MATHEMATICS

GRADE 12: REVISION STUDY GUIDE

WINTER CLASSES

Topic
Differential Calculus

JENN TRAINING & CONSULTANCY: GRADE 12 WINTER CLASSES

Differential Calculus
Outline :
1. An intuitive understanding of the limit concept,
in the context of approximating the rate of
change or gradient of a function at a point.
2. Use limits to define the derivative of a function
f (x  h)  f (x)
f at any x : f '(x)  lim .
h 0 h
Generalise to find the derivative of f at any
point x in the domain of f , i.e., define the
derivative function f ' (x) of the function f (x) .
Understand intuitively that f ' (a)
of the tangent to the graph of f
x -coordinate a .
3. Using the definition (first principle), find the
derivative, f ' (x) for a, b and c constants:

(a) f (x)  ax2  bx  c;

Parts 4.
(b) f (x)  ax3 ;
a
(c) f (x)  ;
(d) f (x)  c
x

Use the formula

d (axn )  anxn  1,
dx
any real number n) together with the rules
(for

Part 1 : First principles and Rules of differentiation d [ f (x)  g(x)]  d [ f (x)]  d [g(x)]
(a)
Part 2 : Sketching dx dx dx
and
Part 3 : Finding parameters/equation
d [kf (x)]  k d [ f (x)]
Part 4 : Reading the graph (b) ( k a constant)
dx dx
Part 5 : Maxima and Minima (optimisation) 5. Find equations of tangents to graphs of
functions.
6. Introduce the second derivative
f ' ' (x)  d ( f ' (x)) of f (x) and how it
dx
determines the concavity of a function.
7. Sketch graphs of cubic polynomial functions
using differentiation to determine the
co-ordinate of stationary points, and points of
inflection (where concavity changes). Also,
determine the x -intercepts of the graph using
the factor theorem and other techniques.
8. Solve practical problems concerning
optimisation and rate of change, including
calculus of motion.

JENN TRAINING & CONSULTANCY: GRADE 12

 SURD FORM TO EXPONENTIAL FORM AND VICE VERSA

The interchange between surd form and exponential form is defined by the following
rule/definition.
n
m
x n
= x m
check examples that follow,
1 2
1 6 6 −3
1. x = x2 ; 2.
3
x2 = x 3 ; 3. = 1  x −1 ; 4. = x 2
x 5 x3 5
CURVE SKETCHING

i. Consider the sign of a and determine the general shape of the graph of
y = ax + bx + cx + d .
3 2

When a  0 , the graph/function will start with When a  0 , the graph/function will start with
a Local maximum. a Local minimum.

ii. Determine the y − int ercepts by letting x =0.

iii. Find the x − values of the turning points of the function by letting f '( x ) = 0 and solve.
iv. Determine the x − int ercepts by letting ax + bx + cx + d = 0 and solve. The great
3 2

emphasis is on Synthetic Division Method for simplicity.

v. Make a sketch on the Cartesian plane, taking consideration of the shape. Indicate all
the coordinates with the axes as well as the coordinates of the turning points.

INTERPRETATION OF THE CUBIC FUNCTIONS

1. Determining the equation of the cubic function when given the x − int ercepts and
any point on the graph. We use the following formula y = a ( x − x1 )( x − x2 )( x − x3 )
2. Given the turning points (stationary points) S ( a; b) and T ( p; q ) . We use the concepts
of finding the turning points as above i.e. determining the first derivative of
y = ax3 + bx 2 + cx + d in the following sense:
f '(0) = a and f (a ) = b

f '(0) = p and f ( p ) = q
MINIMA AND MAXIMA ANALYSIS

The interpretation of the 2-Dimensional shapes, 3-Dimensional shapes and Distance/Height is

depending on the knowledge of the First Derivative and the Formulas for Geometric shapes,
Prisms, Pyramids and/or Spheres. The following information will give guidance in the
interpretation of the questions for minima and/or maxima.

