Western Cape Education Department
Examination Preparation
Learning Resource 2016
Calculus - Differentiation
MATHEMATICS
Grade 12
Razzia Ebrahim
Senior Curriculum Planner for Mathematics
E-mail: Razzia.Ebrahim@wced.info/ Razzia.Ebrahim@westerncape.gov.za
Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835grade-12?Itemid=-1
Website: http://wcedeportal.co.za
Mathematics P1/ Differentiation
Index
2
Page
1. 2016 November Paper 1
3
2. 2016 June Paper 1
4
3. 2016 Feb-March Paper 1
4
4. 2015 November Paper 1
4
5. 2015 June Paper 1
5
6. 2015 Feb-March Paper 1
5
7. 2014 November Paper 1
5
8. 2014 Exemplar Paper 1
5
Compiled by R.Ebrahim Senior Curriculum Planner FET Mathematics WCED
page 2
Mathematics P1/ Differentiation
1.2
Given:
1.2.3
DBE/November 2016
f  x   x 2  3x  4
Determine the values of x for which f / (x) ≥ 0
QUESTION 5
Sketched below is the parabola f, with equation
with equation  x  p   y  t   3.


3
(2)
f x    x 2  4 x  3 and a hyperbola g,
B, the turning point of f, lies at the point of intersection of the asymptotes of g.
A ( 1 ; 0) is the x-intercept of g.
y
g
B
x
0
A( 1 ; 0)
g
f
5.6
Write down the values of x for which f (x). g / (x ) ≥ 0
(4)
QUESTION 8
8.1
Determine f / (x) from first principles if f (x)  3x 2
8.2
John determines g / (a) , the derivative of a certain function g at x = a, and arrives
(5)
4h 2
h
Write down the equation of g and the value of a.
at the answer: lim
h 0
dy
if y 
dx
x3 
(2)
5
x3
8.3
Determine
8.4
g  x   8x  20 is a tangent to
values of a and b.
(4)
f  x   x 3  ax 2  bx 18 at x = 1. Calculate the
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Mathematics P1/ Differentiation
4
DBE/2016
QUESTION 7
7.1
Determine f (x ) from first principles if f ( x)  3 x 2  5
7.2
Determine
dy
if:
dx
7.2.1
y  2x5 
7.2.2
y

4
x3
xx2
(3)

2
(4)
[12]
DBE/Feb.-Mar. 2016
QUESTIONS
8.1
Determine /'(x) from first principles if f(x) = -x 2 + 4.
8.2
Determine the derivative of:
8.2.1
(5)
(5)
y = 3x 2 + I Ox
(2)
(3)
8.2.2
.
DBE/November 2015
QUESTION 8
8.1
If f(x) = x 2 − 3x , determine f ′(x) from first principles.
8.2
Determine:
8.2.1
1 
dy

if y =  x 2 − 2 
dx
x 

8.2.2
 x3 − 1 

Dx 
 x −1 
(5)
2
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page 4
Mathematics P1/ Differentiation
5
QUESTION 8
DBE/2015
i,
8.1
If .f(x) = x determine /' (x) from first principles.
8.2
Determine:
8.2.1
dy
( 5)
if y = 5x 2 + 5x + 2
dx
(2)
8.2.2
8.3
(3)
Given: p(x) = x3 + 2x
Show, using relevant calculations, why it is not possible for a tangent drawn to the
graph of p to have a negative gradient.
DBE/Feb-Mar 2015
QUESTION 8
8.1
Determine the derivative of
8.2
Differentiate:
8.2.1
8.2.2
(3)
(13)
f(x)=2x2 + 4 from first principles.
(4)
f(x) = -3x 2 + 5.Jx
p(x)=U +4x)
(3)
2
(4)
DBE/November 2014
QUESTION 8
8.1
Determine f / (x) from first principles if f (x)  x 3 .
8.2
Determine the derivative of: f (x)  2x 2 
8.3
If y  x 6 1 , prove that
8.4
Given: f (x)  2x 3  2x 2  4x  1. Determine the interval on which f is concave up.
2
(5)
1 4
x 3
2
(2)
dy
 12 x 5 y , if x  1.
dx
(3)
QUESTION 8
8.1
Determine f ′(x) from first principles if f (x) = 3x 2 − 2 .
8.2
dy
x
Determine
if y = 2x −4 − .
5
dx
Compiled by R.Ebrahim Senior Curriculum Planner FET Mathematics WCED
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[14]
NSC – Grade 12 Exemplar
(5)
(2)
[7]
page 5
Western Cape Education Department
Examination Preparation
Learning Resource 2016
Calculus - Graphs
MATHEMATICS
Grade 12
Razzia Ebrahim
Senior Curriculum Planner for Mathematics
E-mail: Razzia.Ebrahim@wced.info/ Razzia.Ebrahim@westerncape.gov.za
Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835grade-12?Itemid=-1
Website: http://wcedeportal.co.za
Gr 12 Mathematics P1/Calculus Graphs
Index
2
Page
1. 2016 November Paper 1
3
2. 2016 June Paper 1
4
3. 2016 Feb-March Paper 1
4
4. 2015 November Paper 1
5-6
5. 2014 November Paper 1
6
6. 2014 Exemplar Paper 1
6
7. 2015 June Paper 1
7
8. 2015 Feb-March Paper 1
8
Compiled by R.Ebrahim Senior Curriculum Planner FET Mathematics WCED
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Gr 12 Mathematics P1/Calculus Graphs
3
DBE/November 2016
QUESTION 5
Sketched below is the parabola f, with equation
with equation  x  p   y  t   3.


