EXPONENTIAL FUNCTIONS
24.94
Sketch the graph of
Fig. 24-14
205
See Fig. 24-13.
The critical numbers are
The first-derivative test shows that
yields a relative maximum and
a relative minimum. There is a vertical asymptote at *=-!. y>0 for x<-\ and
as
and
since
as
since
and
Fig. 24-13
Problems 24.95-24.110 concern the hyperbolic sine and cosine functions sinh x = |(e* — e *) and
cosh* = \(e* + e~
x ).
24.95
24.96
24.97
Find D (sinh x) and D,(cosh x).
Find Dl(sinh x) and £>^(cosh x).
By Problem 24.95, D*(sinh x) = D x (cosh x) = sinh x and £>*(cosh x) = Z),(sinh x) = cosh x.
Graph y = sinh x.
See Fig. 24-14. Since D x (sinh *) = cosh x>0, sinhx is an increasing function. It is clearly an odd
function, so sinh 0 = 0. Since D x (sinh x) = sinh x, the graph has an inflection point at (0,0), where the
slope of the tangent line is cosh 0=1.