2-Dimensional: Perimeter and Area (Squares, Circles, Rectangles…)

3-Dimensional: Surface Area and Volume (Prisms, Pyramids and Spheres)

3

JENN TRAINING & CONSULTANCY: GRADE 12

CONTENTS PAGE

PROGRAMME 5

PART 1: First Principles and Rules of Differentiation

6-7
➢ First Principles and Rules of Differentiation

PART2: Sketching 8

PART3: Finding Parameters/Equation

9-13

PART3: Reading the Graph 14

PART3: Maxima and Minima 15-22

JENN TRAINING & CONSULTANCY: GRADE 12

 WINTER CLASSES

DAY ACTIVITY TIME

Calculus 2 Hours
6

7 Calculus 2 Hours

8 Calculus 2 Hours

9 Calculus 2 Hours

Calculus 1 Hour 35 Mins

10

Post Test 25 Min

JENN TRAINING & CONSULTANCY: GRADE 12

Part 1
QUESTION 1

1.1 Use the definition of the derivative (first principles) to determine f ′(x) if f ( x) = 2 x 3

2
1.2 Given: f ( x) = −
x

1.2.1 Determine f ′(x) from first principles.

1.2.2 For which value(s) of x will f ′( x) > 0 ? Justify your answer.

1.3 Determine f ′(x) from first principles if f ( x) = 9 − x 2 .

1.4 Differentiate f by first principles where f ( x) = x 2 − 2 x .

1.5 Determine f ′( x) from first principles if f ( x) = − 4 x 2 .

QUESTION 2

2.1 Determine, using the rules of differentiation:

dy x 1
if y = − 3
dx 2 6x
Show ALL calculations.

2.2 Use the rules of differentiation to determine the following:

Dx [( x − 2)( x + 3) ]

dy 1
2.3 Evaluate: if y = x 2 − 3
dx 2x

dy x6
2.4 Evaluate if y = +4 x.
dx 2

dy 3 x2
2.5.1 if y = −
dx 2x 2

2.5.2 f ′(1) if f ( x) = (7 x + 1) 2
dy x
2.6 Evaluate if y = x − 4 + 2 x 3 − .
dx 5
x2 + x − 2
2.7 Given: g ( x) =
x −1
2.7.1 Calculate g ′(x) for x ≠ 1 .
6
2.7.2 Explain why it is not possible to determine g ′(1) .
JENN TRAINING & CONSULTANCY: GRADE 12
2.8 Evaluate:

2.8.1 D x [( x 3 − 3) 2 ]

dy 4 x3
2.8.2 if y = −
dx x 9
dy
if y = (2 − 5 x )
2
2.9 Use the rules of differentiation to determine
dx

4 1
2.10 Calculate Dx 4  3  4  .
 x x 

2.11 Determine
dy
dx

if y  1  x . 2

2.12 Evaluate:

2.12.1 D x [1 + 6 x ]

dy 8 − 3x 6
2.12.2 if y =
dx 8x 5

dy 2 x +1
2.13 Determine if y =
dx x2

dy 1
2.14 Evaluate if y = x 2 − 2 x.
dx 4

2.15 ( ) and
Given: y = 4 3 x 2 x = w− 3

dy
Determine .
dw

JENN TRAINING & CONSULTANCY: GRADE 12

Part 2
QUESTION 1
Given: f ( x) = − x 3 + x 2 + 8 x − 12
1.1 Calculate the x-intercepts of the graph of f.

1.2 Calculate the coordinates of the turning points of the graph of f.

1.3 Sketch the graph of f, showing clearly all the intercepts with the axes and turning
points.

1.4 Write down the x-coordinate of the point of inflection of f.

1.5 Write down the coordinates of the turning points of h( x) = f ( x ) − 3 .

QUESTION 2
Given: f ( x) = − x 3 − x 2 + x + 10

2. 1 Write down the coordinates of the y-intercept of f.

2. 2 Show that (2 ; 0) is the only x-intercept of f.

2. 3 Calculate the coordinates of the turning points of f.

2. 4 Sketch the graph of f in your ANSWER BOOK. Show all intercepts with the axes
and all turning points.
QUESTION 3

Given: f ( x) = − x 3 + 3x 2 − 4.

3.1 Show that f (−1) = 0

3.2 Hence, or otherwise, determine the x- and y-intercepts of the graph of f..

3.3 Determine the coordinates of the turning points of the graph of f.

3.4 Draw a neat sketch graph of f. Clearly show all the intercepts with the
axes and the turning points on the graph.

3.5 For which values of x is f increasing?

QUESTION 4
Given: g ( x)  ( x  6)( x  3)( x  2)
4.1 Calculate the y-intercept of g.

4.2 Write down the x-intercepts of g.

4.3 Determine the turning points of g.

4.4 Sketch the graph of g.
4.5 For which values of x is g(x) . g / (x)  0? 8

JENN TRAINING & CONSULTANCY: GRADE 12

Part 3
QUESTION 1

The function f ( x ) = − 2 x 3 + ax 2 + bx + c is sketched below.

