I LOVE MATHS SERIES.
BOOK 6
MATHEMATICS
GRADE 12
2022 REVISION MATERIAL
DIFFERENTIAL CALCULUS
A collection of questions from previous
question papers (2016 to 2021).
Prepared by T Faya.
This booklet is divided into 8 SECTIONS.
1. SECTION A – Differentiation ……………………………………….Page 2
2. SECTION B – Tangents and average gradient………………………. Page 21
3. SECTION C – Sketching a cubic function…………………………...Page 24
4. SECTION D – Interpretation of a cubic function………………….... Page 30
5. SECTION E – The graph of the derivative…………………………. Page 59
6. SECTION F – Maxima and Minima………………………………... Page 64
7. SECTION G – Rate of change……………………………………… Page 86
8. SECTION H – Calculus of motion…………………………………. Page 90
1
SECTION A - Differentiation
QUESTION 1 KZNM16
−5
1.1
Given: f ( x) =
, determine f ′ (x) from first principles.
x
1.2
(5)
Determine:1.2.1
dy
if y = x 2 + 3
dx
1.2.2
f ′ (x) if
(
)
f ( x) =
(4)
x
x3 − 8
2−x
(4)
[13]
QUESTION 7 KZNJ16
7.1
Determine the derivative of
7.2
(5)
f ( x) = −2 x 2 + 5 from first principles.
Determine the derivative of the following using the rules for differentiation.
7.2.1
7.2.2
y=
(4)
x3 − 27
3− x
(4)
 
7 
Dx  x  2 x − 3 
x 
 
ECS16
QUESTION 8 FSS16
8.1
8.2
Determine 𝑓𝑓 ′ (𝑥𝑥) from first principles if f ( x ) = x
3
(5)
Determine:
8.2.1
8.2.2
1 2
(3)
𝑓𝑓 ′ (𝑥𝑥) if 𝑓𝑓(𝑥𝑥) = �𝑥𝑥 + �
3
dy
if y = 2 − πx
dx
3
𝑥𝑥
(3)
x
[11]
2
GPS16
QUESTION 8 LPS16
8.1
8.2
Determine the derivative of 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 − 𝑥𝑥 2 from first principle.
(5)
Differentiate the following with respect to x:
3
8.2.1
(3)
𝑓𝑓(𝑥𝑥) = 6𝑥𝑥 4 − √𝑥𝑥 −2
8.2.2
(3)
𝑦𝑦 = (4𝑥𝑥 2 )3 + 2𝑥𝑥
[11]
MPS16
3
NWS16
QUESTION 6
6.1
6.2
Determine f '( x) from first principles if f ( x) =
Determine
(3)
y π x−
6.2.1 =
QUESTION 8
8.1
8.2
(5)
dy
if:
dx
3
6.2.2
3
.
x
3
x
(2)
7 x5 − 3x
y=
4x
[10]
WCS16
2
Determine 𝑓𝑓′(𝑥𝑥) using first principles if 𝑓𝑓(𝑥𝑥) = .
𝑥𝑥
Determine the following:
8.2.1
ℎ′ (𝑥𝑥) if ℎ(𝑥𝑥) = 2𝑥𝑥 3 −
8.2.2
𝐷𝐷𝑥𝑥 �
QUESTION 8 NM16
8𝑥𝑥 3 − 27
�
2𝑥𝑥 − 3
4
+ 3√𝑥𝑥
𝑥𝑥
8.1
Determine f ′(x ) from first principles if f ( x) = − x 2 + 4 .
8.2
Determine the derivative of:
8.2.1
y = 3 x 2 + 10 x
8.2.2
3

f ( x) =  x − 
x

(5)
(4)
(3)
(5)
(2)
2
(3)
4
NJ16
NN16
QUESTION 1 KZNM17
1.1
Determine the derivative of f ( x) = x 2 + 3 x from first principles.
1.2
Evaluate:
1.2.1
dy
if y = 3 x 2 . 3 8 x 4
dx
1.2.2
f ′( x) if f ( x) =
(5)
(4)
x3 − 5x 2 + 4 x
; x≠4
x−4
(5)
[14]
5
QUESTION 8 KZNJ17
8.1
8.2
Given f ( x) =
2
Determine the derivative of f from first principles.
3x
(5)
8.2.1
f ′( x) if f ( x) = 3( x 2 − 5)2
(4)
8.2.2
dy
if
dx
(5)
x2 − 3 = 3 y
[14]
QUESTION 7 KZNS17
7.1
Determine f / ( x) from first principles if f ( x) = 3 x 2 − x .
7.2
Determine
ECJ17
(5)
dy
if :
dx
(
7.2.1
y = x + x −2
7.2.2
y = 3 x4 −
)
2
(4)
1 5
x
10
(3)
ECS17
6
FSS17
LPS17
MPS17
7
NWS17
WCS17
8
NM17
QUESTION 8 NJ17
8.1
Given f ( x ) = 3 − 2 x 2 . Determine f ′( x ) , using first principles.
8.2
Determine
8.3
The function f ( x ) = x 3 + bx 2 + cx − 4 has a point of inflection at (2 ; 4). Calculate the
values of b and c.
