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Graphs. What can we find out from the function itself? Take the function To find the roots.

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Presentation on theme: "Graphs. What can we find out from the function itself? Take the function To find the roots."— Presentation transcript:

1 Graphs

2 What can we find out from the function itself? Take the function To find the roots

3 Function -5 3

4 Stationary Points Find where the first derivative is zero Substitute x-values to find y-values (1.31, -24.6), (-3.31, 24.6)

5 (1.31, -24.6) (-3.31, 24.6)

6 (1.31, -24.6) (-3.31, 24.6) Gradient function is positive i.e. Function is increasing

7 (1.31, -24.6) (-3.31, 24.6) Gradient function is positive i.e. Function is increasing

8 (1.31, -24.6) (-3.31, 24.6) Gradient function is negative i.e. Function is decreasing

9 Nature of turning points Function First derivative Second derivative Substitute the x-values of the stationary points Positive indicates minimum Negative indicates maximum

10

11 is a maximum is negative is a minimum is positive

12 is concave down is negative

13 is concave up is positive

14 Concave Up - 2nd derivative positive Concave Down - 2nd derivative negative

15 has a point of inflection is zero There is a change in curvature

16 Example 1 Find the stationary points of the following function and determine their nature. To find the roots Roots are: (-3.63, 0) (-1, 0) Using solver on graphics calculator

17 x = -3.63

18 Example 1 To find the stationary points. Differentiate Factorise Stationary Points are: (0, 1), (-1, 0), (-3, 28)

19 -3, 28 -1, 0 0, 1

20 The first derivative tells us where the function is increasing/decreasing and where it is stationary.

21 Function is stationary Function is stationary Function is stationary

22 The first derivative tells us where the function is increasing/decreasing and where it is stationary. Gradient is positive

23 The first derivative tells us where the function is increasing/decreasing … Function is increasing Function is increasing Function is increasing

24 The first derivative tells us where the function is increasing/decreasing … Function is decreasing

25 To determine the nature of the turning points: Differentiate again:

26

27 x = -3

28 x = -1

29 x = 0 Let’s take a closer look!

30 x = 0 This means we need to look at the gradient function.

31 x = 0 Before ‘0’, the gradient is negative.

32 x = 0 After ‘0’, the gradient is positive.

33 To determine the nature of the turning points: Differentiate again: Gradient is negative just before “0” and positive just after “0” minimum

34 Practice: Concavity Find where the following function is concave down. Differentiate twice:

35 Practice: Find where the function is increasing Draw the graph

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