Edexcel IGCSE Math A 3.4 calculus

Use this calculus page to differentiate curves, find gradients or stationary points and connect displacement, velocity and acceleration in motion questions.

Exam points

Differentiate polynomial curves to find gradients at stated points or values.
Solve derivative equations to locate stationary points or tangents with a given gradient.
Apply differentiation to displacement-time formulae to find velocity or acceleration.

Covered syllabus points

3.4.HA Variable rates of change 3.4.HB Differentiate integer powers of x 3.4.HC Gradients, rates of change, stationary points, turning 3.4.HD Distinguish between maxima and minima by considering 3.4.HE Calculus in kinematics and practical problems

[Maximum number: 6]

The curve C has equation $y=5 x^{3}-x^{2}-6 x+4$

(a)

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$

\frac{\mathrm{d} y}{\mathrm{~d} x}=

There are two points on the curve C at which the gradient of the curve is 2

[ 2 ]

Final answer:
$\frac{dy}{dx}=15x^{2}-2x-6$

B2 for correct differentiation.
B1 for two of $15x^{2}$ , -2x, and -6 correct.

(b)

Find the x coordinate of each of these two points.

Show clear algebraic working.

[ 4 ]

Set the gradient equal to 2:
$15x^{2}-2x-6=2$ $15x^{2}-2x-8=0$

Factorise:
(3x+2)(5x-4)=0

Final answers:
$x=-\frac23,\qquad x=\frac45$

Allow -0.66 or better for $-\frac23$ .

[Maximum number: 4]

A particle is moving along a straight line that passes through the fixed point O The displacement, s metres, of the particle from O at time t seconds is given by

s=2 t^{3}-5 t^{2}+6 t-5

Find the value of t when the acceleration of the particle is $5 \mathrm{~m} / \mathrm{s}^{2}$

s=2t^3-5t^2+6t-5

]

v=\frac{ds}{dt}=6t^2-10t+6

]

a=\frac{dv}{dt}=12t-10

]

12t-10=5

]

12t=15

]

Final answer:

t=1.25

]

M1 for $\frac{ds}{dt}=6t^2-10t+6$ , with at least 2 terms correct.
M1ft for $\frac{dv}{dt}=12t-10$ , following from a 3-term quadratic.
M1ft for setting 12t-10=5, dependent on the previous M1.
A1 oe for 1.25.

All question bank results loaded

Covered syllabus points

3.4 Calculus question 1

Question (a)

Question (b)

3.4 Calculus question 2