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

11.4 Higher derivatives 403
y
C
B
y
C
A
y
B
A
y
(a)
A
B
x
A
(b)
x
(c)
C
x
(d)
B
C
x
Figure11.2
(a)yis concave up (y ′ > 0,y ′′ > 0);(b)yisconcave down (y ′ > 0,y ′′ < 0);(c)yis concave down (y ′ < 0,y ′′ < 0);(d)y
is concave up (y ′ < 0,y ′′ > 0).
is, the function is concave down. For the function in (d)y ′ < 0 andy ′′ > 0; that is, the
function isconcave up. Insummary, we can state:
Wheny ′ > 0,yisincreasing.Wheny ′ < 0,yisdecreasing.
Wheny ′ isincreasingthe functionisconcave up. Inthiscasey ′′ > 0.
Wheny ′ isdecreasing the functionisconcave down. Inthiscasey ′′ < 0.
EXERCISES11.4
An easy way of determining the concavity of a curve is to note that as the curve is
traced from left to right, an anticlockwise motion reveals that the curve is concave up.
Aclockwisemotionmeansthatthe curve is concave down.
Aswillbeseeninthenextchapter,higherderivativesareusedtodeterminethelocationandnatureofimportantpointscalledmaximumpoints,minimumpointsandpoints
ofinflexion.
1 Calculate dy
dt and d2 y
dt 2 given
(a) y=t 2 +t
2 If
(b) y=2t 3 −t 2 +1
(c) y = sin2t
(d) y = sinkt
(e) y=2e 3t −t 2 +1
(f) y=
t
t +1
(g) y=4cos t 2
(h) y =e t t
(i) y = sinh4t
(j) y = sin 2 t
k constant
y=2x 3 +3x 2 −12x+1
findvalues ofxforwhichy ′′ = 0.
3 If dy
dt =3t2 +t,find
(a)
d 2 y
dt 2
(b)
d 3 y
dt 3
4 Find values oft atwhichy ′′ = 0,where
y = t3 3 − 7t2
2 +12t−1
5 Determine whether the following functionsare
concave up orconcave down.
(a) y=e t (b) y=t 2 (c) y=1+t−t 2
6 Determine the interval onwhichy =t 3 is
(a) concave up,
(b) concave down.
7 Evaluatey ′′ atthe specifiedvalue oft.
(a) y=2cost−t 2 t=1
(b) y= sint+cost
2
t=π/2
(c) y=(1+t)e t t=0
8 Find d2 y
dx 2 givenxy +x2 =y 2 .
9 Find dx
dt whenx3 + x t =t2 +x 2 t.