12 x 2 + 2 x + 1
dy
if y =
.
dx
6x
(5)
(4)
(7)
[16]
QUESTION 7 NN17
7.1
Given: f ( x) = 2 x 2 − x
Determine f / ( x) from first principles.
7.2
(6)
Determine:
7.2.1
Dx [( x + 1)(3 x − 7 )]
7.2.2
5 1
dy
if y = x 3 − + π
dx
x 2
(2)
(4)
[12]
9
KZNJ18
QUESTION 9
KZNS18
9.1
Determine the derivative of f ( x) = −5 x 2 + 3 x from first principles.
9.2
Calculate g / (4) if g ( x) =
9.3
3
Determine D x [(2 x − 3) ]
1
(5)
(4)
2 x
(4)
[13]
ECS18
10
QUESTION 8 FSS18
8.1
Determine f ′(x) from first principles if f (x ) = −3x 2
8.2
Determine
8.3
It is given that g ( x ) = ax 3 − 24 x + b has a local minimum turning point at (− 2; 17 ) .
Determine the values of a and b.
(5)
dy
3
if y = 7 x 4 − 5 x −
x
dx
(4)
(4)
GPS18
GPS18
LPS18
11
[13]
MPS18
NWS18
WCS18
12
QUESTION 8 NM18
8.1
Determine f ′(x ) from first principles if f ( x ) = 4x 2 .
8.2
Determine:
8.2.1
 x 2 − 2x − 3
Dx 

 x +1 
8.2.2
f
//
(x )
(5)
(3)
if f ( x ) = x
(3)
[11]
QUESTION 7 NJ18
7.1
Given: f ( x) = 2 − 3x 2
Determine f / ( x) from first principles.
7.2
(5)
Determine:
[
]
7.2.1
D x (4 x + 5) 2
7.2.2
dy
x2 − 8
if y = 4 x +
dx
x2
(3)
(4)
[12]
QUESTION 8 NN18
8.1
Determine f / (x) from first principles if it is given f ( x) = x 2 − 5.
8.2
Determine
dy
if:
dx
8.2.1
y = 3x 3 + 6 x 2 + x − 4
8.2.2
yx − y = 2 x 2 − 2 x
(3)
;
x ≠1
(4)
[12]
QUESTION 9 KZNJ19
9.1
9.2
(5)
Determine 𝑓𝑓 / (3) from first principles given𝑓𝑓(𝑥𝑥) = 5𝑥𝑥 2 + 4.
(5)
Differentiate:
9.2.1
1 

g ( x) =  2 x −

2x 

9.2.2
 x 3 − 1
Dx 

 1− x 
2
(4)
(4)
13
[13]
QUESTION 9 KZNS19
9.1
Determine f /(x) from first principles given f ( x) = x 2 −
9.2
Determine:
9.2.1
ECS19
1
x.
2
𝑑𝑑
5
�3𝑥𝑥 4 + √𝑥𝑥 + 𝑎𝑎2 �
𝑑𝑑𝑑𝑑
(5)
(3)
9.2.2 𝑑𝑑𝑑𝑑
, if 𝑥𝑥𝑥𝑥 = 𝑥𝑥 + 𝑥𝑥 2 − 1.
𝑑𝑑𝑑𝑑
(4)
[12]
QUESTION 8 FSS19
8.1
8.2
Determine:
8.1.1
f /(x) from first principles if f ( x ) = 3x 2
(5)
8.1.2
d  3
3
 x −x+ 2
dx 
x 
(3)
Given that g(x) = – 4x + 12 and g(x) = f /(x).
8.2.1
Calculate the x coordinate of the turning point of f.
8.2.2
Determine the values of x for which the graph of f will be decreasing.
GPS19
14
(2)
(2)
[12]
LPS19
QUESTION 7 NWS19
7.1
7.2
− x 2 + 3x − 7
Given: f ( x) =
Determine f / ( x) from first principles.
(6)

Determine: D x 15

(6)
5
x4 −
3x7 + x 
4 x3 
[12]
QUESTION 7 NJ19
7.1
7.2
Given f ( x ) = x 2 + 2 .
Determine f / (x ) from first principles.
dy
Determine
if:
dx
2
y = 4x 3 +
7.2.1
x
7.2.2
(4)
(3)
y = 4.3 x + (3 x 3 ) 2
(4)
15
7.3
If g is a linear function with g (1) = 5 and g / (3) = 2 , determine the equation of g in
the form y = …..
QUESTION 7
7.1
Determine f / ( x) from first principles if it is given that f ( x) = 4 − 7 x.
7.2
Determine
7.3
Given: y = ax 2 + a
dy
if y = 4 x 8 + x 3
dx
(3)
[14]
(4)
(3)
Determine:
7.3.1
dy
dx
(1)
7.3.2
dy
da
(2)
KZNJ20
KZNS20
16
ECS20
FS20
GPS20
17
LPS20
MPS20
NWS20
18
WCS20
NN20
KZNJ21
19
KZN MOCK 21
NN21
20
SECTION B – Tangents and Average gradient
ECS16
QUESTION 2 KZNM16
Given the equation of the curve h( x) = x 3 + 3x 2 + 15 x
2.1
Determine the equation of the tangent to h at x = −3 .
