Pure Mathematics Practice: Differentiation

Original practice material

These four structured questions total 27 marks. They practise exact differentiation, curve geometry and modelling suitable for Paper 1 or Paper 2 Section A; they are not a complete paper.

Answers: Differentiation Answers.

Question 1: Product and Chain Rules

Let

$$y=(x^2+1{)}^3\sin x.$$

(a) Find $dy/dx$. [3]

(b) Hence find the gradient of the curve at $x=0$. [1]

[Total: 4]

Question 2: Tangent and Normal

The curve $C$ has equation

$$y=x^3-3x+4.$$

(a) Find the equation of the tangent to $C$ at the point where $x=2$. [4]

(b) Find the equation of the normal to $C$ at the same point. [3]

[Total: 7]

Question 3: Implicit and Parametric Differentiation

(a) Given

$$x^2+xy+y^2=7,$$

find $dy/dx$ in terms of $x$ and $y$. [4]

(b) A curve is given parametrically by

$$x=t^2+1,y=t^3-t.$$

Find $dy/dx$ when $t=2$. [3]

[Total: 7]

Question 4: Optimisation and Connected Rates

A rectangle changes shape while its perimeter remains fixed at $40$ cm. At time $t$ seconds, its width is $x$ cm and its area is $A$ cm${}^2$.

(a) Show that

$$A=20x-x^2.$$

[2]

(b) Find the maximum possible area and justify that it is a maximum. [4]

(c) At an instant when $x=6$, the width is increasing at $0.2$ cm s${}^{-1}$. Find $dA/dt$ at this instant and interpret its sign. [3]

[Total: 9]