The vertical profile of a sculpted roller coaster rail is modeled by the function $h(x) = (x - 4)(2x + 5)^2$ for $\ge -3$ , where $h$ is the height in decimetres and $x$ is the horizontal distance from a sensor.

The rail touches the baseline at point $P$ and crosses the baseline at point $Q$ .

State the coordinates of the point $P$ .

Determine $h^{'} (x)$ .

Hence show that the equation of the tangent to the rail at the point where $\frac{11}{6}$ can be expressed in the form $y = k$ , where $k$ is a constant to be found.

A modification shifts the track horizontally so the equation becomes $y = h (x + b)$ , where $b$ is a constant. The modified track now passes through the sensor's origin $O (0, 0)$ .

State the possible values of $b$ .

The rail touches the baseline at point $P$ and crosses the baseline at point $Q$ .

State the coordinates of the point $P$ .

Determine $h^{'} (x)$ .

Hence show that the equation of the tangent to the rail at the point where $\frac{11}{6}$ can be expressed in the form $y = k$ , where $k$ is a constant to be found.

A modification shifts the track horizontally so the equation becomes $y = h (x + b)$ , where $b$ is a constant. The modified track now passes through the sensor's origin $O (0, 0)$ .

State the possible values of $b$ .

3.5.2 Differentiation (A-level only)

Question 81

Markscheme

3.5.2 Differentiation (A-level only)

Question 81

Markscheme

3.5.2 Differentiation (A-level only)

Question 81

3.5.2 Differentiation (A-level only) Questions

3.5.2 Differentiation (A-level only)

Question 81

3.5.2 Differentiation (A-level only) Questions