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

418 Chapter 12 Applications of differentiation
EXERCISES12.3
1 Locate the maximumpoints,minimum pointsand (e) y=t 5
pointsofinflexion of
(f) y=t 6
(a) y=3t 2 +6t−1
(g) y=x 4 −2x 2
(b) y=4−t−t 2
(h) z=t+ 1
(c) y= x3
3 − x2
t
2 +10
(i) y=x 5 − 5x3
(d) y= x3
3 + x2
3
2 −20x+7 (j) y =t 1/3
Solutions
1 (a) (−1,−4) minimum
(b) (−0.5,4.25) maximum
(c) (0,10) maximum,
( 1
2 , 119
12
(
(d) 4, − 131
3
(
1, 59
6
)
point ofinflexion
)
minimum,
)
minimum, (−5,77.83) maximum,
(− 1 2 ,17.08 )
point ofinflexion
(e) (0,0) point ofinflexion
(f) (0,0) minimum
(g) (0,0) maximum, (1,−1) minimum,
( ) ( )
1
(−1,−1) minimum, √ , − 5 , −√ 1 , − 5
3 9 3 9
pointsofinflexion
(h) (1,2) minimum, (−1, −2)maximum
(
(i) 1, − 2 ) (
minimum, −1, 2 )
maximum,
3 3
( ) ( )
1
(0,0), √ , − 7
2 12 √ , −√ 1 7
,
2 2 12 √ are
2
alsopointsofinflexion
(j) (0,0) point ofinflexion
TechnicalComputingExercises12.3
(a) Useatechnicalcomputing language such as
MATLAB ® to produce agraph ofy = 3t 1/5 .
(b) Fromyour graphfind the position ofany maxima,
minimaorpointsofinflexion.
12.4 THENEWTON--RAPHSONMETHODFORSOLVING
EQUATIONS
We often needtosolve equationssuchas
f(x)=2x 4 −x 3 +x 2 −10=0
f(t) =2e −3t −t 2 = 0
f(t)=t−sint=0
The Newton--Raphson technique is a method of obtaining an approximate solution, or
root, ofsuchequations.Itinvolves the use ofdifferentiation.