Derivatives: exam-style practice Problem set A

A language app records its users (thousands) x months after launch. The data are in the table. A quadratic model f(x) = ax² + bx + c is fitted to the data.

1. (a) Use quadratic regression to find a, b and c to 3 significant figures.
2. (b) The coefficient of determination is R² = 0.987 (3 s.f.). State whether R² supports a quadratic model and justify your answer.
3. (c) Comment on the validity of the model with reference to the value of c.

A packing crate has a square base of side l m. Its height d m is 1 m less than twice the side of the base. The crate holds 112 m³.

1. (a) Write down an expression for d in terms of l.
2. (b) Find l and d.
3. (c) Find the total external surface area of the crate.

A paint tin is a closed cylinder of radius r cm and height h cm. It holds 350 cm³ of paint. Let A cm² be the total surface area of the tin.

1. (a) Find an expression for h in terms of r.
2. (b) Show that \(A=2\pi r^2+\frac{700}{r}\).

A soup can is a closed cylinder that holds 600 cm³. Its total surface area is \(A=2\pi r^2+\frac{1200}{r}\) cm², where r cm is the radius, r > 0.

1. (a) Find \(\frac{dA}{dr}\).
2. (b) Find the value of r that minimises A.
3. (c) Find the minimum value of A.

An open tool box has a square base of side x m and volume 32 m³. Its surface area is \(A=x^2+\frac{128}{x}\) m², where x > 0.

1. (a) Find \(\frac{d^2A}{dx^2}\).
2. (b) Show that the graph of A is concave up for all x > 0.
3. (c) Given that dA/dx = 0 when x = 4, state what the answer to part (b) tells you about this value of x.

Consider the curve \(y=2x(5-e^{3x})\).

1. (a) Find \(\frac{dy}{dx}\).
2. (b) Find \(\frac{d^2y}{dx^2}\).
3. (c) The curve has a point of inflexion at x = a. Find a.

A conservation group counts a colony of seals (hundreds) every 10 years. Let x be the number of years after 1960. The data are in the table. A quadratic model f(x) = ax² + bx + c is fitted.

1. (a) Use quadratic regression to find a, b and c to 3 significant figures.
2. (b) Find f′(60) and interpret your answer in context.
3. (c) Use the change in the table between x = 50 and x = 60 to comment on the validity of the model.

The number of users of a new website, P thousand, after n weeks is modelled by \(P=\frac{15}{1+4e^{-0.3n}}\).

1. (a) Find \(\frac{dP}{dn}\) and explain why the model predicts that the number of users is always increasing.
2. (b) Find the initial number of users and the initial rate of growth.
3. (c) Find the value of P as n becomes very large.
4. (d) Hence sketch P against n.

A juice company packs 640 cm³ of juice in a closed cylinder of radius r cm and height h cm. Let A cm² be its total surface area.

1. (a) Find an expression for h in terms of r.
2. (b) Show that \(A=2\pi r^2+\frac{1280}{r}\).
3. (c) Find \(\frac{dA}{dr}\).
4. (d) Find the value of r that minimises A, and the minimum value of A.
5. (e) A cube-shaped box holds the same 640 cm³. Find its total surface area and say whether the cylinder or the cube needs less material.

Consider the curve \(y=x^{2x}\) for x > 0.

1. (a) Show that \(x^{2x}=e^{2x\ln x}\).
2. (b) Hence find the equation of the tangent to the curve at x = 1.
3. (c) Find the exact value of x at which \(\frac{dy}{dx}=0\).