SAJC Chapter 6 Apps of Differentiation (Tutor)_2022

SAJC 2022 JC 2 H1 Mathematics Page 1 of 39 Chapter 6 (Pure Mathematics) Applications of Differentiation Objectives At the end of the chapter, you should be able to: (a) interpret f′(x) > 0, f′(x) = 0 and f′(x) < 0 graphically; (b) Determine the maximum and minimum points (local maxima a nd minima) and stationary points of inflexion analytically, in simple cases, using the first derivative test; [Note: The use of the second derivative test is not required.] (c) locate the maximum and minimum points with the help of a graphic calculator; (d) find and use the equations of tangents to a curve; (e) interpret the derivative as an instantaneous rate of change of a physical quantity, and use the concept to solve a variety of maxima and minima problems and problems involving connected rates of change. Content 6.1 Some Notation and Terminology 6.1.1 Gradient of a straight line 6.1.2 Equation of a straight line 6.1.3 Gradient of perpendicular lines 6.1.4 Gradient of a curve 6.2 Equation of Tangents to a curve 6.3 Increasing and Decreasing Functions 6.4 Stationary Points 6.4.1 Tests for types of Stationary Point 6.5 Maxima and Minima Problems 6.6 Rate of Change References New Syllabus Additional Mathematics (8th Edition), Shinglee Publishers Pte Ltd. Relevant Resources • http://mathinsite.bmth.ac.uk/applet/difffns/difffns.html (Applet to explore graphically the differentiation (1st and 2nd) of various functions)

SAJC 2022 JC 2 H1 Mathematics Page 2 of 39 Introduction There are many applications of differentiation in science and engineering. Differentiation is also used in analysis of finance and economics. One important application of differentiation is in the area of optimisation, which means finding the condition for a maximum (or minimum) to occur. This is important in business e.g cost reduction, profit increase and engineering e.g maximum strength, minimum cost. Example : Solar Two sustainable energy project in California A power tower produces electricity from sunlight by focusing thousand s of sun -tracking mirrors, called heliostats, on a single receiver sitting on top of a tower. The receiver captures the thermal energy of the sun and stores it in tanks of molten salt (to the right of the tower) at temperatures greater than 500 degrees centigrade. When electricity is needed, the energy in the molten salt is used to create steam, which drives a conventional electricity-generating turbine (to the left of the tower). Differentiation is used to maximise the efficiency of the process. Source: http://www.intmath.com/Calculus/Calculus-intro.php 6.1 Some Notation and Terminology y 2y 1y 1x 2x x 22( , )Q x y 11( , )P x y 0

SAJC 2022 JC 2 H1 Mathematics Page 3 of 39 6.1.1 Gradient of a straight line The gradient of a straight line from point 11( , )P x y to 22(