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

12.3 Points of inflexion 415
Solutions
1 (a)
( 1
2 ,53 4)
,minimum
(b) (−0.75,−0.125),minimum
(c) none
(d) (0.25,1.125),maximum
(e) (2,−16) minimum, (−2,16) maximum
(f) none
2 (a)
(
2, 23 ) (
maximum, 4, 19 )
minimum
3 3
(b) (1,0.368),maximum
(c) (0,0) maximum, (1,−1) minimum, (−1,−1)
minimum
TechnicalComputingExercises12.2
Computerlanguagessuch asMATLAB ® are matrix
orientatedanddonotalways providethe ability to
differentiate functions.Others such asWolfram
Mathematica andMaplesoftMaple have thiscapability by
default. Ifyou are attempting the followingexercisesin
MATLAB ® you mayrequire the Symbolic Math Toolbox
whichis an add-on forthe main program.
(a) Useatechnicalcomputinglanguagetofindy ′ and
y ′′ wheny = e −0.2t cost.
(b) Solvey ′ = 0 andhence locate any turningpoints
in the interval [0,6]anddeterminetheirtype.
(c) Plotagraphofyandcheck the position of
the turningpointswith the resultsobtainedin
part (b).
12.3 POINTSOFINFLEXION
Recall from Section 11.4 that when the gradient of a curve, that is y ′ , is increasing,
the second derivative y ′′ is positive and the curve is said to be concave up. When the
gradient is decreasing the second derivative y ′′ is negative and the curve is said to be
concave down. A point at which the concavity of a curve changes from concave up to
concave down orviceversa iscalled a point ofinflexion.
Apointofinflexionisapointonacurvewheretheconcavitychangesfromconcave
up to concave down or vice versa. It follows that y ′′ = 0 at such a point or, in
exceptionalcases,y ′′ doesnotexist.
Figure 12.10(a) shows a graph for which a point of inflexion occurs at the point
markedA.Notethatatthispointthegradientofthegraphiszero.Figure12.10(b)shows
a graph with points of inflexion occurring at A and B. Note that at these points the
gradientofthe graph isnotzero.
Tolocateapointofinflexionwemustlookforapointwherey ′′ = 0ordoesnotexist.
We mustthenexamine the concavity ofthe curve oneither sideofsuchapoint.