Derivatives: exam-style practice Problem set B

A museum records the number of visitors (thousands per year) x years after 1990. 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 = 0.99996 (5 d.p.). 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 a.

A water tank has a square base of side w m. Its height h m is 4 m more than the side of the base. The tank holds 63 m³ of water when full.

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

A biscuit tin is a closed cylinder of radius r cm and height h cm. It holds 280 cm³. 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 \(A=2\pi r^2+\frac{560}{r}\).

A pot of face cream is a closed cylinder that holds 360 cm³. Its total surface area is \(A=2\pi r^2+\frac{720}{r}\) 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 fish tank has a square base of side x dm and volume 108 dm³. Its glass area is \(A=x^2+\frac{432}{x}\) dm², 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 = 6, state what the answer to part (b) tells you about this value of x.

Consider the curve \(y=3x(1-e^{4x})\).

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 cycling club records its number of members every 5 years. Let x be the number of years after 1995. 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′(30) and interpret your answer in context.
3. (c) Use the change in the table between x = 25 and x = 30 to comment on the validity of the model.

The number of birds on an island, P hundred, n years after they arrive is modelled by \(P=\frac{18}{1+8e^{-0.5n}}\).

1. (a) Find \(\frac{dP}{dn}\) and explain why the model predicts that the number of birds is always increasing.
2. (b) Find the initial number of birds 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 pet-food company packs 480 cm³ of food 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 \(A=2\pi r^2+\frac{960}{r}\).
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 480 cm³. Find its total surface area and say whether the cylinder or the cube needs less material.

Consider the curve \(y=x^{4x}\) for x > 0.

1. (a) Show that \(x^{4x}=e^{4x\ln x}\).
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 \(\frac{dy}{dx}=0\).