5. Higher Order Derivatives & Graphing the Derivative Function f (x) f’ (x) Independent Practice: Oxford Text: Exercises p221: 7M (higher derivatives); Pearson Chapter 11

Differentiation with Numerical Values 7. Consider the functions 𝑓 𝑥 , 𝑔 𝑥 as well as 𝑝 𝑥 and 𝑞(𝑥). The following table shows some values associated with the functions 𝑓 𝑥 and 𝑔(𝑥). 𝑥 𝑓(𝑥) 𝑓′(𝑥) g(𝑥) g′(𝑥) -2 8 -8 2 -3 1 4 − 1 4 − 3 4 a. Write down the values of 𝑓 1 and 𝑔 ′ −2 b. If 𝑝 𝑥 =𝑓(𝑥)∙𝑔(𝑥) 𝑖) write down the values of 𝑝(−2) and 𝑝(1) 𝑖𝑖) calculate the value of 𝑝′(−2) c. If 𝑞 𝑥 = 𝑓(𝑥) 𝑔(𝑥) 𝑖) write down the values of 𝑞(−2) and 𝑞(1) 𝑖𝑖) calculate the value of 𝑞′(1)

Higher Order Derivatives and Notation first derivative: Higher than 3rd derivative: 𝑓 ′ ′ 𝑥 or 𝑑 2 𝑦 𝑑 𝑥 2 Example: second derivative: 𝑓 ′ ′′ 𝑥 or 𝑑 3 𝑦 𝑑 𝑥 3 third derivative: 1. Find 𝑓 ′′ 𝑥 given that 𝑓 𝑥 = 𝑥 3 − 3 𝑥 2. Given that 𝑓 𝑥 =cos2𝑥 show that 𝑓′′′ 𝜋 8 =4 2

Finding Higher Order Derivatives and General Patterns (nth derivative) 3. Find the first four derivatives and then find the general pattern for the nth derivative. 𝑓 𝑥 = 2 𝑥 2 a) b) 𝑦= 𝑒 −3𝑥

Finding a Point on the Curve with a Given Gradient 4. Find the point(s) on the curve 𝑦= 𝑥 3 −𝑥+2 where the gradient is 11. 5. Find the point(s) on the curve 𝑓 𝑥 = 𝑥 3 − 𝑥 2 −𝑥+1 where the tangent(s) are horizontal.

Sketching the Gradient Function f ’(x) Given the graph of f(x), it is possible to make a sketch of the derivative function f’(x). 1. Identify all the points where the tangent is horizontal (occurs when f(x) is a local maximum, local minimum or an inflexion point) 2. The graph is increasing where the slope of the tangent is positive The graph is decreasing where the slope of the tangent is negative 3. When sketching f ’(x): the points where the tangent is horizontal, will be = 0 (on the x-axis) the areas where f(x) is increasing will be positive (above the x axis) the areas where f(x) is decreasing will be negative (below the x axis) look for areas of greatest slope – these will become maximums & minimums

Sketching the Gradient Function f ’(x) 6. Sketch the graph of the gradient function of the curves shown below.

Using the GDC (Ti84) Finding the derivative at a point. (TiNspire instructions on p208) Press ALPHA F2 and choose 3:nDerive Enter the function, variable & value of x. Finding the derivative at a point from a graph Graph f(x) Press 2nd CALC choose 6: dy/dx and enter the x value Gradient at the point will be displayed. Graphing the derivative function (TiNspire instructions on p208) Press Y= and then Press ALPHA F2 and choose 3:nDerive Enter x, then the function and x = x, ENTER The derivative function is graphed Switch to TABLE to see the values of the derivative at all the values of x you need. If you need to find the values of x where the derivative = a value (eg 3), Press Y= and type the value in line Y2=. There will be a horizontal line drawn, find the coordinates of where f’(x) = 3 by using the intersection menu.