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Ref No: 240.134.1 task_title_page_report_191 Page

Approved by: TS Letho Issue Date: 12.03.2015 Rev: 1 12.03.2015

TASK TITLE PAGE

SUBJECT : MATHEMATICS LEVEL : N5

TASK : ASSIGNMENT 3 DATE : 1 JUNE 2020
ASSESSOR : L. THOBEJANE DURATION : 3 HRS
MODERATOR : B. FUZILE MARKS : 000
CAMPUS : SASOLBURG

INSTRUCTIONS AND INFORMATION

1. Answer ALL the questions.

2. Read ALL the questions carefully.

3. Number the answers according to the numbering system used in this question paper.

4. Show ALL intermediate steps and simplify where possible.

5. ALL final answers must be rounded off to THREE decimal places.

6. Questions may be answered in any order, but subsections of questions must be kept
together.

7. Use only BLUE or BLACK ink.

8. Sketches must be large, neat and fully labelled.

9. Write neatly and legibly.

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QUESTION 1 : APPLICATIONS OF DEFFERENTIATION

1.1 Given: f x   2 x 3  9 x 2  17 x  20

1.1.1 Determine the coordinate of the point of inflection of f x  . (2)

1.1.2 Draw up a table of x and f x  , where 𝑥 is ranging from x   2 to x  2

𝑥 = −2. (2)

1.1.3 Draw a neat graph of f x  between these values show to the point of
inflection on it. (2)

1.1.4 One root of the equation 2 x 3  9 x 2  17 x  20 is close to -0,827. Use this

value and one approximation of Taylor's/Newton's method to determine a
better approximation of this root (Root correct to THREE decimal figures). (4)

1.2 A rectangular storage area is to be constructed along the side of a tall building. A
security fence is required along the remaining 3 sides of the area. What is the maximum
area that can be enclosed with 800 m of fencing?

𝒚
(5)
[15]
QUESTION 2:

2.1 Given: 𝑓(𝑥) = 7𝑥 3 − 8𝑥 + 4

2.1.1 Determine the coordinate of the point of inflection of 𝑓(𝑥). (3)

2.1.2 Draw up a table of 𝑥 and 𝑓(𝑥), where 𝑥 is ranging from 𝑥 = −2 to 𝑥 = 2. (2)

2.1.3 Draw a neat graph of 𝑓(𝑥) between these values showing the turning
points on it. (2)

2.1.4 One root of the equation 𝑓(𝑥) = 7𝑥 3 − 8𝑥 + 4 is close to −1. Use this
value and one approximation of Taylor's/ Newton's method to determine
a better approximation of this root (Root correct to THREE decimal
figures). (3)

2.2 The length of one side of a rectangle is three times the length of the other side. At
what rate is the enclosed area decreasing when the shortest side is 6 𝑚 long and is
decreasing at a rate of 2 𝑚/𝑠? (5)

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2.3 An object moves in a straight line so that after 𝑡 seconds its distance is 𝑥 metres from
a fixed point on the line given by 𝑥 = 𝑡 3 − 7𝑡 2 + 8𝑡 + 2. Obtain an expression for
velocity and acceleration of the object after 𝑡 seconds and then calculate the values
of 𝑡 when the object is at rest. (5)
[20]
QUESTION 3:

3.1 Given:

𝑓(𝑥) = 𝑥 3 − 7𝑥 2 + 8𝑥 − 3

3.1.1 Determine the coordinates of the turning points of 𝑓(𝑥). (2)

3.1.2 Draw up a table of 𝑥 and 𝑓(𝑥), where 𝑥 is ranging from 𝑥 = −2 to 𝑥 = 7. (2)

3.1.3 Draw a neat graph of 𝑓(𝑥) between these values and show the turning
points on it. (2)

3.1.4 One root of the equation 𝑓(𝑥) = 𝑥 3 − 7𝑥 2 + 8𝑥 − 3 is close to 5.

Use this value and one approximation of Taylor's/Newton's method to

determine a better approximation of this root (correct to THREE decimal
figures). (4)

3.2 A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the
area of the sheet is decreasing at a rate of 0,5 𝑚2 /𝑠 at what rate is the radius decreasing
when the area of the sheet is 12 m2 ? (5)

3.3 A cylindrical can with bottom but no top with a volume of 30 cm3 must be constructed.

Determine the dimensions of the can that will minimise the amount of material needed
to construct the can.