(5)
2.2
Show that the function h is increasing for all real value(s) of x.
(5)
[10]
GPS16
WCS16
8.3
NN16
Explain, by doing the necessary calculations, why the tangent to the curve of
𝑓𝑓(𝑥𝑥) = −𝑥𝑥 3 − 2𝑥𝑥 will never have a positive gradient.
QUESTION 7 KZNS17
7.3
Given : f ( x) = x 2 −
(3)
4
.
x2
7.3.1
Determine the gradient of the tangent to f at the point where x = 2.
(3)
7.3.2
Determine the equation of the tangent to f at x = 2.
(3)
FSS17
21
NWS17
WCS18
GPS19
LPS19
QUESTION 7
7.4
NN19
12
passes through the point A(2 ; b). Determine the
x
equation of the line perpendicular to the tangent to the curve at A.
The curve with equation y = x +
GPS20
22
(4)
GPS20
LPS20
WCS20
23
SECTION C – Sketching a cubic function
KZNJ16
7.3
Given g ( x) = x 3 − x 2 − 8 x + 12
7.3.1
7.3.2
7.3.3
7.3.4
7.3.5
Determine g (2) .
(2)
(3)
Solve g ( x) = x 3 − x 2 − 8 x + 12 = 0
Calculate the coordinates of the stationary points of g.
Sketch g indicating the intercepts and the stationary points.
For which value(s) of k will x 3 − x 2 − 8 x + 12 = k have three unequal
roots.
(4)
(4)
(2)
[28]
QUESTION 8 NM16
Given: f ( x) = 2 x 3 − 23x 2 + 80 x − 84
8.3
8.3.1
Prove that ( x − 2) is a factor of f.
(2)
8.3.2
Hence, or otherwise, factorise f ( x ) fully.
(2)
8.3.3
Determine the x-coordinates of the turning points of f.
(4)
8.3.4
Sketch the graph of f , 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.
24
(6)
[27]
QUESTION 9 KZNJ17
9.1
3
Given g ( x) = ax + bx + c, a ≠ 0 , which has the following properties.
f / ( x) > 0 if x < −3 or x > 3
f / ( x) < 0 if − 3 < x < 3
f / (−3) = f / (3) = 0
f (0) > 0
Use the above information to draw a sketch of g.
ECS17
WCS17
25
(4)
NM17
QUESTION 9 NJ17
Given:
f ( x) = x 3 − x 2 − x + 1
9.1
Write down the coordinates of the y-intercept of f.
(1)
9.2
Calculate the coordinates of the x-intercepts of f.
(5)
9.3
Calculate the coordinates of the turning points of f.
(6)
9.4
Sketch the graph of f in your ANSWER BOOK. Clearly indicate all intercepts with
the axes and the turning points.
(3)
9.5
Write down the values of x for which f ′( x) < 0 .
QUESTION 8 NN17
Given: f ( x ) = x( x − 3) with f
2
/
(1) = f / (3) = 0
(2)
[17]
and f (1) = 4
8.1
Show that f has a point of inflection at x = 2.
(5)
8.2
Sketch the graph of f, clearly indicating the intercepts with the axes and the turning
points.
(4)
8.3
For which values of x will y = − f ( x ) be concave down?
(2)
8.4
Use your graph to answer the following questions:
8.4.1
8.4.2
Determine the coordinates of the local maximum of h if h( x ) = f ( x − 2 ) + 3
.
Claire claims that f
/
(2)
(2) = 1 .
Do you agree with Claire? Justify your answer.
26
(2)
[15]
KZNJ18
QUESTION 9 FSS18
Given: f ( x ) = − x 3 + 10 x 2 − 17 x − 28
9.1
Calculate the coordinates of the intercepts of f with the axes.
(5)
9.2
Calculate the coordinates of the turning points of f.
(5)
9.3
Sketch the graph of f, clearly indicating the intercepts with the axes and the
turning points.
(3)
9.4
Determine the values of k for which − x + 10 x − 17 x = 25 + k will have only one
real root.
(2)
3
2
27
[15]
GPS18
QUESTION 9
NWS18
QUESTION 10 KZNJ19
Given: 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 + 2)2 (𝑥𝑥 − 3) is a cubic function.
10.1
Write down the x - intercepts and the y - intercept of f.
(3)
10.2
Determine the local maximum and minimum turning points of f.
(4)
10.3
Sketch f showing the coordinates of the turning points and the intercepts with the axes.
(4)
10.4
If f (x) = k has one root equal to 0, write down the value of k.
(2)
10.5
Given g(x) = f (x − 4), write down the coordinates of the new maximum point.
(2)
10.6
Write down the values of x for which f is concave up.
(2)
10.7
Determine the equation of 𝑔𝑔(𝑥𝑥) if 𝑔𝑔(𝑥𝑥) = 𝑓𝑓 �𝑥𝑥 + �.
28
1
2
(2)
[19]
GPS19
FSS20
29
SECTION D – Interpretation of a cubic function
QUESTION 2 KZNM16
Given the equation of the curve h( x) = x 3 + 3x 2 + 15 x
2.1
Determine the equation of the tangent to h at x = −3 .