The turning points of the graph of f are T(2 ; – 9) and S(5 ; 18).
y

S(5 ; 18)

f
x
O

T(2 ; − 9)

1.1 Show that a = 21 , b = – 60 and c = 43.

1.2 Determine an equation of the tangent to the graph of f at x = 1.
1.3 Determine the x-value at which the graph of f has a point of inflection.

JENN TRAINING & CONSULTANCY: GRADE 12

QUESTION 2

The graph of h( x) = − x3 + ax 2 + bx is shown below. A(– 1 ; 3,5 ) and B(2 ; 10 ) are the turning
points of h. The graph passes through the origin and further cuts the x-axis at C and D.

B(2 ; 10)
•

C D x
0

•
A(– 1 ; –3,5)

3
2.1 Show that a = and b = 6.
2

2.2 Calculate the average gradient between A and B.

2.3 Determine the equation of the tangent to h at x = – 2.

2.4 Determine the x-value of the point of inflection of h.

2.5 Use the graph to determine the values of p for which the equation
3
− x 3 + x 2 + 6 x + p = 0 will have ONE real root.
2

10

JENN TRAINING & CONSULTANCY: GRADE 12

QUESTION 3

The graph below represents the functions f and g with f ( x) = ax 3 − cx − 2 and g ( x) = x − 2 .

A and (– 1 ; 0 ) are the x-intercepts of f . The graphs of f and g intersect at A and C.

y
f
g

(-1 ; 0) x
O A

–2

C
B

3.1 Determine the coordinates of A.

3.2 Show by calculation that a = 1 and c = 3.

3.3 Determine the coordinates of B, a turning point of f.

3.4 Show that the line BC is parallel to the x-axis.

3.5 Find the x-coordinate of the point of inflection of f.

3.6 Write down the values of k for which f ( x) = k will have only ONE root.

3.7 Write down the values of x for which f / ( x ) < 0 .

11

JENN TRAINING & CONSULTANCY: GRADE 12

 QUESTION 4

The graphs of f ( x) = ax 3 + bx 2 + cx + d and g ( x) = 6 x − 6 are sketched below.

A(– 1 ; 0) and C(3 ; 0) are the x-intercepts of f.
The graph of f has turning points at A and B.
D(0 ; – 6) is the y-intercept of f.
E and D are points of intersection of the graphs of f and g.
y

g
A(−1 ; 0) C(3 ; 0)
x
O

D(0 ; −6)
f

E B

4.1 Show that a = 2 ; b = – 2 ; c = – 10 and d = – 6.

4.2 Calculate the coordinates of the turning point B.

4.3 h(x) is the vertical distance between f(x) and g(x), that is h( x) = f ( x ) − g ( x ) .
Calculate x such that h(x) is a maximum, where x < 0.

12

JENN TRAINING & CONSULTANCY: GRADE 12

QUESTION 5

The graph of f ( x) = − x 3 + ax 2 + bx + c is sketched below. The x-intercepts are indicated.

x
0 1 2 4
A

5.1 Calculate the values of a, b and c.

5.2 Calculate the x-coordinates of A and B, the turning points of f.

5.3 For which values of x will f ′( x) < 0 ?

QUESTION 6

The tangent to the curve of g ( x) = 2 x 3 + px 2 + qx − 7 at x = 1 has the equation y = 5x – 8.

6.1 Show that (1 ; – 3) is the point of contact of the tangent to the graph.

6.2 Hence or otherwise, calculate the values of p and q.

QUESTION 7
7.1 Calculate the values of a and b if f ( x) = ax 2 + bx + 5 has a tangent at x = –1 which
is defined by the equation y = −7 x + 3

13

JENN TRAINING & CONSULTANCY: GRADE 12

Part 4
QUESTION 1
A cubic function f has the following properties:
⎛1⎞
• f ⎜ ⎟ = f (3) = f (−1) = 0
⎝2⎠
⎛ 1⎞
• f ′(2) = f ′⎜ − ⎟ = 0
⎝ 3⎠
⎡ 1 ⎤
• f decreases for x ∈ ⎢− ; 2⎥ only
⎣ 3 ⎦
Draw a possible sketch graph of f, clearly indicating the x-coordinates of the turning points
and ALL the x-intercepts.
QUESTION 2

