Differentiation

Find the gradient of the curve at $P$.

Find the coordinates of $M$. Give each coordinate correct to 3 significant figures.

A point $P$ is moving along the curve in such a way that its $y$-coordinate is decreasing at 5 units per second.

Find the rate at which the $x$-coordinate of point $P$ is changing when $x = 2$.

Find the coordinates of the stationary points of the curve and determine their nature.

Find the rate at which the $y$-coordinate of $P$ is changing when $x = 9$. Give your answer in terms of the constant $a$.

Given that the curve has a minimum point when $x = \frac{1}{4}$, find the value of $a$.

Find the set of values of $x$ for which $y$ decreases as $x$ increases.

It is given that the maximum point of the curve has $y$-coordinate 27.

Find the equation of the curve.

Find the values of the constants $a$ and $b$.

Find the coordinates of the other stationary point.

A point $P$ is moving along part of the curve in such a way that the $y$-coordinate of $P$ is increasing at a constant rate of 6 units per second.

Find the rate at which the $x$-coordinate of $P$ is increasing when $x = 5$.

Find the value of $k$. Give your answer correct to 6 decimal places.

The table below shows the gradients of the chords $AB$ and $AC$, given correct to 4 decimal places.

Find the gradient of the chord $AD$. Give your answer correct to 4 decimal places.

State what the values in the table suggest about the value of $f'(8)$.

Find the coordinates of the point at which the tangent to the curve at the point $(3, 5)$ intersects the line $y = -8x$.

Find the $x$-coordinates of each of the stationary points of the curve.

Find $\frac{d^2y}{dx^2}$ and hence determine the nature of each of the stationary points.

Show that the volume, $V\text{ cm}^3$, of the tank is given by

Find the exact value of $r$ which corresponds to the maximum value of $V$.

Hence, find the maximum value of $V$.