082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

100 Chapter 2 Engineering functions
TechnicalComputingExercises2.4.4
1 Useatechnical computing language to draw
y =log(kx)for0.5 x 50fork =1,2,3and4.
( 1
2 Drawy=lnxandy=ln for
x)
0.5 x20. What doyou observe? Can you explain
your observation usingthe laws oflogarithms?
3 Drawy=lnxandy=1− x 3 for
0.5 x4.Fromyourgraphs statean approximate
solution to
lnx=1− x 3
2.4.5 Thehyperbolicfunctions
The hyperbolic functions are y(x) = coshx, y(x) = sinhx, y(x) = tanhx, y(x) =
sechx, y(x) = cosechx and y(x) = cothx. Cosh is a contracted form of ‘hyperbolic
cosine’, sinh of‘hyperbolic sine’ and so on. We definecoshx and sinhx by
y(x) =coshx = ex +e −x
2
y(x) =sinhx = ex −e −x
2
Note:
cosh(−x) = e−x +e x
= coshx
2
sinh(−x) = e−x −e x
=−sinhx
2
so, for example, cosh1.7 = cosh(−1.7) and sinh(−1.7) = −sinh1.7. Clearly, hyperbolicfunctionsarenothingotherthancombinations
oftheexponential functionse x and
e −x .However,theseparticularcombinationsoccursofrequentlyinengineeringthatitis
worth introducing the coshx and sinhx functions. The remaining hyperbolic functions
are defined interms of coshx and sinhx.
y(x) =tanhx = sinhx
coshx = ex −e −x
e x +e −x
y(x) =sechx = 1
coshx = 2
e x +e −x
y(x) = cosechx = 1
sinhx = 2
e x −e −x
y(x) =cothx = coshx
sinhx = 1
tanhx = ex +e −x
e x −e −x
Values of the hyperbolic functions for various x values can be found from a scientific
calculator. UsuallyaHyp button followed by a Sin, Cos or Tan button isused.
Example2.19 Evaluate
(a) cosh3
(c) tanh1.6
(e) coth1
(b) sinh(−2)
(d) sech(−2.5)
(f) cosech(−1)