7/23/2019 91578-ass-2014-3

1/7

NCEA Level 3 Calculus (91578) 2014 page 1 of 7

Assessment Schedule 2014

Calculus: Apply differentiation methods in solving problems (1!"#$

%vidence Statement

Q1 Expected Coverage Achievement

u

Merit

r

Excellence

t

(a)15sin 3x( ) A correct

expression for

the derivative.

(b)

dy

dx= 2 3x2 5x( ) 6x 5( )

Atx= 1,dy

dx= 2 21=4

Gradient of normal =1

4 thro!h (1,4)

A correct

soltion.

(c)

x=2sint y= cos2tdx

dt=2cost

dy

dt= 2sin2t

dy

dx=

2sin2t2cost

=2 2sintcost

2cost

= 2sint

"orrect

expressions for

dx

dtand

dy

dt.

A correct

soltion.

(d)

y= 4

e2x2+ #x=4e2x+2 + #x

dy

dx= #e2x+2 + #

$arallel tox%axisdy

dx=&

#e2x+2 = #

e2x+2 = 12x+ 2 = &x= 1

A correct

expression for

dydx

.

A correct

soltion.

(e) "orrect

derivative foran incorrect bt

relevant

expression for

V.

A correct

expression

fordr

dV.

A correct

soltion.

'nits not

reired.

7/23/2019 91578-ass-2014-3

2/7

NCEA Level 3 Calculus (91578) 2014 page 2 of 7

V 3225 cm3

Alternative *or+in!

N N1 N2 A3 A4 M5 M6 E E!

-o response

no relevant

evidence.

/-0 ans*er

demonstratin!

limited+no*led!e of

differentiation

technies.

/-0 correct

derivative

2 3 1r 2r 1t *ith minor

error(s).

1t

7/23/2019 91578-ass-2014-3

3/7

NCEA Level 3 Calculus (91578) 2014 page 3 of 7

Q2 Expected Coverage Achievement

u

Merit

r

Excellence

t

(a)

f x( ) =2x1( ) 4e4x e4x.2

2x1( )2

A correct

expression for

the derivative.

(b)

y= #ln 3x 2( )dy

dx=

24

3x 2( )

Atx=2 dy

dx=6

A correct

soltion.

(c)(i)

(ii)(iii)

1. 2, 1, 2

2. x< 23. 2

43

2 correct

ans*ers

4 correct

ans*ers.

(d)

C=4v+1&&&&&&

v

dC

dv=4

1&&&&&&

v2

inimm *hendC

dv=&

v2 =25&&&&v= 5&&

C=4 5&& +1&&&&&&

5&&=4&&&

"orrect vale

for v*ith

correct

derivative.

A correct

soltion.

'nits not

reired.

7/23/2019 91578-ass-2014-3

4/7

NCEA Level 3 Calculus (91578) 2014 page 4 of 7

(e)

tan3& =h

y

h=ytan3&

cos3& =y+ b

5&

y+ b= 5&cos3&

b= 5&cos3& yArea=base hei!ht

A= 5&cos3& y( ) ytan3&( )= 5&ysin3& y2 tan3&

=25yy2

3

dA

dy=25

2y

3

Aty=2&dA

dy=25

4&

3

dA

dt=

dA

dy

dy

dt

= 254&

3

3

= 5.2 cm2 s1

"orrect

derivative for

an incorrect bt

relevant

expression for

A.

A correct

expression

fordA

dy

A correct

soltion.

'nits not

eired.

N N1 N2 A3 A4 M5 M6 E E!

-o response

no relevantevidence.

/-0 ans*er

demonstratin!limited

+no*led!e ofdifferentiation

technies.

/-0 correct

derivative

2 3 1r 2r 1t *ith minor

error(s).

1t

7/23/2019 91578-ass-2014-3

5/7

NCEA Level 3 Calculus (91578) 2014 page 5 of 7

Q3 Expected Coverage Achievement

u

Merit

r

Excellence

t

(a)

y= x2 + 4x3( )2

= x2 + 4x( )2

3

dydx

= 23x2 + 4x( )

1

3 . 2x+ 4( )

A correct

expression for

the derivative.

(b)

y=x+32

x2

dy

dx= 1

64

x3

tationarpoints*hendy

dx= &

x3 =64x=4

A correct

soltion.

(c)

f(x) = 5xxlnx

f x( ) = 5 lnxx

x

= 4 lnx

7ncreasin! f x( ) > &4 lnx> &

lnx< 4

x< e4

x< 54.6

8t ifx & then lnxis not defined, so &

7/23/2019 91578-ass-2014-3

6/7

NCEA Level 3 Calculus (91578) 2014 page 6 of 7

(e) "orrect

derivative for

an incorrect bt

relevant

expression for

V.

A correct

expression

fordV

dr

A correct

soltion.

'nits not

reired.

N N1 N2 A3 A4 M5 M6 E E!

-o response

no relevant

evidence.

/-0 ans*er

demonstratin!

limited+no*led!e of

differentiation

technies.

/-0 correct

derivative

2 3 1r 2r 1t *ith minor

error(s).

1t

Cut Scores

&ot Achieved AchievementAchievement 'ith

erit

Achievement'ith

%)cellen

ce

Score range 0 7 8 12 13 20 21 24

7/23/2019 91578-ass-2014-3

7/7

NCEA Level 3 Calculus (91578) 2014 page 7 of 7

Mar"ing code#

"odes that ma6 have been sed in mar+in! this examination paper have meanin! as follo*s.

$ 9ash % *here a candidate obtains a correct ans*er bt contines *ith frther, nnecessar6, material that is

incorrect bt does not sho* a lac+ of nderstandin! or a contradiction.

C "onsistenc6 % *here a candidate has obtained an incorrect vale *ithin a estion and sbseentl6 ses that

vale.

NC -on%consistenc6 % *here a candidate has obtained an incorrect vale or expression *ithin a estion and does

not se that vale or expression *here it is sbseentl6 reired.

%A&& i!ht ans*er, *ron! *or+in! % *here a candidate presents a correct ans*er bt the *or+in! or reasonin! leadin!

to it is irrelevant, incomplete or contains one or more errors.

% ondin! error % *here a candidate prodces a correct seence of calclations, bt the ans*er does not a!ree to

2 si!nificant fi!res *ith the ans*er !iven in the assessment schedle as a reslt of rondin! a nmber in the

seence of calclations.

mei inor error i!nored % *here a candidate ma+e a minor error and this has been i!nored.

'(o an# :*o ans*ers !iven % *here a candidate *rites t*o ans*ers, one correct and the other incorrect, and neither has

been deleted (the correct ans*er is not accepted).