f x    x 2  4 x  3 and a hyperbola g,
B, the turning point of f, lies at the point of intersection of the asymptotes of g.
A ( 1 ; 0) is the x-intercept of g.
y
g
B
A( 1 ; 0)
x
0
g
f
5.6
Write down the values of x for which f (x). g / (x ≥)0
8.4
g  x   8x  20 is a tangent to
values of a and b.
(4)
f  x   x 3  ax 2  bx 18 at x = 1. Calculate the
(5)
[16]
QUESTION 9
For a certain function f, the first derivative is given as f
/
x   3x 2  8x  3
9.1
Calculate the x-coordinates of the stationary points of f.
(3)
9.2
For which values of x is f concave down?
(3)
9.3
Determine the values of x for which f is strictly increasing.
(2)
9.4
If it is further given that f  x   ax3  bx 2  cx  d
equation of f.
Compiled by R.Ebrahim Senior Curriculum Planner FET Mathematics WCED
and f 0  18, determine the
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[13]
Gr 12 Mathematics P1/Calculus Graphs
4
DBE/2016
QUESTION 8
Sketched below are the graphs of f ( x)  ( x  2) 2 ( x  k ) and g ( x)  mx  12




A and D are the x-intercepts of f.
B is the common y-intercept of f and g.
C and D are turning points of f.
The straight line g passes through A.
C
y
g
B
f
A
x
O
D
8.1
Write down the y-coordinate of B.
(1)
8.2
Calculate the x-coordinate of A.
(3)
8.3
If k = –3, calculate the coordinates of C.
(6)
8.4
For which values of x will f be concave down?
(3)
[13]
8.3
Given: f(x) = 2x 3 -23x 2 + 80x-84
8.3.1
Prove that (x - 2) is a factor off
J(x)
DBE/Feb.-Mar. 2016
(2)
fully.
8.3.2
Hence, or otherwise, factorise
8.3.3
Determine the x-coordinates of the turning points off
(4)
8.3.4
Sketch the graph off, clearly labelling ALL turning points and intercepts
with the axes.
(3)
8.3 .5
Determine the coordinates of the y-intercept of the tangent to f that has
a slope of 40 and touches f at a point where the x-coordinate is
an integer.
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Gr 12 Mathematics P1/Calculus Graphs
5
DBE/November 2015
QUESTION 5
Given: h( x ) = 2 x − 3 for − 2 ≤ x ≤ 4 . The x-intercept of h is Q.
5.1
Determine the coordinates of Q.
(2)
5.2
Write down the domain of h −1 .
(3)
5.3
Sketch the graph of h −1 in your ANSWER BOOK, clearly indicating the y-intercept
and the end points.
(3)
5.4
For which value(s) of x will h( x ) = h −1 ( x ) ?
(3)
5.5
P(x ; y) is the point on the graph of h that is closest to the origin. Calculate the
distance OP.
(5)
5.6
Given: h( x) = f ′( x) where f is a function defined for − 2 ≤ x ≤ 4 .
5.6.1
Explain why f has a local minimum.
5.6.2
Write down the value of the maximum gradient of the tangent to the
graph of f.
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[19]
Gr 12 Mathematics P1/Calculus Graphs
6
DBE/November 2015
QUESTION 9
Given: h( x) = − x 3 + ax 2 + bx and g ( x) = −12 x . P and Q(2 ; 10) are the turning points of h.
The graph of h passes through the origin.
3
and b = 6 .
2
9.1
Show that a =
9.2
Calculate the average gradient of
x = –1 at P.
9.3
Show that the concavity of h changes at x =
9.4
Explain the significance of the change in QUESTION 9.3 with respect to h.
(1)
9.5
Determine the value of x, given x < 0, at which the tangent to h is parallel to g.
(4)
[17]
(5)
h
between
P
and
Q, if it is given that
1
.
2
QUESTION 9
(4)
(3)
DBE/November 2014
Given: f (x)  (x  2)( x 2  6x  9)
 x3  4x 2  3x  18
9.1
Calculate the coordinates of the turning points of the graph of f.
(6)
9.2
Sketch the graph of
turning points.
(4)
9.3
