Question Bank

Differentiate $y = \tan(x^2 + 1)$.

*You do not need to simplify your answer.*

Find the gradient of the tangent to the function $f(x)=\ln(3x-e^x)$ at the point where $x=0$.

Find the $x$ values of any points of inflection on the graph of the function $y = e^{(6 - x^2)}$.

Show any derivatives that you need to find when solving this problem.

A curve is defined by the parametric equations:

$x = 5\sin t$ and $y = 3\tan t$

Find the gradient of the normal to the curve at the point where $t = \dfrac{\pi}{3}$.

*Show any derivatives that you need to find when solving this problem.*

A closed cylindrical tank is to have a surface area of $20\ \mathrm{m}^2$.

Find the radius the tank needs to have so that the volume it can hold is as large as possible.

*You do not have to prove that your solution gives the maximum volume.*

*Show any derivatives that you need to find when solving this problem.*

\nDifferentiate $y = \sqrt[3]{\pi - x^2}$.\n\n*You do not need to simplify your answer.*

A curve has the equation $y = (x^3 - 2x)^3$.

Find the equation of the tangent to the curve at the point where $x = 1$.

*Show any derivatives that you need to find when solving this problem.*

For what value of $k$ does the function $f(x)=x-e^x-\frac{k}{x}$ have a stationary point at $x=-1$?

Show any derivatives that you need to find when solving this problem.

Differentiate $y = \dfrac{\sin(2x)}{x^2}$.

*You do not need to simplify your answer.*

For the function $f(x)=x+\dfrac{16}{x-2}$, find the $x$-values of any stationary points.

*You must use calculus and clearly show your working, including any derivatives you need to find when solving this problem.*