(5)
2.2
Show that the function h is increasing for all real value(s) of x.
(5)
[10]
QUESTION 10 KZNS16
10.1
10.2
Given f ( x) = x 2 − 8
10.1.1 Calculate f (−3).
(1)
10.1.2 Calculate f / (−3).
(1)
10.1.3 Determine the equation of the tangent to f ( x) = x 2 − 8 at x = – 3.
The graph of a cubic function with equation f ( x) = x 3 − 3 x − 2 and g ( x) = 2 x − 2
(2)
is drawn. A and B are the turning points of f. P is a point on g and Q is a point on f
such that PQ is perpendicular to the x – axis.
10.2.1 Calculate the coordinates of A and B.
(4)
30
10.2.2 If PQ is perpendicular to the x – axis, calculate the maximum length of PQ,
(4)
10.2.3 Determine the values of k for which f(x) = k has only two real roots.
(2)
10.2.4 Determine the values of x for which f is concave up.
(3)
[17]
ECS16
31
QUESTION 9 FSS16
The sketch below shows the graph of g ( x ) = x 3 − 6 x − 1. P and Q are the turning points
and R the y-intercept of g.
9.4
Determine the coordinates of R.
(2)
9.5
Determine the coordinates of the turning points P and Q.
(6)
9.6
Calculate the values of x for which g strictly increases as x increases.
(2)
9.7
If h( x ) = g ' ( x ) , determine for which values of x is h( x) ≤ 0.
(2)
9.8
Determine the equation of the tangent to g at R.
(4)
9.9 Write down the equation of the line perpendicular to the tangent at P.
[18]
32
(2)
LPS16
QUESTION 9
Given: 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 3 + 𝑏𝑏𝑥𝑥 2 + 3𝑥𝑥 + 3 and 𝑔𝑔(𝑥𝑥) = 𝑓𝑓ꞋꞋ (𝑥𝑥) where 𝑔𝑔(𝑥𝑥) = 12𝑥𝑥 + 4.
9.1
9.2
9.3
9.4
Show that 𝑎𝑎 = 2 and 𝑏𝑏 = 2.
(4)
Determine the minimum gradient of 𝑓𝑓.
(4)
Prove that 𝑓𝑓 will never decrease for any real value of 𝑥𝑥.
Explain the concavity of 𝑓𝑓 for all values of 𝑥𝑥 where 𝑔𝑔(𝑥𝑥) < 0.
MPS16
33
(5)
(1)
[14]
NWS16
QUESTION 7
The function defined by f ( x) = x 3 + px 2 + qx − 12 is sketched below. A(− 4; 36)
and B are turning points of f. g is a tangent to the graph of f at D.
y
A(− 4;36)
g
C
O
x
D
f
B
7.1
Show that p = 5 and q = − 8 .
(6)
7.2
If C(− 1; 0) is an x-intercept of f, calculate the other x-intercepts of f.
(4)
7.3
7.4
Determine the equation of g, the tangent to f at D(1 ; ‒ 14).
For which values of k will f=
( x) g ( x) + k have two positive roots?
(4)
(2)
[16]
QUESTION 9
The diagram below shows the graph of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 3 + 𝑥𝑥 2 − 𝑥𝑥 − 1 .
𝑦𝑦
B
A
C
9.1
9.2
9.3
9.4
D
Calculate the distance between A and B, the x-intercepts.
Calculate the coordinates of D, a turning point of f.
1
Show that the concavity of f changes at 𝑥𝑥 = − 3
For which values of 𝑥𝑥 is:
𝑓𝑓(𝑥𝑥) > 0
9.4.1
𝑓𝑓(𝑥𝑥). 𝑓𝑓′(𝑥𝑥) < 0
9.4.2
34
𝑥𝑥
(5)
(3)
(3)
(1)
(3)
[15]
NJ16
35
QUESTION 2 KZNM17
2.1
Determine co-ordinates of the points on the curve y =
tangent to the curve is – 1 .
2.2
4
where the gradient of the
x
(5)
The graph of a cubic function with equation f ( x) = x 3 + ax 2 + bx + c is drawn.
 f (1) = f (4) = 0
 f has a local maximum at B and a local minimum at x = 4.
y
x
2.2.1
Show that a = −9, b = 24 and c = −16.
(4)
2.2.2
2.2.3
Calculate the coordinates of B.
Write down the value(s) of k for which f(x) = k has negative roots only.
(5)
(2)
2.2.4
Determine the value(s) of x for which f is concave up.
(3)
[19]
36
QUESTION 9 KZNJ17
9.2
The graph of a cubic function with equation f ( x) = − x 3 + mx 2 + nx + 26 has
turning points D(2 ; 54) and E. f intersects the x – axis at A,B and C and the
y – axis at F.
9.2.1
Write down the coordinates of F.
(1)
9.2.2
Show that m = –3 and n = 24.
(6)
9.2.3
Determine the coordinates of E.
(4)
9.2.4
Calculate the point of inflection of the graph of f.
(3)
9.2.5
Determine the values of x for which f is concave up.
