Calculus Techniques for Solved Assignments, Essays, Dissertations

Running head: CALCULUS TECHNIQUES 1
Calculus Techniques
Name
Institution

CALCULUS TECHNIQUES 2
Task 1-Part a
Question a
y=4 x3−9 x2 +3 x−4
dy
dx =4 (3 ) x3−1−9 ( 2 ) x2−1+3 ( 1 ) x1−1−0=12 x2−18 x+3
Question b
s=4 e3 t −2 e−3 t
ds
dt =4 ( 3 ) e3 t−2 ( −3 ) e−3 t =12 e3 t −6 e−3 t
Question c
y=4 cos (5 x )−3 sin (2 x)
dy
dx =4 d
dx cos ( 5 x ) −3 d
dx sin (2 x)
In evaluating d
dx ( 4 cos ( 5 x ) ) ,let 5 x=u
cos (5 x )=cosu
du
dx =5 , d
du ( cosu ) =−sinu
d
dx (4 cos ( 5 x ) )=4 du
dx × d
du =−4 (5)sinu=−20 sin (5 x )
In evaluating d
dx ¿let 2 x=m

CALCULUS TECHNIQUES 3
sin ( 2 x )=sin ( m )
dm
dx =2, d
dm ( sin (m) ) =cos (m)
d
dx ¿
dy
dx =−20 sin ( 5 x ) −6 cos (2 x)
Task 1-Part b
Question d
∫(8¿ x3 +5 x2−4 x )dx=8∫ x3 dx +5∫ x2 dx−4∫ x dx ¿
¿ 8 x3 +1
3+1 +5 x2+1
2+1 −4 x1+1
1+1
¿ 8 x4
4 +5 x3
3 −4 x2
2
¿ 2 x4 + 5
3 x3−2 x2 +C
Question e
∫(¿ e2 t−3 e6 t )d t=∫e2 t dt−3∫ e6 t dt ¿
In evaluating ∫ e2 t dt ,let 2 t=u , dt=1
2 du
∫ e2 t dt= 1
2 ∫eu du= 1
2 eu= 1
2 e2 t