f, clearly indicating the intercepts with the axes and the
For which value(s) of x will x. f /(x) < 0?
QUESTION 9
NSC – Grade 12 Exemplar
(3)
[13]
DBE/2014
Given: f (x) = x 3 − 4x 2 −11x + 30 .
9.1
Use the fact that f (2) = 0 to write down a factor of f(x).
(1)
9.2
Calculate the coordinates of the x-intercepts of f.
(4)
9.3
Calculate the coordinates of the stationary points of f.
(5)
9.4
Sketch the curve of f in your ANSWER BOOK. Show all intercepts with the axes
and turning points clearly.
(3)
9.5
For which value(s) of x will f ′(x) < 0 ?
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Gr 12 Mathematics P1/Calculus Graphs
7
June/2015
QUESTI0N9
The graph of y = ax2 +bx+ c below represents the derivative off.
It is given that /'(1)=0, /'(3)=0 and /'(0)=6.
y
9.1
Write down the x-coordinates of the stationary points of f.
(2)
9.2
For which value(s) of x is f strictly decreasing?
(2)
9.3
Explain at which value of x the stationary point off will be a local minimum.
(2)
9.4
Determine the x-coordinate of the point of inflection of f.
(1)
9.5
For which value(s) of x is f concave up?
(2)
(9)
June/2015
8.3
Given: p(x) = x3 + 2x
Show, using relevant calculations, why it is not possible for a tangent drawn to the
graph of p to have a negative gradient.
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Gr 12 Mathematics P1/Calculus Graphs
8
DBE/Feb.-Mar. 2015
8.3
The sketch below shows the graph of h(x) = x 3
point A is 1. C is another x-intercept of h.
- 7x
2
+ 14x-8 . The x-coordinate of
y
h
B
C
8.3.1
Detem1ine h'(x).
(1)
8.3.2
Determine the x-coordinate of the turning point B.
(3)
8.3.3
Calculate the coordinates of C.
(4)
8.3.4
The graph of h is concave down for x < k. Calculate the value of k.
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[22]
Western Cape Education Department
Examination Preparation
Learning Resource 2016
Calculus - Application
MATHEMATICS
Grade 12
Razzia Ebrahim
Senior Curriculum Planner for Mathematics
E-mail: Razzia.Ebrahim@wced.info/ Razzia.Ebrahim@westerncape.gov.za
Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835grade-12?Itemid=-1
Website: http://wcedeportal.co.za
Mathematics P1/ Calculus Application
Index
2
Page
1. 2016 November Paper 1
2
2. 2016 June Paper 1
3
3. 2016 Feb-March Paper 1
3
4. 2015 November Paper 1
4
5. 2015 June Paper 1
4
6. 2015 Feb-March Paper 1
5
7. 2014 November Paper 1
6
8. 2014 Exemplar Paper 1
6
DBE/November 2016
QUESTION 10
The number of molecules of a certain drug in the bloodstream t hours after it has been taken
is represented by the equation Mt   t 3  3t 2  72t , 0  t  10 .
10.1
Determine the number of molecules of the drug in the bloodstream 3 hours after the
drug was taken.
(2)
10.2
Determine the rate at which the number of molecules of the drug in the bloodstream is
changing at exactly 2 hours after the drug was taken.
(3)
10.3
How many hours after taking the drug will the rate at which the number of molecules
of the drug in the bloodstream is changing, be a maximum?
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[8]
Mathematics P1/ Calculus Application
3
QUESTION 9
DBE/2016
A 340 mℓ can with height h cm and radius r cm is shown below.
1 mℓ = 1 cm
r
3
h
9.1
Determine the height of the can in terms of the radius r.
9.2
Calculate the length of the radius of the can, in cm, if the surface area is to be a
minimum.
(3)
(6)
[9]
DBE/Feb.–Mar./Feb.–Mrt. 2016
QUESTION 9
...................