(2)
[20]
37
QUESTION 8 KZNS17
The graphs of f ( x) = ( x + 3)( x − 1)( x − p ) and g ( x) = mx + 15 are sketched below.
Both graphs are passing through points A and B. C and D are local maximum and
minimum points of f respectively. A is an x - intercept of f and g.
B is the y - intercept of f and g.
8.1
Explain why the value(s) of p = 5 and m = 5?
(2)
8.2
Determine the coordinates D.
(5)
8.3
For which value(s) of x will f / ( x).g ( x) ≤ 0 ?
8.4
For which values of x will f be concave down?
(3)
38
(3)
[13]
ECJ17
FSS17
39
LPS17
40
MPS17
41
QUESTION 10
KZNS18
3
h( x) = x 3 − x 2 + cx + d is sketched below. A and B are the turning points of h at x = − 2 and x = 3
2
respectively. C is the y – intercept of h. D is the point (4 ; 0).
y
A
h
C
3
O
D
x
B
10.1
Show that c = − 18 and d = 32.
(5)
10.2
Calculate the co-ordinates of A.
(2)
10.3
Determine the x – value of the point of inflection.
(2)
10.4
Write down the interval for which h is concave up.
(1)
10.5
If g ( x) = h(− x) , write down the co-ordinates of the turning point of g that is the
image of A.
(2)
Determine the values of k for which h( x) = k has 2 unequal negative real roots and
one positive real root.
(2)
10.6
[14]
42
ECS18
LPS18
43
MPS18
44
NWS18
QUESTION 9 NM18
The sketch below represents the curve of f ( x) = x 3 + bx 2 + cx + d . The solutions of the equation
f ( x ) = 0 are –2 ; 1 and 4.
y
B
f
x
O
9.1
Calculate the values of b, c and d.
(4)
9.2
Calculate the x-coordinate of B, the maximum turning point of f.
(4)
9.3
Determine an equation for the tangent to the graph of f at x = –1.
(4)
9.4
In the ANSWER BOOK, sketch the graph of
y-intercepts on your sketch.
9.5
For which value(s) of x is f (x) concave upwards?
45
f
//
(x ).
Clearly indicate the x- and
(3)
(2)
[17]
QUESTION 8 NJ18
The graph of f ( x) = − x 3 + 13x + 12 is sketched below.
A, B and D(–1 ; 0) are the x-intercepts of f.
C is the y-intercept of f.
y
f
C
A
D
–1 O
B
x
8.1
Write down the coordinates of C.
(1)
8.2
Calculate the coordinates of A and B.
(5)
8.3
Determine the point of inflection of g if it is given that g(x) = –f(x).
(4)
8.4
Calculate the value(s) of x for which the tangent to f is parallel to the line
= –14x + c.
46
y
(4)
[14]
QUESTION 9 NN18
9.1
The graph of g ( x) = x 3 + bx 2 + cx + d is sketched below.
The graph of g intersects the x-axis at (–5 ; 0) and at P, and the y-axis at (0 ; 20).
P and R are turning points of g.
R
y
g
20
–5
9.2
x
O
P
9.1.1
Show that b = 1, c = –16 and d = 20.
(4)
9.1.2
Calculate the coordinates of P and R.
(5)
9.1.3
Is the graph concave up or concave down at (0 ; 20)? Show ALL your
calculations.
(3)
If g is a cubic function with:
• g(3) = g / (3) = 0
•
g(0) = 27
•
g // (x) > 0 when x < 3 and g//(x) < 0 when x > 3,
draw a sketch graph of g indicating ALL relevant points.
47
(3)
[15]
QUESTION 10
KZNS19
In the diagram, the graph of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 3 + 5𝑥𝑥 2 − 8𝑥𝑥 − 12 is drawn. A and B are the
turning points and C the y - intercept of f. 𝑔𝑔(𝑥𝑥) = 𝑚𝑚𝑚𝑚 + 𝑐𝑐 is a tangent to the graph of f at C.
D is the intersection of f and g.
10.1
Calculate the:
10.1.1
10.1.2
10.1.3
10.2
co-ordinates of the x-intercepts of f.
co-ordinates of B.
x – coordinate of the point of inflection of f.
(6)
(4)
(2)
Determine the:
10.2.1
10.2.2
equation of the g.
values of x for which f /(x). g /(x) > 0.
48
(2)
(3)
[17]
QUESTION 9 FSS19
In the diagram, the graph of f ( x ) = − x 3 + 10 x 2 − 17 x − 28 intersects the y-axis at A.
B and C are the turning points of f .
y
C
x
O
f
A
B
9.1
Write down the coordinates of A.
(1)
9.2
Calculate the coordinates of B and C.
(6)
9.3
For which value(s) of x is f concave up?
(4)
9.4
Determine the value(s) of p for which f ( x ) = p has only one positive root.
49
(2)
[13]
LPS19
QUESTION 8 NJ19
 3 
A cubic function h( x ) = −2 x 3 + bx 2 + cx + d cuts the x-axis at (–3 ; 0);  − ; 0  and (1 ; 0).
 2 
8.1
Show that h( x ) = −2 x 3 − 7 x 2 + 9 .
