Question Bank

The graph shows the curve $y = \dfrac{2}{(x+1)^3}$ , along with the tangent to the curve drawn at $x = 1$.

A second tangent to this curve is drawn which is parallel to the first tangent shown in the diagram.

Find the $x$-coordinate of the point where this second tangent touches the curve.

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

The diagram below shows a tangent passing through the point P $(p,q)$ which lies on the circle
with parametric equations $x = 4\cos\theta$ and $y = 4\sin\theta$.

Show that the equation of the tangent line is $px + qy = p^2 + q^2$.

The graph of $y = x(x - 2m)^2$, where $m > 0$, is shown.

The total shaded area between the curve and the $x$-axis
from $x = 0$ to $x = 2m$ is given by $A = \dfrac{4m^4}{3}$.

A right-angled triangle is now constructed with one
vertex at $(0,0)$ and another on the curve $y = x(x - 2m)^2$,
as shown below.

Show that the maximum area of such a triangle is $\dfrac{3}{8}$ of the total shaded area.

*You must use calculus and show any derivatives that you need to find when solving this problem.*
*You do not have to prove that the area you have found is a maximum.*

A police helicopter is flying above a straight horizontal section of motorway chasing a speeding
car.

The helicopter is flying at a constant speed of $72\ \mathrm{m\ s^{-1}}$ and at a constant height of $400$ metres
above the ground. The helicopter is attempting to catch up with the car.

When the direct distance from the helicopter to the car is $2500$ metres, the angle of depression, $\theta$,
between the horizontal and the line of sight from the helicopter to the car is increasing at a rate of
$0.002\ \mathrm{rad\ s^{-1}}$.

Calculate the speed of the car at this instant.

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

The graph below shows the function $y = f(x)$.

For the function above:
(i) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.

(ii) Find the value(s) of $x$ where $f'(x) = 0$ and $f''(x) < 0$ are both true.

(iii) What is the value of $\lim_{x \to 6} f(x)$?

State clearly if the value does not exist.