Question Bank

Differentiate $y = e^{3x}\sin 2x$.

*You do not need to simplify your answer.*

A curve has the equation $y = (2x + 3)e^{x^2}$.

Find the $x$-coordinate(s) of any stationary point(s) on the curve.

*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 $x = t^2 + 3t$ and $y = t^2 \ln(2t - 3)$, for $t > \dfrac{3}{2}$.

Find the gradient of the tangent to the curve at the point $(10,0)$.

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

Differentiate $f(x) = (1 - x^2)^5$.

*You do not need to simplify your answer.*

A curve has the equation $y = \dfrac{x^2}{x+1}$.

Find the $x$-coordinate(s) of any stationary point(s) on the curve.

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

A curve has the equation $y = (x^2 + 3x + 2)\cos 3x$.

Find the equation of the normal to the curve at the point where the curve crosses the $y$-axis.

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

The volume of a spherical balloon is increasing at a constant rate of $60\ \mathrm{cm}^3$ per second.

Find the rate of increase of the radius when the radius is $15\ \mathrm{cm}$.

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

Differentiate $y = \dfrac{\cot x}{x^2 + 1}$.

You do not need to simplify your answer.

The graph of the function $y = 4\sqrt{x} - x + 2$, where $x > 0$, has a stationary point at point Q.

Find the coordinates of point Q.

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