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

402 Chapter 11 Techniques of differentiation
notation is used. The third derivative is written d3 y
dx 3 ory′′′ ory (3) . The fourth derivative
iswritten d4 y
dx 4 oryiv ory (4) . The fifth derivative iswritten d5 y
dx 5 oryv ory (5) .
Example11.21 Findthe first five derivatives ofz(t) = 2t 3 +sint.
Solution z ′ =6t 2 +cost z iv =sint
z ′′ =12t−sint z v =cost
z ′′′ =12 −cost
Example11.22 Calculate the values ofxforwhichy ′′ = 0,giveny =x 4 −x 3 .
Solution y=x 4 −x 3 y ′ =4x 3 −3x 2 y ′′ =12x 2 −6x
Puttingy ′′ = 0 gives
Hence
12x 2 −6x =0 andso 6x(2x −1) =0
x=0, 1 2
The first and second derivatives can be used to describe the nature of increasing and
decreasing functions. In Figure 11.2(a, b) the tangents to the curves have positive gradients,
that isy ′ > 0. As can be seen, asxincreases the value of the function increases.
Conversely, in Figure 11.2(c, d) the tangents have negative gradients (y ′ < 0) and as
x increases the value of the function decreases. The sign of the first derivative tells us
whetheryis increasing or decreasing. However, the curves in (a) and (b) both showy
increasingbut, clearly, thereisadifference inthe wayychanges.
ConsideragainFigure11.2(a).ThetangentsatA,BandCareshown.Asxincreases
thegradientofthetangentincreases,thatisy ′ increasesasxincreases.Sincey ′ increases
as x increases then the derivative of y ′ is positive, that is y ′′ > 0. (Compare with: y
increaseswhenitsderivativeispositive.)SoforthecurveshowninFigure11.2(a),y ′ > 0
andy ′′ > 0.
For that shown in Figure 11.2(b) the situation is different. The value ofy ′ decreases
asxincreases, as can be seen by considering the gradients of the tangents at A, B and
C, thatisthe derivative ofy ′ mustbenegative. For thiscurvey ′ > 0 andy ′′ < 0.
A function is concave down wheny ′ decreases and concave up wheny ′ increases.
HenceFigure11.2(a)illustratesaconcaveupfunction;Figure11.2(b)illustratesaconcavedownfunction.Thesignofthesecondderivativecanbeusedtodistinguishbetween
concave upand concave down functions.
Consider now the functions shown in Figure 11.2(c) and Figure 11.2(d). In both (c)
and(d),yisdecreasingandsoy ′ < 0.In(c)thegradientofthetangentbecomesincreasinglynegative;thatis,itisdecreasing.Hence,forthefunctionin(c)y
′′ < 0.Conversely,
for the function in (d) the gradient of the tangent is increasing asxincreases, although
it is always negative, that isy ′′ > 0. So for the function in (c)y ′ < 0 andy ′′ < 0; that