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

384 Chapter 10 Differentiation
(g)y= 2w3
3 + e4w
2
(h)y= √ ( √x )
x+ln .[Hint: √ x =x 1/2 ,and use the
laws oflogarithms.]
2 Evaluate the derivatives ofthe functionsat the given
value:
(a) y=2t+9+e t/2 t=1
(b) y= t2 −4t+6
t = 2
3
(c) y=sint+cost t=1
( t
(d) y=3e 2t −2sin t = 0
2)
(e) y = 5tan(2x) + 1
e 2x x=0.5
(f) y =3lnt +sin(3t) t =0.25
3 Find dx
dt , if
(a) x = e ωt (b) x = e −ωt
where ω isaconstant.
4 Find the derivative of
(a) y = 3sin −1 (2t) −5cos −1 (3t)
(b) y= 1 2 tan−1 (t +2) +4cos −1 (2t −1)
( )
t −3
(c) y = 2sinh(3t −1) −4cosh
2
cosech(4t) +3sech(6t)
(d) y=
2
( ) ( )
(e) y = 2sinh −1 t +1
−3cosh −1 1 −t
2 2
(f) y = 3tanh −1 (2t +3) −2tanh −1 (3t +2)
5 Afunction,y(t),is given by
y(t) = t3 3 − 5t2
2 +4t+1
(a) Find dy
dt .
(b) Forwhich values oft isthe derivative zero?
6 Findthe equationofthe tangent to the curve
y(x)=x 3 +7x 2 −9
atthe point (2,27).
7 Findvalues oft in the interval [0,π] forwhichthe
tangent tox(t) = sin2t has zerogradient.
8 Findthe rate ofchangeof
when
(a) t=0
(b) t=3
z(t) =2e t/2 −t 2
Solutions
1 (a) 12x 2 −10x (b) 15cos5t +8e 4t
(c) 4cos4t −6sin2t −1
(d) 3sec 2 3z
(e) 6e 3t −8cos2t (f) − 3
t 4 −2.5sin5t
(g) 2w 2 +2e 4w (h) 0.5x −1/2 + 1 2x
2 (a) 2.8244 (b) 0 (c) −0.3012
(d) 5 (e) 33.5194 (f) 14.195
3 (a) ωe ωt (b) −ωe −ωt
4 (a)
(b)
6
√ + 15
√
1−4t 2 1−9t 2
1
2[1+ (t +2) 2 ] − 4
√ t(1−t)
(c) 6cosh(3t −1) −2sinh t −3
2
(d) −2cosech4tcoth4t −9sech6ttanh6t
(e)
(f)
1
√ (1
2 (t+1) ) 2
+1
+
6
1−(2t+3) 2 − 6
1−(3t+2) 2
5 (a) t 2 −5t+4 (b) 1,4
6 y=40x−53
7 π/4,3π/4
8 (a) 1 (b) −1.5183
3
2√ (1
2 (1−t) ) 2
−1