Download presentation
Presentation is loading. Please wait.
Published bySharyl Ford Modified over 9 years ago
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
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:
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
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.