C 2 Chapter 9 Differentiation Dr J Frost

C 2 Chapter 9 Differentiation Dr J Frost (jfrost@tiffin. kingston. sch. uk) Last modified: 5 th October 2013

Increasing Functions How could we use differentiation to tell us if this is a strictly increasing function? . . . if the gradient is always ? positive, i. e. f’(x) > 0 for all x. f(x 2) f(x 1) x 1 x 2 A function is increasing if for any two values of x, x 1 and x 2 where x 2 > x 1, then f(x 2) ≥ f(x 1) A function is strictly increasing if f(x 2) > f(x 1)

Example Exam Question Edexcel C 2 June 2010 ? ?

Showing a function is increasing/decreasing Show that f(x) = x 3 + 24 x + 3 (x ϵ ℝ) is an increasing function. ?

Increasing/Decreasing in an Interval This is a decreasing function in the interval (a, b) i. e. where a < x < b a b 1 Find the values of x for which the function f(x) = x 3 + 3 x 2 – 9 x is a decreasing function. 3 x 2 + 6 x – 9 < 0 Thus -3 < x < 1? 2 Find the values of x for which the function f(x) = x + (25/x) is a decreasing function. 1 – (25/x 2) < 0 Thus -5 < x < 5?

Questions C 2 pg 130 Exercise 9 A

Stationary Points Features you’ve previously used to sketch graphs? f’(x) = 0 Maximum point Stationary points are those for which f’(x) = 0 Minimum point f’(x) = 0 Maximum/minimum points are known as ‘turning points’.

Finding turning points Edexcel C 2 May 2013 (Retracted) (2, 9) ? Although it might be interest to know if this is a minimum point or a maximum point. . .

Do we have a minimum or maximum point? Method 1: Consider the points immediately before and after the stationary point. ? Method 2: Use the second-order derivative to see whether the gradient is ? increasing or decreasing.

Do we have a minimum or maximum point? Find the coordinates of the turning point on the curve with equation y = x 4 – 32 x. Determine whether this is a minimum or maximum point. Method 1 Method 2 Value of x x < 2 e. g. x = 1. 9 x = 2 x > 2 e. g. x = 2. 1 Gradient e. g. -4. 56 ? ? 0 e. g. 5. 04 ? ? Shape We can see from this shape that this is a minimum point. ? ? What are the advantages of each method? ? (2, -48)

Points of inflection Stationary point of inflection (“saddle point”) Non-stationary point of inflection A point of inflection is a point where the curve changes from concave to convex (or vice versa).

Stationary Points of inflection So how can we tell if a stationary point is a point of inflection?

Non-Stationary Points of inflection (not in the A Level syllabus) At this point: > 0 (i. e. not stationary) ? = 0 (i. e. gradient is not ? changing at this point)

Stationary Point Summary d 2 y / dx 2 Type of Point < 0 Maximum > 0 Minimum = 0 Could be maximum, minimum or point of inflection. Use ‘Method 1’ to find gradient just before and after. y = x 4 has a turning point at x = 0. Show that this is a minimum point. dy/dx = 4 x 3. d 2 y/dx 2 = 12 x 2 When x = 0, d 2 y/dx 2 = 0, so we can’t classify immediately. ? When x = -0. 1, dy/dx = -0. 004. When x = +0. 1, dy/dx = +0. 004. Gradient goes from negative to positive, so minimum point.

Further Examples Find the stationary points of y = 2 x 3 – 15 x 2 + 24 x + 6 and determine which of the points are maximum/minimum/points of inflection. (1, 17) is a maximum point. (4, -10) is a maximum point. ? State the range of outputs of 6 x – x 2 ?

Exercise 9 B

Dr Frost Differentiation – Practical applications Objectives: Use differentiation in real-life problems that involve optimisation of some variable.

Optimisation Problems We have a sheet of A 4 paper, which we want to fold into a cuboid. What height should we choose for the cuboid to maximise the volume? These are examples of optimisation problems: we’re trying to maximise/minimise some quantity by choosing an appropriate value of a variable that we can control. x y We have 50 m of fencing, and want to make a bear pen of the following shape, but that maximises the area. What should we choose x and y to be?

Breaking down optimisation problems Suppose that we have 100 cm of rope, that we want to put in the shape of a minor segment. We want to choose a radius for this minor segment that maximises the area covered by the rope. What radius do we choose? M N r cm Strategy 1. Form two equations: one representing the thing we’re trying to maximise (here the area) and the other representing the constraint (here the length of rope) Typically we’d need to write out two equations (e. g. perimeter and area, or volume and area) and combine them together, using given information, to form the one equation we’d need. O

Example Exam Question Edexcel C 2 May 2011

Example 1 M N Suppose that we have 100 cm of rope, that we want to put in the shape of a minor segment. We want to choose a radius for this minor segment that maximises the area covered by the rope. What radius do we choose? a) Show that A = 50 r – r 2 r cm O Given that r varies, find: b) The value of r for which A is a maximum and show that A is a maximum. c) Find the value of angle MON for this maximum area. d) Find the maximum area of the sector OMN.

Example 2