HINT: 𝑉 = 𝜋𝑟 2 ℎ and 𝐴 = 2𝜋𝑟ℎ + 𝜋𝑟 2 (5)

[20]
QUESTION 4:

4.1 Given: 𝑓(𝑥) = 2𝑥 3 + 5𝑥 2 + 4𝑥 − 3

4.1.1 Determine the coordinates of the turning points of 𝑓(𝑥). (3)

4.1.2 Draw up a table of 𝑥 and 𝑓(𝑥), where x is ranging from 𝑥 = −2 to 𝑥 = 2. (2)

4.1.3 Draw a neat graph of 𝑓(𝑥) between these values and show the turning
points on it. (2)

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4.1.4 Use the table and the graph to estimate a value for the best root between
𝑥 = 0 and 𝑥 = 1 of the equation 2𝑥 3 + 5𝑥 2 + 4𝑥 − 3 and then use
Taylor's/Newton's method to determine a better approximation of this
root. (Root correct to THREE decimal figures) (4)

4.2 A stone is dropped into a pond, the ripples forming concentric circles, which expand.

At what rate is the area of one of those circles increasing when the radius is 4 𝑚 and
increasing at the rate of 0, 5 𝑚/𝑠.

HINT: 𝐴 = 𝜋𝑟 2 (3)

4.3 You are going to fence in a rectangular field. Looking at the field from above, the cost
of the vertical sides is R10/𝑚, the cost of the bottom is R2/𝑚 and the cost of the top
is R7/𝑚.

If you have R700, determine the dimensions of the field that will maximise the
enclosed area.

R7/m

R10/m R10/m

R2/m
(5)
[19]
QUESTION 5:

5.1 Given: 𝑓(𝑥) = 𝑥 3 − 2𝑥 − 5

5.1.1 Determine the coordinates of the turning points of 𝑓(𝑥). (2)

5.1.2 Draw up a table of 𝑥 and 𝑓(𝑥), where 𝑥 is ranging from

𝑥 = −2 to 𝑥 = 4. (2)

5.1.3 Draw a neat graph of 𝑓(𝑥) between these values in QUESTION 3.1.2
showing the turning points on it. (2)

5.1.4 One root of the equation 𝑓(𝑥) = 𝑥 3 − 2𝑥 − 5 is close to 2.

Use this value and ONE approximation of Taylor's/Newton's method to

determine a better approximation of this root, correct to THREE decimals. (4)

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5.2 A water tank has the shape of an inverted circular cone with a base radius of 2 m and a
height of 4 m.

If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the
water level is rising when the water is 3 m deep.
1 ℎ
HINT: 𝑉 = 3 𝜋𝑟 2 ℎ and the radius of the water level is given by 𝑟 = 2 (5)

5.3 The position function of a particle is given by 𝑠 = 𝑡 3 − 4.5𝑡 2 − 7𝑡, 𝑡 ≥ 0, where 𝑠 is

the displacement in metres and 𝑡 is time in seconds.

5.3.1 When does the particle reach a velocity of 5 m/s? (3)

5.3.2 When is the acceleration zero? (2)

[20]

QUESTION 6:

6.1 Given: f ( x)  x 3  x 2  2 x  1

6.1.1 Determine the coordinates of the point of inflection of f (x) . (2)

6.1.2 Draw up a table of values for x and f(x) with x ranging from x = 1,1 to
x = 1,4 using intervals of 0,1. Round answers off to ONE decimal.

Draw a neat graph of f(x) between these values. (4)

6.1.3 An estimated value of the root of the equation x 3  x 2  2 x  1  0 is

x  1,23 .

Use Taylor/Newton’s method to determine a better approximation of this

root. (4)

6.2 An open cardboard box with a square base is required to hold 205 dm3.

Determine the dimensions of the box if the area of the cardboard used should be as
small as possible. (5)

6.3 An object moves in a straight line. After t seconds the distance x meters from a fixed
point on the line is given by

1
x  t 3  t 2  8t  1
3

Obtain an expression for the velocity of the object after t seconds and then calculate the
values of t when the object is at rest. (4)
[19]

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QUESTION 7:

7.1 Given: f ( x)  3x3  4 x 2  2 x  2

7.1.1 Determine the coordinates of the turning points of f (x). (3)

7.1.2 Verify by using a table that the equation 0  3x3  4 x 2  2 x  2 has a root
between the points x =1 and x =2.

Use values on the table:  1  x  2 (2)

7.1.3 Hence, make a neat sketch of the graph of the function f (x). (3)

7.1.4 If the positive root of f (x) is estimated as 1,7, use Taylor's/Newton's method
to determine a better approximation of this root. (2)

7.2 A farmer wants to enclose a field with length and breadth x meter and y meter
respectively.

The cost of fencing is R36,00/m for the length and R45,00/m for the breadth. He has
an amount of R56 000,00 available.