(3)
8.2
Calculate the x-coordinates of the turning points of h.
(3)
8.3
Determine the value(s) of x for which h will be decreasing.
(2)
8.4
For which value(s) of x will there be a tangent to the curve of h that is parallel to the
line y − 4 x = 7 .
(4)
50
[12]
QUESTION 9 NN19
Given: f ( x) = 3 x 3
9.1
Solve f (x) = f / (x)
9.2
The graphs f , f
(3)
/
and f
//
all pass through the point (0 ; 0).
9.2.1
For which of the graphs will (0 ; 0) be a stationary point?
(1)
9.2.2
Explain the difference, if any, in the stationary points referred to in
QUESTION 9.2.1.
(2)
9.3
Determine the vertical distance between the graphs of f
9.4
For which value(s) of x is f ( x) − f / ( x) < 0 ?
KZNJ20
51
/
and f
//
at x = 1.
(3)
(4)
[13]
ECS20
GPS20
52
LPS20
53
MPS20
NWS20
54
WCS20
55
NN20
56
KZN MOCK 21
57
NN21
58
SECTION E – Graph of the derivative
QUESTION 3 KZNM16
The sketch below represents g ′ ( x) = ax 2 + bx + c with
g ′(1) = g ′(3) = 0 and g ′(0) = 6 .
6
1
3
3.1
Write down the x-coordinates of the stationary points of g.
(2)
3.2
Determine the value of x, for which the gradient of the curve is 6, except x = 0.
(3)
3.3
For which value(s) of x is the graph of g strictly increasing?
(3)
3.4
Calculate the x – coordinate of the point of inflection of g.
(1)
3.5
If it is further given that g (−1) =
4
and g is a cubic function.
3
g ( x) = ax 3 + bx 2 + cx + d , calculate the value of the y – intercept of g.
(7)
Determine the value(s) of x for which g is concave up.
(2)
3.6
[18]
59
NWS17
WCS18
60
ECS19
61
QUESTION 8 NWS19
The graph of f / ( x) = x 2 + bx + c , where f is a cubic function, is sketched below.
The derivative function f / cuts the x-axis at x = − 3 and x = 2 .
y
f
−3
/
x
2
8.1
For which values of x is graphf increasing?
(2)
8.2
At which value of x does graphf have a local maximum value?
(1)
Determine the equation of f / ( x) .
(2)
8.3
8.4
8.5
If f ( x) = px3 + qx 2 + rx + 10 , show that p =
1
1
,q=
and r = − 6 .
3
2
For which value(s) of xis graphf concave down?
KZNS20
62
(4)
(3)
[12]
63
SECTION F – Applications of Maxima and Minima
QUESTION 4 KZNM16
A right circular cone with perpendicular height (h cm) , radius ( r cm) has a slant height(s) of 12 cm.
r
s
1
V = π r 2h
3
4
V = π r3
3
A = π r 2 + π rs
h
A = 4π r 2
4.1
Express the radius of the cone in terms h.
(2)
4.2
1
Show that the Volume of the cone is given by V = 48 π h − π h 3 .
3
(2)
4.3
Calculate the height of the cone for which the volume is at a maximum.
QUESTION 8 KZNJ16
A water tank in the shape of a cylindrical prism has a volume of 330 ml, height of h
cm and radius of r cm..
r
h
64
(5)
[9]
8.1
8.2
8.3
330
π r2
Show that the Surface Area of the cylinder(A) is given by
660
A = 2π r 2 +
r
Calculate the value of the radius of the prism if its surface area is at a minimum.
Show that height of the cylinder is given by h =
(2)
(2)
(5)
[9]
QUESTION 10 FSS16
The diagram below shows a cylindrical can that fits into circle O with radius
R = 2 3 units.
10.1
Express r in terms of x.
(2)
10.2
Calculate the value of x for which the volume of the cylinder is a
maximum.
(5)
10.3
Calculate the maximum volume of the can in terms of π.
65
(2)
[9]
GPS16
MPS16
66
QUESTION 9 NM16
A soft drink can has a volume of 340 cm3, a height of h cm
and a radius of r cm.
9.1
Express h in terms of r.
(2)
9.2
Show that the surface area of the can is given by A(r ) = 2π r 2 + 680 r −1 .
(2)
9.3
Determine the radius of the can that will ensure that the surface area is a minimum.
(4)
[8]
NJ16
67
QUESTION 3 KZNM17
A rectangular box has a length of 5x units, breadth of (9 − 2 x) units and its height
of x units.
x
(9 – 2x)
5x
3.1
Show that the volume (V) of the box is given by V = 45 x 2 − 10 x 3 .
(2)
3.2
Calculate the value of x for which the box will have maximum volume.
(5)
[7]
QUESTION 9 KZNS17
A 450 cm 3 cylindrical piece of wood with height, h, and radius, r, is shown below.
r
h
9.1
9.2
Determine the height of the wood in terms of the radius 𝑟𝑟.
Show that the surface area of the wood is A = 2πr 2 +
900
.
r
9.3
Calculate the radius of the wood (in cm), if the surface area of the wood has to be
as small as possible.