2.1 Given: f ( x ) = ax 3 + bx 2 + cx + d

Draw a possible sketch of y = f ′(x) if a, b and c are all NEGATIVE real

numbers.
QUESTION 3
3.1 The graph of the function f ( x) = − x 3 − x 2 + 16 x + 16 is sketched below.

x
0

3.1.1 Calculate the x-coordinates of the turning points of f.

3.1.2 Calculate the x-coordinate of the point at which f ′(x) is a maximum.

3.2 Consider the graph of g ( x) = −2 x 2 − 9 x + 5 .
3.2.1 Determine the equation of the tangent to the graph of g at x = –1.
3.2.2 For which values of q will the line y = –5x + q not intersect the parabola?
3.3 Given: h( x) = 4 x 3 + 5 x
Explain if it is possible to draw a tangent to the graph of h that has a negative
gradient. Show ALL your calculations.
 QUESTION 4
The graph of y = f ′(x) , where f is a cubic function, is sketched below.
y

−4 0 1 x

y = f /(x)

Use the graph to answer the following questions:

4.1 For which values of x is the graph of y = f ′(x) decreasing?

4.2 At which value of x does the graph of f have a local minimum? Give reasons for
your answer.
QUESTION 5
In the sketch below, the graph y = ax 2 + bx + c represents the derivative, f ′ , of f where f is a
cubic function.

12

Graph of y = f ′(x)

x
0 1 2

5.1 Write down the x-coordinates of the stationary points of f.

5.2 State whether each stationary point in QUESTION 10.1 is a local minimum or a local
maximum. Substantiate your answer.
5.3 Determine the x-coordinate of the point of inflection of f.
5.4 Hence, or otherwise, draw a sketch graph of f
JENN TRAINING & CONSULTANCY: GRADE 12
.
QUESTION 6

The graphs of y = g ′( x) = ax 2 + bx + c and h( x) = 2 x − 4 are sketched below. The graph of

y = g ′( x) = ax 2 + bx + c is the derivative graph of a cubic function g.

The graphs of h and g ′ have a common y-intercept at E.

C(– 2 ; 0) and D(6 ; 0) are the x-intercepts of the graph of g ′ .
A is the x-intercept of h and B is the turning point of g ′ .
AB || y-axis.

h y = g / (x)

C(– 2 ; 0) A D(6 ; 0) x
O

6.1 Write down the coordinates of E.

6.2 Determine the equation of the graph of g ′ in the form y = ax 2 + bx + c .

6.3 Write down the x-coordinates of the turning points of g

6.4 Write down the x-coordinate of the point of inflection of the graph of g

6.5 Explain why g has a local maximum at x = −2 .

16

JENN TRAINING & CONSULTANCY: GRADE 12

Part 5
QUESTION 1
1.1 The depth h of petrol in a large tank, t days after the tank was refilled, is given by
t t3
h(t ) = 12 − − metres for 0 ≤ t ≤ 4 .
4 6
1.1.1 What is the depth after 3 days?

1.1.2 What is the rate of decrease in the depth after 2 days? (Give your answer
in the correct units.)

b
1.2 A function g ( x) = ax 2 + has a minimum value at x = 4. The function value at
x
x = 4 is 96. Calculate the values of a and b.

QUESTION 2
A tourist travels in a car over a mountainous pass during his trip. The height above sea level of
the car, after t minutes, is given as s (t ) = 5t 3 − 65t 2 + 200t + 100 metres. The journey lasts
8 minutes.
2.1 How high is the car above sea level when it starts its journey on the mountainous
pass?

2.2 Calculate the car's rate of change of height above sea level with respect to time,
4 minutes after starting the journey on the mountainous pass.

2.3 Interpret your answer to QUESTION 12.2.

2.4 How many minutes after the journey has started will the rate of change of height with
respect to time be a minimum?
QUESTION 3
Water is flowing into a tank at a rate of 5 litres per minute. At the same time water flows out of
the tank at a rate of k litres per minute. The volume (in litres) of water in the tank at time t (in
minutes) is given by the formula V (t ) = 100 − 4t .
3.1 What is the initial volume of the water in the tank?
3.2 Write down TWO different expressions for the rate of change of the volume of water
in the tank.
3.3 Determine the value of k (that is, the rate at which water flows out of the tank).

QUESTION 4

A particle moves along a straight line. The distance, s, (in metres) of the particle from a fixed
point on the line at time t seconds ( t ≥ 0 ) is given by s (t ) = 2t 2 − 18t + 45 .

