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Differential calculus

39 marks in our Paper 1 questions. 12 questions from 2024.

Practise this topic, marked

This topic covers differentiating with the rules, differentiating from first principles, and using derivatives on cubic graphs and in rates of change and optimisation. The 12 questions here are worth 39 marks in the November 2024 Paper 1 set. Of the 39 marks, 24 are for the correct final answer, 12 for method and working and 3 for reading data from diagrams, tables and graphs. The method marks for setting up the derivative are separate from the mark for the final value.

The 2024 Diagnostic Report says candidates handle negative and fractional exponents wrongly before differentiating, and lose first-principles marks mostly through incorrect notation. On cubic graphs they miss that the point of inflection lies halfway between the turning points. In rate questions they set the speed equal to zero, or work out the second derivative and never set it equal to zero. Write what you are setting equal to zero, and why, before you solve.

Where marks are lost: in our words, from Department of Basic Education, NSC 2024 Diagnostic Report, Part 1 (Book 1).

Questions

These questions are not yet checked by a teacher.

The diagram below shows the parabola

f (x) = - x^{2} + 6 x + 7

and the straight line

g (x) = 2 x + 2

. The two graphs cross at A on the x-axis and at C

(5; 12)

. A vertical segment EH joins a point E on

f

to a point H on

g

, somewhere between A and C where the parabola lies above the line. Determine the greatest possible length of EH.

(4)Based on Nov 2024 · Paper 1 · Q6.3

The diagram below shows the parabola

f (x) = - x^{2} + 6 x + 7

and a straight line

g

g

2

and passes through the point

(2; 11)

g (x) = 2 x + 7

k

is obtained by shifting

f

horizontally, so that

k (x) = f (x + m) = - x^{2} - 2 m x - m^{2} + 6 x + 6 m + 7

. Determine the value of

m

g

is a tangent to

k

(5)Based on Nov 2024 · Paper 1 · Q6.4

\frac{d}{d x} [7 x^{2} - 4 x]

(2)Based on Nov 2024 · Paper 1 · Q8.1.1

f ’ (x)

f (x) = \frac{3}{x^{4}} - \sqrt[3]{x^{5}}

(4)Based on Nov 2024 · Paper 1 · Q8.1.2

Find the equation of the tangent to the curve

g (x) = x^{3} - 3 x^{2} + 4 x + 1

at the point where

x = 2

(3)Based on Nov 2024 · Paper 1 · Q8.2

g (x) = - 4 x^{2}

. Use first principles to determine

g ’ (x)

(5)Based on Nov 2024 · Paper 1 · Q8.3.1

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. State the values of

x

g

(2)Based on Nov 2024 · Paper 1 · Q9.1

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. Write down the

x

g ’

, the derivative of

g

(2)Based on Nov 2024 · Paper 1 · Q9.2

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. Determine the values of

x

g

(2)Based on Nov 2024 · Paper 1 · Q9.3

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. Determine all the values of

k

y = g (x) + k

has exactly THREE positive

x

(2)Based on Nov 2024 · Paper 1 · Q9.4

A delivery drone flies from a depot and lands at a warehouse. Its speed, in km/h, after

t

hours of flight is given by

v (t) = - 2 t^{2} + 12 t

. Speed is the rate of change of distance with respect to time. Calculate the greatest speed the drone reaches during this flight.

(3)Based on Nov 2024 · Paper 1 · Q10.1

A runner travels from checkpoint A and stops at checkpoint B. Her speed (in km/h) after

t

hours is given by

s ’ (t) = - 6 t^{2} + 24 t

s (t)

is the distance travelled from A. Note: speed is the rate of change of distance with respect to time, and she starts at A when

t = 0

. Calculate the distance between checkpoint A and checkpoint B.

(5)Based on Nov 2024 · Paper 1 · Q10.2

Updated 12 November 2026