If we apply the decomposition developed in the previous two problems to the expo
ID: 1536868 • Letter: I
Question
If we apply the decomposition developed in the previous two problems to the exponential function, we get the hyperbolic functions, exp(x) exp(-x) cosh(x) exp(x) exp(-x) sinh(x) These functions are covered in your calculus text, but sometimes that section is skipped. They are closely related to the usual trigonometric sinh(x) The next few functions and you can define hyperbolic tangent, secant, cosecant, and cotangent in the obvious way (e.g. tanh(x) cosh(x) problems give a quick overview of some of their properties.Explanation / Answer
22. we have to show if e^(rx) and e^(sx) are linearly independent for s != r
consider
a(e^(rx)) + b(e^(sx)) = F
F = a(e^r * e^x) + b(e^s * e^x)
if we can find F = 0 for some value of a and b such that any value of x will work then the two fucntions are not linearly independent
so, assume F = 0
a(e^rx) + b(e^sx ) = 0
if this equation is true and then so is its differentiation
are^rx + bse^sx = 0
but e^rx = -be^sx / a
ar(-be^sx / a)) + bse^sx = 0
bre^sx = bse^sx
r = s
but r! = s
so the functions are linearly independent unless r = s