Edexcel A-Level Maths: Differentiation

A manufacturer makes an open-topped cylindrical container (a circular base and a curved side, but no lid) from thin sheet metal. The container must hold a fixed volume of $125\pi\ \text{cm}^3$125π cm3.

The container has base radius $r\ \text{cm}$r cm and height $h\ \text{cm}$h cm. The manufacturer wants to use as little metal as possible, i.e. to minimise the total surface area $S\ \text{cm}^2$S cm2 of the base and curved side.

(a) Using the fixed-volume condition to eliminate $h$h, show that the surface area is given by $S = \pi r^2 + \frac{250\pi}{r}.$S=πr2+r250π​. (4 marks)

(b) Find $\dfrac{dS}{dr}$drdS​ and hence determine the value of $r$r for which $S$S is stationary. (5 marks)

(c) Justify that this value of $r$r gives a minimum surface area, and find that minimum area, giving your answer as an exact multiple of $\pi$π. (3 marks)

A curve $C$C is defined by the parametric equations $x = t^2, \qquad y = t^3 - 3t, \qquad t \in \mathbb{R}.$x=t2,y=t3−3t,t∈R.

(a) Find $\dfrac{dy}{dx}$dxdy​ in terms of $t$t, simplifying your answer. (4 marks)

(b) The point $P$P on $C$C has parameter $t = 2$t=2. Find an equation of the tangent to $C$C at $P$P, giving your answer in the form $y = mx + c$y=mx+c. (3 marks)

(c) Find the coordinates of the points on $C$C at which the tangent is parallel to the $x$x-axis. (3 marks)

A curve has equation $y = x^2 \ln x$y=x2lnx, where $x > 0$x>0.

(a) Use the product rule to find $\dfrac{dy}{dx}$dxdy​, giving your answer in a fully factorised form. (3 marks)

(b) Hence find the exact coordinates of the stationary point of the curve, giving each coordinate in terms of $e$e. (5 marks)

The function $f$f is defined by $f(x) = 2x^2 + 3x$f(x)=2x2+3x.

Using differentiation from first principles, and showing each step of your limiting argument, prove that $f'(x) = 4x + 3.$f′(x)=4x+3.

(6 marks)

A spherical balloon is being inflated. Air is pumped in at a constant rate of $50\ \text{cm}^3\,\text{s}^{-1}$50 cm3s−1.

The volume of a sphere of radius $r$r is $V = \dfrac{4}{3}\pi r^3$V=34​πr3.

Find the rate at which the radius of the balloon is increasing at the instant when $r = 5\ \text{cm}$r=5 cm, giving your answer as an exact value in $\text{cm}\,\text{s}^{-1}$cms−1.

(5 marks)

A curve has equation $y = \frac{4}{x^2} + 3\sqrt{x}, \qquad x > 0.$y=x24​+3x​,x>0.

Find the exact value of the gradient of the curve at the point where $x = 4$x=4.

(4 marks)