7.2.1 Give the formula of the area and the cost in terms of x and y. (2)

7.2.2 Calculate the dimensions of the field with maximum area. (5)

7.3 A fluid flows into a cylindrical tank of radius 1,5 m at a rate of 3 m3/s.

Calculate how fast the surface is rising.

r  1,5m

HINT: V   .r 2 .h (4)
[21]

QUESTION 8:

AUG 2016 (MISSING)

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QUESTION 9:

9.1 Given: f ( x)  x( x2  5)  4

9.1.1 Determine the co-ordinates of the turning points of f (x). (3)

9.1.2 Verify, using a table, that the equation 0  x( x2  5)  4 has a root

between the points x = 2 and x = 3.

Use values on the table: 0  x  4 (4)

9.1.3 Hence, make a neat sketch of the graph of the function f (x). (2)

9.1.4 If the positive root of f (x) is estimated as 2,7, use Taylor's/Newton's

method to determine a better approximation of this root. (4)

9.2 Two sides of a rectangle are lengthened at a rate of 3 cm/s while the other two sides
are being shortened in such a way that the figure remains a rectangle with a constant
area of 50 cm2.

9.2.1 Calculate the rate of change of the perimeter of the rectangle when the
length of an increasing side is 7 cm. (5)

9.2.2 Prove that when the rate of change of the perimeter is zero, the figure
must be a square. (2)

9.3 A particle moves in a straight line according to the distance formula

s(t )  t (3  3t  t 2 ).

9.3.1 Calculate the velocity of the particle after 3,5 seconds. (4)

9.3.2 Calculate the acceleration after 2 seconds. (3)

[27]

QUESTION 10:

10.1 Given: f ( x)  2 x 3  4 x 2  8x  8

10.1.1 Determine the coordinates of the turning points of f (x) (2)

10.1.2 Draw up a table of values for x and f(x) with x ranging from x = -2 to x
= 4. (2)

10.1.3 Draw a neat graph of f(x) between the values determined in QUESTION
3.1.2. (2)

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10.1.4 An estimated root of the graph is 0,7.

Use Taylor's/Newton's method to determine a better approximation of this

root. (4)

10.2 An open cardboard box with a square base is required to hold 108 m 3 .

Determine the dimensions that require the least amount of material, and neglect the
thickness of the material and waste in construction.

V  x 2 .h
A  4 xh  x 2
h

x
x (5)

10.3 An electric current is varying through a resistor of 16  . The rate of change of the
current is 0,6 A/s when the current is 12A.

Find the rate of change of power developed with the current at 12A.
HINT: P  I 2 R (4)
[19]

QUESTION 11:
(THIS QUESTION WAS ASKED IN BOTH NOV AND AUG EXAMS OF 2015)

11.1 Given: f ( x)  2 x 3  4 x 2  8x  8

11.1.1 Determine the coordinates of the turning points of f (x) . (2)

11.1.2 Draw up a table of values for x and f(x) with x ranging from x= -2 to x=4. (2)

11.1.3 Draw a neat graph of f(x) between these values. (2)

11.1.4 An estimated root of the graph is 0,7.

Use Taylor's/Newton method to determine a better approximation of this

root. (4)

1.06.2020 Mathematics N5 : Application of Differentiation Page 8 of 16

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 -1-

11.2 An open cardboard box with a square base is required to hold 108 m 3 .
Find the dimensions that require the least amount of material.
Neglect the thickness of the material and waste in construction.

V  x 2 .h
A  4 xh  x 2
h

x
x
(5)
11.3 An electric current is varying through a resistor of 16  . The rate of change of the
current is 0,6 A/s when the current is 12A.

Find the rate of change of power developed with the current 12A.

