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

Review exercises 10 385
REVIEWEXERCISES10
1 Findtherateofchangeof f(x) = 5 +3x 2 atx = 2by
considering each ofthe intervals [2 − δx,2],
[2,2 + δx],[2 − δx,2 + δx].Showthatthesame
resultis obtained in each case.
2 Useatable ofderivatives and the linearity rules to
differentiate the following:
(a) y=4x 2 +6x−11
(b)y=−x 2 +2x−10
(c) y =x 1/3 −x 1/4
(d) y=5cos4x−3cos2x
(e) y = sin −1 (4x +3)
(f) y= √ 2x 4 − 5
3x 2
(g) y =t 3/2 +cost
(h)y=t 2 −14t+8
(i) y=5lnt+sin4t
(j) y= 1 2 x − 1 3 x2
(k)y= 2t2
3 +e2t
3 Find the equationofthe tangent toy =x 2 +7x −4 at
the point on the graph wherex = 2.
4 Find the rate ofchange of f (t) = 2cost +3sint at
t=1.
5 At any timet,the voltage, v, acrossan inductorof
inductanceLis relatedto the current,i,through the
inductorby v =L di
dt .
(a) Findan expression forthe voltage when
i = 5cosωt where ω isthe constantangular
frequency.
(b) Findan expression forthe voltage when the
current takesthe formofasinewave with
amplitude 10 andperiod 0.01 seconds.
6 Use the shrinking interval methodto findthe rate of
change of f (t) = sint att = 0 by consideringthe
interval [0, δt]. [Hint:use the trigonometricidentities
in Section3.6 andthe small-angle approximation in
Section6.5.]Usetheshrinkingintervalmethodtofind
the rate ofchange of f (t) = sint at ageneral point.
7 Giveny(t) = 3 +sin2t,findthe averagerate of
change ofyast variesfrom 0 to 2.
8 Explain the essentialdifference between δy dy
and
δx dx .
9 Findy ′ forthe followingfunctions:
(a) y =2e −t +6cos(t/2)
(b) y = (−t +2) 2
10 Using derivatives, estimatethe changeinyasx
changes from 1.5 to 1.55 wherey = 2e 2x +x 3 .
Solutions
1 12
2 (a) 8x+6
(b) −2x +2
1
(c) 3 x−2/3 − 1 4 x−3/4
(d) −20sin4x +6sin2x
4
(e) √
1−(4x+3) 2
(f) 2 √ 2x + 10
3x 3
3
(g) 2 t1/2 −sint
(h) 2t −14
(i)
5
t +4cos4t
(j)
(k)
1
2 − 2 3 x
4
3 t+2e2t
3 y=11x−8
4 −0.062
5 (a) −5ωLsin ωt (b) 2000πLcos200πt
7 −0.3784
( )
9 (a) −2e −t t
−3sin
2
10 4.355
(b) 2t−4