Applications: Differentiation test questions - Higher Maths Revision

1

Find the rate of change of \(y = 4\sqrt x + 6x\) at \(x = 16\)

\[112\]

\[14\]

\[\frac{{13}}{2}\]

2

Find the rate of change of \(y = \frac{{{x^3} - x}}{{{x^2}}}\) when x = - 2

\[- \frac{3}{2}\]

\[\frac{5}{4}\]

\[- \frac{{11}}{4}\]

3

Find the rate of change of \(f(x) = \frac{3}{{\sqrt x }}\) at x = 9.

\[\frac{{ - 1}}{{18}}\]

\[1\]

\[\frac{9}{2}\]

4

An open water tank in the shape of a triangular prism has a capacity of 108 litres. The tank is to be lined on the inside in order to make it watertight.

The triangular cross-section of the tank is right-angled and isoceles with equal sides of length x cm.

The tank has a length of l cm.

Find an expression for the surface area of the tank in terms of x.

An open water tank in the shape of a triangular prism, with length l and height x

\[A(x) = \frac{1}{2}{x^2} + \frac{{216000}}{x}\]

\[A(x) = {x^2} + \frac{{432}}{x}\]

\[A(x) = {x^2} + \frac{{432000}}{x}\]

5

A goldsmith has built up a solid which consists of a triangular prism of fixed volume with a regular tetrahedron at each end.

The surface area, A, of the solid is given by:

\[A(x) = \frac{{3\sqrt 3 }}{2}\left( {{x^2} + \frac{{16}}{x}} \right)\]

where x is the length of each edge of the tetrahedron.

Find the value of x which the goldsmith should use to minimise the amount of gold plating required to cover the solid.

A triangular prism with a tetrahedron at either end

- 2

2

3