Differentiation: The Rules and How Exams Use Them — Worksheet & Memo

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 12 · Calculus

Differentiation: The Rules and How Exams Use Them

EXAM-STYLE CLASS TEST

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 35

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
If y = 4x⁵ − 6x⁴ + 3x, then dydx = …
(1)
A)20x⁴ − 24x³ + 3
B)20x⁴ − 24x³ + 3x
C)20x⁵ − 24x⁴ + 3
D)4x⁴ − 6x³ + 3
1.2
The derivative of a CONSTANT is …
(1)
A)0
B)1
C)the constant itself
D)undefined
1.3
If f(x) = √x, then f'(x) = …
(1)
A)12√x
B)1√x
C)2√x
D)12x
1.4
If y = 3x, then dydx = …
(1)
A)−3x²
B)3x²
C)−3x
D)−3x
1.5
Before differentiating (3x + 2)(x − 1), you should …
(1)
A)expand the brackets first
B)differentiate each bracket and multiply the results
C)differentiate the first bracket only
D)take logarithms
1.6
If f(x) = x³ − 3x, then f'(2) = …
(1)
A)9
B)2
C)12
D)6
1.7
D_x notation means …
(1)
A)differentiate with respect to x
B)the value of the function at x
C)the domain of the function
D)the discriminant
1.8
The gradient of the TANGENT to y = x² at x = 3 is …
(1)
A)6
B)9
C)3
D)2
1.9
If f'(a) = 0, the graph of f has …
(1)
A)a stationary point at x = a
B)an x-intercept at x = a
C)a point of inflection at x = a
D)an asymptote at x = a
1.10
If y = (2x)³, then dydx = …
(1)
A)24x²
B)6x²
C)8x²
D)3(2x)²

Answer grid — circle your answers for Question 1

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[13 MARKS]

Answer the questions below. Show ALL your working.

2.1
Determine f′(x) from FIRST PRINCIPLES if f(x) = 2x² − 3x.
(5)
2.2
Determine dydx if y = 4x⁵ − 6x⁴ + 3x.
(3)
2.3
Determine D_x [ (3x2 − 13x)² ].
(5)

Question 3

[12 MARKS]

The graph of h(x) = ax³ + bx² has turning points at the origin O(0 ; 0) and at B(4 ; 32). A is the other x-intercept of h.

3.1
Show that a = −1 and b = 6.
(5)
3.2
Calculate the coordinates of A.
(3)
3.3
Write down the values of x for which h is INCREASING.
(2)
3.4
Write down the values of x for which h is CONCAVE DOWN.
(2)

TOTAL: 35 marks

This question paper consists of 3 questions.