Hint: P  I 2 R (4)
[19]
QUESTION 12:

12.1 Given: f ( x)  x 3  5x  8

12.1.1 Determine the coordinates of the turning points of f (x) -

Show ALL calculations. (3)

12.1.2 Draw up a table with values 0  x  3 for the equation x 3  5x  8  0 .

Use the table and turning points to make a neat sketch of the function of
f (x) . (3)

12.1.3 Let x  3,7 be an estimated value of the root. Use Taylor/Newton’s

method ONCE to determine a better approximation of THIS root to FOUR
decimal numbers. (4)

12.2 Prove that a rectangular field with a perimeter of 100 m will have the greatest area
when it is a square. (5)

12.3 The distance of a particle is given by f ( x)  10 tan x . f (x) gives the distance in
metres after x seconds.

12.3.1 
Determine the velocity of the particle after seconds.
4 (2)

12.3.2 Determine the acceleration of the particle at this moment. (3)

12.3.3 Determine the distance travelled by the particle in 4 seconds. (2)

[22]
1.06.2020 Mathematics N5 : Application of Differentiation Page 9 of 16
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QUESTION 13:

13.1 Given: f ( x)  2 x 3  4 x 2  2 x  1

13.1.1 Determine the coordinates of the turning points of f (x)

Show ALL the calculations (3)

13.1.2 Draw up a table of values of x and f (x) for values of x from -1 to 2 and
make a neat sketch of this part of the curve. (4)

13.1.3 If the value of the negative root is estimated as -0,3 use Taylor/Newton's
method ONCE to determine a better approximation of THIS root correct to
FOUR decimal numbers. (4)

13.2 A metal ball is covered by a layer of ice. The diameter of the ball is 100 mm. When
the layer of ice is 5 mm thick, it melts at a rate of 10 000 mm3/s.

At what rate is the thickness of the ice decreasing?

4
V   .r 3
3 (5)

13.3 A rectangular wall is to be erected on a river bank. Building cost budgeted for is R5
600-00. The building cost of the two parallel walls are R10-00 /m and the cost of the
wall parallel to the river bank is R15-00 / m (see the sketch).

river

x x

Calculate the dimensions of the wall to enclose a maximum area for the money
available. (5)
[21]

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 -1-

QUESTION 14:

14.1 GIVEN: f ( x)  4 x 3  4 x  1

14.1.1 Determine the co-ordinates of the turning points of f (x). (2)

14.1.2 One root of the equation f ( x)  4 x 3  4 x  1 is close to -1,1.

Use this value and ONE approximate of Newton’s method and determine the
root correct to THREE decimal values. (4)

14.2 A gutter is to be made from a long sheet of metal 300 mm wide, by bending up equal
widths along the edges into vertical positions.

What dimensions will give the largest capacity?

HINT: A = lb and V = lbh (5)

14.3 A rectangular shape window has an equilateral triangle on top.

x x

xxx x
l

14.3.1 If the equation of the circumference is given as p  3x  2l , prove that the

3 2
equation of the area is equal to lx  x (3)
4

14.3.2 If the circumference of the window is 12 m, calculate the dimensions of the

rectangle so that it covers the greatest area of the window. (5)
[19]

1.06.2020 Mathematics N5 : Application of Differentiation Page 11 of 16

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 -3-

MATHEMATICS N5

FORMULA SHEET
Any applicable formula may also be used.

TRIGONOMETRY
cos2x = 1 – sin2x
sin2 x + cos2 x = 1
sin2x = 1 – cos2x

1 + tan2 x = sec2 x tan2x = sec2x - 1

1 + cot2 x = cosec2 x cot2x = cosec2x - 1

sin 2A = 2 sin A cos A

cos 2A = cos2A - sin2A

2 tan A
tan 2A =
1  tan 2 A

sin2 A = ½ - ½ cos 2A = ½(1 - cos2A)

cos2 A = ½ + ½ cos 2A = ½ (1 + cos2A)

sin (A ± B) = sin A cos B ± sin B cos A

cos (A ± B) = cos A cos B  sin A sin B

tan A ± tan B
tan (A ± B) =
1  tan A tan B

sin A cos B = ½ [sin (A + B) + sin (A - B)]

cos A sin B = ½ [sin (A + B) - sin (A - B)]

cos A cos B = ½ [cos (A + B) + cos (A - B)]

sin A sin B = ½ [cos (A - B) - cos (A + B)]

sin x 1 1
tan x = ; sin x = ; cos x =
cos x cosec x sec x

cos x 1 1
cot x  ; cos ecx  ; sec x 
sin x sin x cos x

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 BINOMIAL THEOREM

n(n  1) n  2 2 n(n  1)(n  2) n3 3

( x  h) n  x n  nx n 1 h  x h  x h  ...
2! 1 2  3

DIFFERENTIATION

f (a)
e
f ' (a)

r =a+e

PRODUCT RULE
y = u(x) . v(x)

dy dv du
u. v.
dx dx dx
 u  v'  v  u'