68
(3)
(2)
(4)
[9]
ECJ17
ECS17
69
LPS17
MPS17
70
NWS17
NM17
71
QUESTION 10 NJ17
b
r
●
The figure above shows the design of a theatre stage which is in the shape of a semicircle attached
to a rectangle. The semicircle has a radius r and the rectangle has a breadth b.
The perimeter
of the stage is 60 m.
10.1
Determine an expression for b in terms of r.
(2)
10.2
For which value of r will the area of the stage be a maximum?
(6)
[8]
QUESTION 9 NN17
An aerial view of a stretch of road is shown in the diagram below. The road can be described by
the function y = x 2 + 2 , x ≥ 0 if the coordinate axes (dotted lines) are chosen as shown in the
diagram.
Benny sits at a vantage point B(0 ; 3) and observes a car, P, travelling along the road.
y^
□
P
•B(0; 3)
y = x2 + 2; x > 0
O
x
>
Calculate the distance between Benny and the car, when the car is closest to Benny.
72
[7]
ECS18
QUESTION 10 FSS18
In the sketch, ∆ ABC is an equilateral triangle with sides equal to y units. DEFG is a rectangle. BE = FC = x
units.
A
D
B
G
E
F
10.1Prove that the area of the rectangle is A = 3 xy − 2 3 x 2 .
10.2
Determine, in terms of y, the maximum area of the rectangle.
73
C
(4)
(5)
[9]
LPS18
MPS18
74
QUESTION 10 NM18
Given: f ( x ) = −3 x 3 + x .
Calculate the value of q for which f (x ) + q will have a maximum value of
8
.
9
[6]
QUESTION 9 NJ18
A right circular cone with radius p and height t is machined (cut out) from a solid sphere (with
centre C) with a radius of 30 cm, as shown in the sketch.
Sphere: V =
30
t
4 3
πr
3
1
Cone: V = π r 2 h
3
.C
30
A
9.1
B
p
From the given information, express the following:
9.1.1
AC in terms of t.
(1)
9.1.2
p 2 , in its simplest form, in terms of t .
(3)
9.2
1
Show that the volume of the cone can be written as V (t ) = 20π t 2 − π t 3 .
3
(1)
9.3
Calculate the value of t for which the volume of the cone will be a maximum.
(3)
9.4
What percentage of the sphere was used to obtain this cone having maximum volume?
(4)
[12]
75
QUESTION 10 NN18
In ∆ ABC:
• D is a point on AB, E is a point on AC and F is a point on BC such that DECF is a
parallelogram.
• BF : FC = 2 : 3.
• The perpendicular height AG is drawn intersecting DE at H.
• AG = t units.
• BC = (5 – t) units.
A
D
B
E
H
F
C
G
10.1
Write down AH : HG.
10.2
Calculate t if the area of the parallelogram is a maximum.
(NOTE: Area of a parallelogram = base × ⊥ height)
(1)
(5)
[6]
QUESTION 11 KZNJ19
A piece of wire 10 metres long is cut into two pieces. One piece is bent into a square and the other is bent into
the shape of a rectangle. The rectangle has the width the same length as the square.
11.1
The length of the wire used to make the square is x metres. Write down in terms of x the
length of the side of the square.
(1)
11.2
Show that the sum of the areas(S) of the square and the rectangle is given by
(4)
11.3
Determine the value of x for which the sum of the areas is a maximum.
(3)
[8]
1
5
S = − x2 + x
8
4
76
QUESTION 11 KZNS19
In the diagram below, ∆ABC is an equilateral triangle with sides d units long. P and S are points on sides AB
and AC respectively. Q and R are points on BC such that PQRS is a rectangle. BQ = RC = 2y units.
A
d
S
P
B
2y
Q
R
C
2y
11.1
Show that the area of the rectangle PQRS is given by A = 2 3 y (d − 4 y ).
(4)
11.2
Determine the maximum area of the rectangle in terms of d.
(6)
[10]
QUESTION 10 FSS19
In the diagram below, TUVW is a rectangular picture. The picture is framed such that there is a 3 cm space
around the picture. The perimeter of the rectangle PQRS is 70 cm.
PQ = x units and QR = y units.
x
P
Q
T
W
y
3 cm
U
3 cm
S
V
R
[8]
Calculate the maximum area of the picture TUVW.
77
GPS19
LPS19
78
QUESTION 9 NJ19
A
A cone with radius r cm and height AB is inscribed in a
sphere with centre O and a radius of 8 cm. OB = x.
Volume of sphere =
Volume of cone =
8
4 3
πr
3
O
1 2
πr h
3
r
x
B
9.1
Calculate the volume of the sphere.
(1)
9.2
Show that r 2 = 64 − x 2 .
(1)
9.3
Determine the ratio between the largest volume of this cone and the volume of the
sphere.
KZNJ20
79
(7)
[9]
KZNS20
ECS20
80
FSS20
GPS20
81
LPS20
82
MPS20
NWS20
83
WCS20
NN20
84
NN21
85
SECTION G – Rates of change
ECS16
QUESTION 10
WCS16
A water tank with an inlet and an outlet is used to water a garden. The equation
1
1
𝐷𝐷 = 3 + 𝑡𝑡 2 − 𝑡𝑡 3 gives the depth of water in metres where 𝑡𝑡 is the time in hours that has elapsed
2
4
since 09:00.
