Chapter Review 7: Differentiation

Question 1

Given that \(y = x^{\frac{3}{2}} + \frac{48}{x}, x > 0\)

a) Find the value of \(x\) and the value of \(y\) when \(\frac{dy}{dx} = 0\). (5 marks)

b) Show that the value of \(y\) which you found in part a) is a minimum. (2 marks)

Question 2

A curve has equation \(y = x^3 - 5x^2 + 7x - 14\). Determine, by calculation, the coordinates of the stationary points of the curve.

Question 3

The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that:

\(f'(x) = x^2 - 2 + \frac{1}{x^2}\)

a) Find the value of \(f''(x)\) at \(x = 4\). (4 marks)

b) Prove that f is an increasing function. (3 marks)

Question 6

The diagram shows part of the curve with equation \(y = f(x)\), where:

\(f(x) = 200 - \frac{250}{x} - x^2, x > 0\)

The curve cuts the x-axis at the points A and C. The point B is the maximum point of the curve.

a) Find \(f'(x)\). (3 marks)

b) Use your answer to part a) to calculate the coordinates of B. (4 marks)

Question 7

The diagram shows the part of the curve with equation \(y = 5 - \frac{1}{2}x^2\) for which \(y > 0\).

The point P(x, y) lies on the curve and O is the origin.

a) Show that \(OP^2 = \frac{1}{4}x^4 - 4x^2 + 25\). (3 marks)

b) Find the values of \(x\) for which \(f'(x) = 0\). (4 marks)

c) Hence, or otherwise, find the minimum distance from O to the curve. (4 marks)

Question 8

The diagram shows part of the curve with equation \(y = 3 + 5x + x^2 - x^3\). The curve touches the x-axis at A and crosses the x-axis at C. The points A and B are stationary points on the curve.

a) Show that C has coordinates (3, 0). (1 mark)

b) Using calculus and showing all your working, find the coordinates of A and B. (5 marks)

Question 9

The motion of a damped spring is modelled using this graph.

On a separate graph, sketch the gradient function for this model. Choose suitable labels and units for each axis, and indicate the coordinates of any points where the gradient function crosses the horizontal axis.

Question 10

The volume, \(V \text{ cm}^3\), of a tin of radius \(r \text{ cm}\) is given by the formula \(V = \pi(40r - r^2 - r^3)\). Find the positive value of \(r\) for which \(\frac{dV}{dr} = 0\), and find the value of \(V\) which corresponds to this value of \(r\).

Question 11

The total surface area, \(A \text{ cm}^2\), of a cylinder with a fixed volume of \(1000 \text{ cm}^3\) is given by the formula \(A = 2\pi x^2 + \frac{2000}{x}\), where \(x \text{ cm}\) is the radius. Show that when the rate of change of the area with respect to the radius is zero, \(x^3 = \frac{500}{\pi}\).

Question 12

A wire is bent into the plane shape ABCDE as shown. Shape ABDE is a rectangle and BCD is a semicircle with diameter BD. The area of the region enclosed by the wire is \(R \text{ m}^2\), AE = x metres, and AB = ED = y metres. The total length of the wire is 2 m.

a) Find an expression for y in terms of x. (3 marks)

b) Prove that \(R = \frac{x}{8}(8 - 4x - \pi x)\). (4 marks)

c) Find the maximum value of R. (5 marks)

Question 13

A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1 cm, as shown. The radii of the tin and the lid are both \(x \text{ cm}\). The tin and the lid are made from a thin sheet of metal of area \(80\pi \text{ cm}^2\) and there is no wastage. The volume of the tin is \(V \text{ cm}^3\).

a) Show that \(V = \pi(40x - x^2 - x^3)\). (5 marks)

b) Use differentiation to find the positive value of x for which V is stationary. (3 marks)

c) Prove that this value of x gives a maximum value of V. (2 marks)

d) Find this maximum value of V. (1 mark)

e) Determine the percentage of the sheet metal used in the lid when V is a maximum. (2 marks)