QUOTIENT RULE

u ( x)
y=
v( x)
du dv
v. u.
dy dx dx

dx 2
v

v  u'  u  v'
=
v2

CHAIN RULE
y = f(u(x))
dy dy du
 
dx du dx

1.06.2020 Mathematics N5 : Application of Differentiation Page 13 of 16

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 ______________________________________________________________________
d
f(x)
dx
f (x)  f ( x)dx
______________________________________________________________________
x n 1
xn nx n1  c (𝑛 ≠ −1)
n 1

d n n
axn a x a  x dx
dx

d e axb
e ax+b
eax+b
. (ax  b) C
dx d
(ax  b)
dx

d a dxe
adx+e adx+e.lna (dx  e) C
dx d
ln a. (dx  e)
dx

1 d
ln(ax) . ax xln ax – x + C
ax dx

d
e f(x) e f(x) . f(x) __
dx

d
a f(x) a f(x) . ln a. f(x) __
dx

1 d
ln f(x) . f(x) __
f ( x) dx

cos ax
sin ax a cos ax  .C
a
sin ax
cos ax -a sin ax .C
a
1
tan ax a sec2 ax ln[sec( ax)]  C
a
1
cot ax - a cosec2 ax ln[sin( ax)]  C sec ax a sec ax·tan
a
1
ax ln [sec ax + tan ax] + C
a
1   ax 
ln tan   C
a   2 
cosec ax - a cosec ax·cot ax

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 -1-

_________________________________________________________________
d
f(x) f (x) ∫ f ( x)dx
dx
__________________________________________________________________

sin f(x) cos f(x)•f 1(x) ___

cos f(x) - sin f(x)•f 1(x) ___

tan f(x) sec2 f(x)•f 1(x) ___

cot f(x) - cosec2 f(x)•f 1 (x) ___

sec f(x) sec f(x)•tan f(x)•f 1(x) ___

cosec f(x) - cosec f(x)•cot f(x)•f 1(x) ___

f ' ( x)
sin -1 f ( x) __
1  [ f ( x)]2

- f ' ( x)
cos -1 f ( x) __
1  [ f ( x)]2

f ' ( x)
tan -1 f ( x) __
1  [ f ( x)]2
 f ' ( x)
cot -1 f ( x) __
1  [ f ( x)]2
f ' ( x)
sec -1 f ( x) __
f ( x) [ f ( x)]2  1

 f ' ( x)
cosec -1 f ( x) __
f ( x) [ f ( x)]2  1
𝑥 sin(2𝑎𝑥)
sin2(ax) ____ − +𝐶
2 4𝑎

𝑥 sin(2𝑎𝑥)
cos2(ax) ____ + +𝐶
2 4𝑎

1
tan2(ax) ____ tan(𝑎𝑥) − 𝑥 + 𝐶
𝑎

1
cot2(ax) ____ − 𝑎 cot(𝑎𝑥) − 𝑥 + 𝐶

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INTEGRATION

 f( x) g' ( x) dx  f(x) g(x) -  f '(x) g(x) dx

n [ f ( x)]n1
 [ f ( x)]  f ' ( x)dx  C (n≠ −1)
n 1
f ' ( x)

∫
f ( x)
dx  ln f ( x)  C

APPLICATIONS OF INTEGRATION

AREAS

b b
Ax  ∫ydx ; Ax  ∫( y1  y2 ) dx
a a

b b
Ay   xdy ; Ay   ( x1  x2 ) dy
a a

VOLUMES

b
Vx    y 2dx ; Vx   
a
y12  y22 dxb
a

Vy    x 2dy ; Vy    x12  x22 dy

b b
a a

SECOND MOMENT OF AREA

b b
Ix   r 2dA ; I y   r 2dA
a a

MOMENTS OF INERTIA
Mass = density × volume

M = V

DEFINITION: I = m r2

b b
GENERAL: I   r 2dm    r 2dV
a a

1.06.2020 Mathematics N6 : Differentiation Page 16 of 16

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