4.1 Calculate the particle's initial velocity. (Velocity is the rate of change of distance.)

4.2 Determine the rate at which the velocity of the particle is changing at t seconds.

4.3 After how many seconds will the particle be closest to the fixed point?

17

JENN TRAINING & CONSULTANCY: GRADE 12

QUESTION 5

Sketched is the graph of y = x 2 . A(t ; t 2 ) and B(3 ; 0) are shown.

A(t ; t 2)

x
O B(3 ; 0)

5.1 A(t ; t 2 ) is a point on the curve y = x 2 and the point B(3 ; 0) lies on the x-axis.
Show that AB 2 = t 4 + t 2 − 6t + 9 .

5.2 Hence, determine the value of t which minimises the distance AB.

QUESTION 6

A small business currently sells 40 watches per year. Each of the watches is sold at R144. For
each yearly price increase of R4 per watch, there is a drop in sales of one watch per year.

6.1 How many watches are sold x years from now?

6.2 Determine the annual income from the sale of watches in terms of x.

6.3 In what year and at what price should the watches be sold in order for the business to
obtain a maximum income from the sale of watches?

18

JENN TRAINING & CONSULTANCY: GRADE 12

 QUESTION 7
A rectangular box is constructed in such a way that the length (l) of the base is three times as
long as its width. The material used to construct the top and the bottom of the box costs R100
per square metre. The material used to construct the sides of the box costs R50 per square
metre. The box must have a volume of 9 m3. Let the width of the box be x metres.

x
l
7.1 Determine an expression for the height (h) of the box in terms of x.
1 200
7.2 Show that the cost to construct the box can be expressed as C = + 600 x 2 .
x
7.3 Calculate the width of the box (that is the value of x) if the cost is to be a minimum.

QUESTION 8
A farmer has a piece of land in the shape of a right-angled triangle OMN, as shown in the figure
below. He allocates a rectangular piece of land PTOR to his daughter, giving her the freedom
to choose P anywhere along the boundary MN. Let OM = a, ON = b and P(x ; y) be any point
on MN. y
N (0 ; b)

P(x ; y)
T

O R M (a ; 0)

8.1 Determine an equation of MN in terms of a and b.

8.2 Prove that the daughter's land will have a maximum area if she chooses P at the
midpoint of MN.

19

JENN TRAINING & CONSULTANCY: GRADE 12

QUESTION 9
A drinking glass, in the shape of a cylinder, must hold 200 ml of liquid when full.

200
9.1 Show that the height of the glass, h, can be expressed as h = .
π r2
400
9.2 Show that the total surface area of the glass can be expressed as S (r ) = π r 2 + .
r
9.3 Hence determine the value of r for which the total surface area of the glass is a
minimum.
QUESTION 10
A satellite is to be constructed in the shape of a cylinder with a hemisphere at each end. The
radius of the cylinder is r metres and its height is h metres (see diagram below). The outer
surface area of the satellite is to be coated with heat-resistant material which is very expensive.
π
The volume of the satellite has to be cubic metres.
6
Outer surface area of a sphere = 4π r 2
r
● Curved surface area of a cylinder = 2π rh
4
Volume of a sphere = π r 3
3
Volume of a cylinder = π r 2 h
h

r
10.1 Show that h = 2
− .
6r 3

10.2 Hence, show that the outer surface area of the satellite can be given as
4π r 2 π
S= + .
3 3r 1 4

10.3 Calculate the minimum outer surface area of the satellite.

JENN TRAINING & CONSULTA NCY: GRADE 12 20

QUESTION 11

Devan wants to cut two circles out of a rectangular piece of cardboard of 2 metres long and
4x metres wide. The radius of the larger circle is half the width of the cardboard and the
2
smaller circle has a radius that is the radius of the bigger circle.
3

A = lb A = πr2 P = 2 (l + b ) C = 2π r

4x

2 metres

52π 2
11.1 Show that the area of the shaded region is A( x) = 8 x − x .
9

11.2 Determine the value of x, such that the area of the shaded region is a maximum.

11.3 Calculate the total area of the circles, if the area of the shaded region is to be a
maximum.

21

JENN TRAINING & CONSULTANCY: GRADE 12

SOURCES OF INFORMATION(SOIs):

1. Mathematics CAPS document

2. Mathematics Grade 12 previous question papers

23

JENN TRAINING & CONSULTANCY: GRADE 12