10.1
What is the depth of the water at 11:00 (2 hours later)?
(1)
10.2
At what rate does the depth of the water change at 12:00?
(2)
10.3
At what time will the inflow of the water be the same as the outflow of water?
(4)
[7]
QUESTION 10 KZNJ17
Petrol in the Tank (in litres)
A car has 4 litres of petrol in the tank. After driving 40 kilometres, the car has 2 litres of petrol left.
The graph below shows the car’s petrol consumption and distance travelled.
4
3
2
1
0
ℓ
40
80
Distance Travelled (in km)
86
120
10.1
10.2
Determine the gradient of the line ℓ.
Explain the significance of the value of the gradient obtained.
NWS17
(2)
(4)
[6]
QUESTION 11 KZNS18
The depth of water (in metres) left in the dam, t hours, after the sluice gate was opened to
allow the flow of water to drain from the dam is given by the equation.
1
1
D (t ) = 28 − t 2 − t 3 .
9
27
11.1
Calculate the average rate of change in the depth of the water after the first 2 hours.
(4)
11.2
Determine the rate at which the level of the water is decreasing after 16 hours.
(4)
[8]
NWS18
87
WCS18
NWS19
QUESTION 9
A hunter was standing at point A, along the fence of a rectangular game enclosure,
when he spotted a deer standing at point B, the corner of the rectangular enclosure.
The distance from A to B is 1200m. At exactly the same time as the hunter started to
move in an easterly direction towards B, the deer started to move in a southerly
direction towards D. The hunter moves at 4metres per second and the deer moves at
5metres per second. After t seconds, the hunter is at a point H and the deer is at point
D.
A
H
B
D
88
The hunter tries to shoot the deer but with his caliber rifle he must be at most 800m
from the deer.
9.1
Show that the distance between the hunter and the deer (HD) at t seconds after
they both started moving can be written as:
HD(t) = 41t 2 − 9 600t + 1 440 000
(4)
9.2
How long after they started walking, were they the nearest to one another?
Show all calculations.
(3)
9.3
The calibre of the hunter’s rifle allows him to be at most 800m from his target.
Was the hunter within shooting range of the deer at the time when they were
nearest to each other? Show all calculations.
(3)
[10]
QUESTION 8 NN19
After flying a short distance, an insect came to rest on a wall. Thereafter the insect started
crawling on the wall. The path that the insect crawled can be described by
h(t ) = (t − 6)(−2t 2 + 3t − 6) , where h is the height (in cm) above the floor and t is the time (in
minutes) since the insect started crawling.
8.1
At what height above the floor did the insect start to crawl?
(1)
8.2
How many times did the insect reach the floor?
(3)
8.3
Determine the maximum height that the insect reached above the floor.
(4)
[8]
89
SECTION H – Calculus of motion
QUESTION 9
KZNJ16
The height(h) (in metres) of a golf ball t seconds after it is been hit into the air is given by
h(t ) = 20t − 5t 2 . Determine the following:
9.1
the average vertical velocity of the ball during the first two seconds.
(2)
9.2
the vertical velocity of the ball after 1.5 seconds.
(3)
9.3
the time taken for the vertical velocity to be zero.
(2)
9.4
the vertical velocity with which the ball hits the ground.
(5)
[12]
QUESTION 11 KZNS16
A car speeds along a 1 kilometre track in 25 seconds. Its distance (in metres) from the start
after t seconds is given by
s (t ) = t 2 + 15t
11.1
Write down an expression for the speed (the rate of change of distance with respect to time)
of the car after t seconds.
(1)
11.2
Determine the speed of the car when it crosses the finish line.
(1)
11.3
Write down an expression for the acceleration (the rate of change of speed with respect to time)
of the car after t seconds.
(2)
11.4
11.5
Hence, or otherwise, calculate the acceleration of the car after 5 seconds.
Calculate the speed of the car when it is 250 metres down the track.
(1)
(4)
[9]
QUESTION 10 LPS16
A particle moves according to the function 𝐻𝐻(𝑡𝑡) = −2𝑡𝑡 2 + 70𝑡𝑡, where 𝑡𝑡 is time in seconds
and 𝐻𝐻 is height in meters above the ground.
10.1
Calculate the height of the particle after 5 seconds.
(2)
10.2
How much time did it take for the particle to strike the ground again?
(4)
10.3
What was the maximum height that the particle reached?
(4)
10.4
What was the initial speed (velocity) of the particle?
(2)
[12]
90
QUESTION 8 NWS16
A marathon athlete trains between two towns A and C. He starts at point B which lies between towns
A and C. The athlete runs from point B to town C and back to point B. The road between the towns
is in a straight line. The displacement S, in kilometres, from point B after t hours, is given by:
S (t ) =− t 3 + 12t 2 − 32t
8.1
How many hours will it take the athlete to return to point B?
(3)
8.2
Calculate the distance between point B and town C.
(5)
8.3
Calculate the maximum speed that the athlete has reached while training.
GPS18
KZN MOCK 21
91
(4)
[12]