx(sinh1x)=11+x2
    • ddx(cosh1x)=1x21ddx(cosh1x)=1x21
    • ddx(tanh1x)=11x2ddx(tanh1x)=11x2
  • Key Integrals:
    • 11+u2du=sinh1u+C11+u2du=sinh1u+C
    • 1u21du=cosh1u+C1u21du=cosh1u+C
    • 11u2du={tanh1u+Cif |u|<1 coth1u+Cif |u|>111u2du={tanh1u+Cif |u|<1 coth1u+Cif |u|>1
  • Each inverse hyperbolic function has specific domain and range restrictions
  • Derivatives of inverse hyperbolic functions are found using implicit differentiation

Evaluate the following derivatives:

  1. ddx(cosh1(3x))ddx(cosh1(3x))
  2. ddx(coth1x)3

Evaluate the following integrals:

  1. 14x21dx
  2. 12x19x2dx

Evaluate the following integral:

xx2+9dx

Applications of Hyperbolic Functions

The Main Idea 

  • Catenary Curves:
    • A catenary is the curve formed by a hanging cable or chain under uniform gravity
    • Mathematically modeled by: y=acosh(xa)
  • Arc Length Formula:
    • Used to calculate the length of a catenary
    • Arc Length=ba1+[f(x)]2dx
  • Hyperbolic Function Properties:
    • 1+sinh2x=cosh2x
    • This identity is useful in simplifying arc length calculations

Assume a hanging cable has the shape 15cosh(x15) for 20x20. Determine the length of the cable (in feet).

A suspension bridge cable forms a catenary with the equation y=50cosh(x50), where x and y are measured in meters. The cable is anchored at x=100 and x=100. Calculate the length of the cable.