Question Bank

Differentiate $y = \sqrt{x} + \tan(2x)$.

Find the gradient of the tangent to the curve $y = \dfrac{e^{2x}}{x + 2}$ at the point where $x = 0$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

A curve is defined parametrically by the equations $x = \sqrt{t + 1}$ and $y = \sin 2t$.

Find the gradient of the tangent to the curve at the point when $t = 0$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Find the values of $a$ and $b$ such that the curve $y = \dfrac{ax-b}{x^2-1}$ has a turning point at $(3,1)$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Differentiate $y = 2(x^2 - 4x)^5$.

*You do not need to simplify your answer.*

The percentage of seeds germinating depends on the amount of water applied to the seedbed
that the seeds are sown in, and may be modelled by the function:

$P(w) = 96\ln(w + 1.25) - 16w - 12$

where $P$ is the percentage of seeds that germinate and
$w$ is the daily amount of water applied (litres per square metre of seedbed), with $0 \le w \le 15$.

Find the amount of water that should be applied daily to maximise the percentage of seeds
germinating.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Differentiate $y = x\ln(3x - 1)$.

*You do not need to simplify your answer.*

Find the gradient of the curve $y = \frac{1}{x} - \frac{1}{x^2}$ at the point $\left(2, \frac{1}{4}\right)$.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*