Application of Derivatives

Application of Derivatives shows how the derivative, which you learnt to compute in the previous chapter, becomes a powerful tool for solving real problems in science, engineering and economics. In this chapter you will interpret dy/dx as the rate of change of one quantity with respect to another, for example how fast the area of a ripple grows as its radius increases, and use related rates through the chain rule. You will apply derivatives in economics to find marginal cost and marginal revenue. You will learn to find the intervals in which a function is increasing or decreasing from the sign of f'(x). The main part of the chapter deals with maxima and minima: critical points, the first derivative test, the second derivative test, and absolute maximum and minimum values of a continuous function on a closed interval. You will then solve optimisation problems of the kind set in board exams, such as finding the dimensions of a box of maximum volume, a field of maximum area or a cylinder of minimum surface area.

Detailed Worksheet: Application of Derivatives Section A - Definitions (10 marks) 1. Define a strictly increasing function on an interval. State the condition on the derivative for f to be strictly increasing on an open interval. (2 marks) 2. What is a critical point of a function? Find the critical points of f(x) = x^3 - 3x. (2 marks) 3. State the second derivative test for local maxima and minima. (2 marks) 4. Define marginal revenue. If R(x) = 3x^2 + 36x + 5, find the marginal revenue when x = 5. (2 marks) 5. Show that f(x) = e^(2x) is strictly increasing on R. (2 marks) Section B - Calculations and Applications (15 marks) 6. The radius of a circle is increasing at the rate of 0.7 cm/s. Find the rate of increase of its circumference. (3 marks) 7. A stone dropped into a still lake produces circular ripples. The radius of the outer ripple increases at 3 cm/s. How fast is the enclosed area increasing when the radius is 5 cm? (3 marks) 8. Find the intervals in which f(x) = 2x^3 - 3x^2 - 36x + 7 is strictly increasing and strictly decreasing. (3 marks) 9. Find the local maximum and local minimum values of f(x) = x^3 - 6x^2 + 9x + 15 using the second derivative test. (3 marks) 10. Find the absolute maximum and absolute minimum values of f(x) = 2x^3 - 15x^2 + 36x + 1 on the interval [1, 5]. (3 marks) Section C - Diagrams (10 marks) 11. Sketch the graph of f(x) = x^3 - 6x^2 + 9x + 15. Mark the local maximum, local minimum and the intervals where the function is increasing and decreasing. (4 marks) 12. Draw a diagram of a square sheet of side 18 cm from which equal squares of side x are cut from each corner and the sides folded up to form an open box. Write the volume V(x) in terms of x. (3 marks) 13. Draw a ladder of length 5 m leaning against a vertical wall, with its foot at distance x and top at height y. Write the relation between x and y and between their rates of change. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A square piece of tin of side 18 cm is to be made into a box without a top by cutting a square from each corner and folding up the flaps. Find the side of the square to be cut off so that the volume of the box is maximum, and find the maximum volume. (5 marks) 15. Show that of all the rectangles inscribed in a given fixed circle of radius r, the square has the maximum area. (5 marks) 16. Show that the right circular cylinder of given total surface area and maximum volume is such that its height is equal to the diameter of the base. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all steps of differentiation and state the test used for maxima and minima.