C1
1
Answers - Worksheet E
DIFFERENTIATION
a −1 + k + 3 = 0
2
k = −2
b f ′(x) = 3x2 − 4x
∴ 3x2 − 4x ≥ 0
x(3x − 4) ≥ 0
x ≤ 0 and x ≥
a
dy
dx
= 3x2 − 3
3x2 − 3 = 0
x2 = 1
x=±1
∴ (−1, 3) and (1, −1)
b PQ2 = 22 + 42 = 20
SP:
4
3
∴ PQ =
3
a y = 0 ⇒ x(x + 3)2 = 0
4
a volume = 2x2h = 288
x = −3, 0
b A = 2x2 + 2(2xh) + 2(xh)
= 2x2 + 6xh
3x2 + 12x + 9 ≤ 0
= 2x2 + (6x ×
3(x + 3)(x + 1) ≤ 0
= 2x2 +
∴ −3 ≤ x ≤ −1
c
144
x2
∴ h=
∴ (−3, 0), (0, 0)
b f ′(x) = 3x2 + 12x + 9
decreasing when
20 = 2 5
c
dA
dx
864
x
= 4x − 864x−2
SP:
O
x
e
(−1, −4)
d2 A
dx 2
= 4 + 1728x−3
when x = 6,
d2 A
dx 2
a 2x − 5 +
2
x
=0
6
a
(2x − 1)(x − 2) = 0
1
2
b SP:
,2
= 12
> 0 ∴ minimum
=
1
2
x
− 12
= − 14 x
− 12
− 4x−2
− 32
+ 8x−3
− 4x−2 = 0
1
2
x
1
2
x−2( x 2 − 8) = 0
3
x2 = 8
x=4
∴ (4, 3)
x=±1
when x = 4,
d2 y
dx 2
y
( 12 , 0)
O
d2 A
dx 2
3
b f ′(x) = 2 − 2x−2
∴ 2 − 2x−2 = 0
x2 = 1
c
dy
dx
d2 y
dx 2
2x2 − 5x + 2 = 0
x=
)
4x − 864x−2 = 0
x3 = 216
x = 3 216 = 6
d min A = 216
y
(−3, 0)
5
144
x2
(2, 0)
(1, −1)
x
(−1, −9)
 Solomon Press
d2 y
dx 2
=
3
32
> 0 ∴ minimum
C1
7
DIFFERENTIATION
a
dh
dt
Answers - Worksheet E
= 8t3 − 24t2 + 16t
1
4
b when t =
dh
dt
8
−
=
1
8
=
2 85
3
3
2
x = −3k, k
when x = k, y = k3 + 3k3 − 9k3 = −5k3
∴ stationary at (k, −5k3)
when x = −3k,
c y = −27k3 + 27k3 + 27k3 = 27k3
∴ (−3k, 27k3)
8t − 24t + 16t = 0
8t(t − 1)(t − 2) = 0
t = 0, 1, 2
from graph, max when t = 1
∴ max height = 3 cm
x 2 − ( 12 x) 2
a height of XS =
3
4
=
1000
3x 2
∴ l=
3
4
b A = (2 ×
c
×x×
or
x2 + (3x ×
=
3
2
(x2 +
SP:
3
2
3
2
x
x × l = 250
x2) + 3xl
3
2
=
x2 =
1000 3
3x 2
=
dA
dx
1000 3
3x 2
)
2000
)
x
3
2
(2x − 2000x−2)
3
2
(2x − 2000x−2) = 0
x3 = 1000
x = 10
d min A = 150 3
e
d2 A
dx 2
=
3
2
(2 + 4000x−3)
when x = 10,
d2 A
dx 2
x2 + 2kx − 3k2 = 0
b (x + 3k)(x − k) = 0
+4
2
1
2
= 3x2 + 6kx − 9k2
⇒
cm per second
volume =
dy
dx
stationary when 3x2 + 6kx − 9k2 = 0
,
= 8( 14 )3 − 24( 14 )2 + 16( 14 )
c SP:
9
a
page 2
d2 A
dx 2
=3 3
> 0 ∴ minimum
 Solomon Press