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

10.6 Existence of derivatives 371
x = a. The function shown in Figure 10.14(b) is continuous but has a cusp or corner
atx = a. In both cases it is impossible to draw a tangent atx = a, and so dy does not
dx
existatx =a.Itisimpossibletodrawatangent toacurve atapointwhere thecurve is
not smooth. Note from Figure 10.14(b) that continuity is not sufficient to guarantee the
existence of a derivative.
Example10.10 Sketch the following functions. State the values oft for which the derivative does not
exist.
(a) y=|t| (b) y=tant (c) y=1/t
Solution (a) Thegraphofy = |t|isshowninFigure10.15(a).Acornerexistsatt = 0andsothe
derivative doesnotexist here.
(b) A graph ofy = tant is shown in Figure 10.15(b). There is a discontinuity in tant
when t = ... −3π/2,−π/2,π/2,3π/2,.... No derivative exists at these
points.
(c) Figure 10.15(c) shows a graph of y = 1/t. The function has one discontinuity at
t = 0,and so the derivative doesnotexist here.
y
y = tan t
y
y
y = |t|
–
p
–
p
–
2
2
t
y = 1_ t
t
(a)
t
(b)
Figure10.15
(a) There is acorner att = 0; (b)tant has discontinuities; (c)y = 1/t has a discontinuity att = 0.
(c)
EXERCISES10.6
1 Sketchthe functionsanddetermineany pointswhere
aderivative does notexist.
(a) y= 1
t −1
(b) y=|sint|
(c) y=e t
(d) y = |1/t|
{ 1 t0
(e) The unitstepfunctionu(t) =
0 t<0
{ ct t0
(f) The ramp function f (t) =
0 t<0