A soft drink can has a volume of 340 cm3 , a height of
h cm and a radius of r cm.
h
9.1
Express h in terms of r.
(2)
9. 2
Show that the surface area of the can is given by A(r) = 2,r r 2 + 680r- 1 •
(2)
9.3
Determine the radius of the can that will ensure that the surface area is a minimum.
(4)
(8)
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Mathematics P1/ Calculus Application
4
DBE/November 2015
QUESTION 10
A rain gauge is in the shape of a cone. Water flows into the gauge. The height of the water
is h cm when the radius is r cm. The angle between the cone edge and the radius is 60°, as
shown in the diagram below.
Formulae for volume:
V = π r 2h
1
V = π r 2h
3
V = lbh
4
V = π r3
3
r
60°
h
10.1
Determine r in terms of h. Leave your answer in surd form.
10.2
Determine the derivative of the volume of water with respect to h when h is equal
to 9 cm.
QUESTION 10
(2)
(5)
[7]
DBE/2015
The mass of a baby in the first 30 days of life is given by
M(t)=t 3 -9t 2 +3000
0�L�30.
t is the time in days and M is the mass of the baby in grams.
IO.I
Write down the mass of the baby at birth.
(1)
10.2
A baby's mass usually decreases in the first few days after birth.
On which day will the baby's mass return to its birth mass?
(4)
10.3
On which day will this baby have a minimum mass?
(4)
10.4
On which day will the baby's mass be decreasing the fastest?
(2)
[11)
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Mathematics P1/ Calculus Application
5
DBE/Feb.-Mar. 2015
QUESTI0N9
A necklace is made by using 10 wooden spheres and 10 wooden cylinders. The radii, r, of the
spheres and the cylinders are exactly the same. The height of each cylinder is h. The wooden
spheres and cylinders are to be painted. (Ignore the holes in the spheres and cylinders.)
V=rcr 2 h
S = 2rc r 2 + 2rc r h
4
V=-rcr 3
S=4rcr 2
9.1
If the volume of a cylinder is 6 cm3 , write h in terms of r.
(1)
9.2
Show that the total surface area (S ) of all the painted surfaces of the necklace is equal
120
to S = 60rcr 2 +
r
(4)
9.3
Determine the value of r so that the least amount of paint will be used.
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[9]
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Mathematics P1/ Calculus Application
6
DBE/November 2014
QUESTION 10
100 cm
h
40 cm
l
h
h
b
h
b
A box is made from a rectangular piece of cardboard, 100 cm by 40 cm, by cutting out the
shaded areas and folding along the dotted lines as shown in the diagram above.
Express the length l in terms of the height h.
(1)
10.2
Hence prove that the volume of the box is given by V  h(50  h)(40  2h)
(3)
10.3
For which value of h will the volume of the box be a maximum?
(5)
[9]
40 cm
10.1
DBE/2014
NSC – Grade 12 Exemplar
QUESTION 10
Two cyclists start to cycle at the same time. One starts at point B and is heading due north to
point A, whilst the other starts at point D and is heading due west to point B. The cyclist
starting from B cycles at 30 km/h while the cyclist starting from D cycles at 40 km/h.
The distance between B and D is 100 km. After time t (measured in hours), they reach points F
and C respectively.
A
F
•
B
•C
D
10.1
Determine the distance between F and C in terms of t.
(4)
10.2
After how long will the two cyclists be closest to each other?
(4)
10.3
What will the distance between the cyclists be at the time determined in
QUESTION 10.2?
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