NCEA Level 3 Calculus (91578) 2015 — page 1 of 6
Assessment Schedule – 2015
Calculus: Apply differentiation methods in solving problems (91578)
Evidence
Q1
Expected Coverage
(a)
30sec 2 (5x)
(b)
2
dy
= 3 4x − 3x 2 ( 4 − 6x )
dx
dy
At x = 1,
= 3 × 1× −2 = −6
dx
(c)
)
2
x
+
( 1)2
f ′(x) > 0 ⇒ 8 >
( x + 1)2 >
Merit
r
A correct
expression for
the derivative.
(
f ′( x) = 8 −
Achievement
u
Correct
solution with
correct
derivative.
Correct
derivative.
Correct
solution
with
correct
derivative.
Correct
derivative.
Correct
solution
with
correct
derivative.
2
( x + 1)2
1
4
1
−1
or x + 1 <
2
2
−1
−3
x>
or x <
2
2
Either x + 1 >
(d)
f ′(x) =
x(x − 5) − (x + 4)(2x − 5)
2
x2 ( x − 5)
f ′(x) = 0 ⇒ x(x − 5) − (x + 4)(2x − 5) = 0
(
)
x 2 − 5x − 2x 2 + 3x − 20 = 0
−x − 8x + 20 = 0
2
x 2 + 8x − 20 = 0
(x + 10)(x − 2) = 0
x = −10 or + 2
Excellence
t
NCEA Level 3 Calculus (91578) 2015 — page 2 of 6
(e)
Let V = volume (m3)
S = slant height (m)
h = height (m)
r = radius (m)
cos 30 =
Valid
statement
of the
relationship
between
rates.
dS dV
or
dr dr
correct.
r
S
Correct
solution
with correct
derivatives.
r
cos 30
dS
1
=
dr cos 30
h
tan 30 =
r
h = r tan 30
1
V = π r 2h
3
1 3
= π r tan 30
3
dV
= π r 2 tan 30
dr
dS dS dr dV
=
×
×
dt dr dV dt
1
1
=
×
×2
cos 30 π r 2 tan 30
When r = 10 m,
dS
1
1
=
×
×2
dt cos 30 π10 2 × tan 30
= 0.01273 m / minute
S=
NØ
N1
N2
A3
A4
M5
M6
E7
E8
No response;
no relevant
evidence.
ONE answer
demonstrating
limited
knowledge of
differentiation
techniques.
1u
2u
3u
1r
2r
1t with minor
error(s).
1t
NCEA Level 3 Calculus (91578) 2015 — page 3 of 6
Q2
Expected Coverage
(a)
1
x − 3x 2
5
(b)
dy
16
= 1+ 2
dx
x
dy
At x = 4,
=2
dx
(
(c)(i)
1. x = 1
2. x = –1, 1, 2
3. –1 < x < 1
(ii)
3
(iii)
Does not exist.
(d)
Excellence
t
Correct
solution with
correct
derivative.
−1
2
Two correct
answers.
dx
correct.
dL
x+L
L
=
5
1.5
1.5x + 1.5L = 5L
1.5x = 3.5L
7L
x=
3
dx 7
=
dL 3
dx
=2
dt
dL dL dx
=
×
dt dx dt
3
= ×2
7
6
= = 0.857 m s –1
7
Merit
r
A correct
expression for
the derivative.
−4
5
) ⋅ (1− 6x )
∴Gradient of normal =
Achievement
u
Four correct
answers.
Correct
solution
with correct
derivatives.
(Units not
required.)
NCEA Level 3 Calculus (91578) 2015 — page 4 of 6
(e)
Depth of water = x
h = x + 20
1
1
V = h 3 − 20 3
3
3
1
1
3
= ( x + 20 ) − 20 3
3
3
dV
2
= ( x + 20 )
dx
A = ( x + 20 )
Correct
dV
dx
OR
Correct
dV
dx
AND
dA
dx
dA
dx
Correct
solution.
2
dA
= 2 ( x + 20 )
dx
dV
= 3000
dt
dA dA dx dV
=
×
×
dt dx dV dt
1
= 2 ( x + 20 ) ×
× 3000
( x + 20 )2
When x = 15
dA
1
= 2 × 35 × 2 × 3000 = 171.4 cm 2 min −1
dt
35
NØ
N1
N2
A3
A4
M5
M6
E7
E8
No response;
no relevant
evidence.
ONE answer
demonstrating
limited
knowledge of
differentiation
techniques.
1u
2u
3u
1r
2r
1t with minor
error(s).
1t
NCEA Level 3 Calculus (91578) 2015 — page 5 of 6
Q3
(a)
Expected Coverage
f ′(x) =
5
10
×2=
2x − 3
2x − 3
10
=4
2x − 3
8x − 12 = 10
Achievement
u
Merit
r
Correct
solution with
correct
derivative.
x = 2.75
(b)
f ′(x) =
=
e 3x − x.3e 3x
(e )
3x 2
1− 3x
e 3x
f ′(x) = 0 ⇒ x =
(c)
Correct
solution with
correct
derivative.
1
3
dx
= −3sint
dt
dy
= 3cos 3t
dt
dy 3cos 3t − cos 3t
=
=
dx −3sint
sint
⎛ 3π ⎞
− cos ⎜ ⎟
⎝ 4⎠
π dy
At t = ,
=
=1
4 dx
⎛ π⎞
sin ⎜ ⎟
⎝ 4⎠
Correct
and
dx
dt
Correct
solution
with
correct
derivatives.
dx
dt
Parts (i)
and (ii)
both
correct.
dy
dt
∴Gradient of normal = −1
(d)(i)
dx
= −Ak sin kt + Bk cos kt
dt
d2 x
= −Ak 2 cos kt − Bk 2 sin kt
dt 2
= −k 2 ( A cos kt + Bsin kt )
= −k 2 x
(ii)
x(0) = 0 ⇒ A cos 0 + Bsin 0 = 0
A=0
v(0) = 2k ⇒ 2k = −Ak sin(0) + Bk cos(0)
B= 2
Correct
d2 x
dt 2
Consistent with
dx
incorrect dt
Or
Excellence
t
NCEA Level 3 Calculus (91578) 2015 — page 6 of 6
(e)
Differentiate
corretedly
related but
incorrect
expression for
w.
cos A =
Correct
dw
dA
2
f
2
cos A
w
sin A =
5− f
w= (5 − f )sin A
f=
2 ⎞
⎛
= ⎜5−
⎟ sin A
⎝
cos A ⎠
= 5sin A − 2 tan A
dw
= 5 cos A − 2sec 2 A
dA
dw
= 0 ⇒ 5 cos A − 2sec 2 A = 0
dA
5 cos 3 A − 2 = 0
2
cos 3 A =
5
A = 42.5°
Correct
solution.
w = 1.55 m
NØ
N1
N2
A3
A4
M5
M6
E7
E8
No response;
no relevant
evidence.
ONE answer
demonstrating
limited
knowledge of
differentiation
techniques.
1u
2u
3u
1r
2r
1t with minor
error(s).
1t
Cut Scores
Not Achieved
Achievement
Achievement with Merit
Achievement with Excellence
0–7
8 – 12
13 – 18
19 – 24