Material Taken From:
Mathematics
for the international student
Mathematical Studies SL
Mal Coad, Glen Whiffen, John Owen, Robert Haese,
Sandra Haese and Mark Bruce
Haese and Haese Publications, 2004
Objectives:
• To find the second derivative of a function.
• To identify where functions are increasing and
decreasing.
• To interpret when the f ’(x)=0, f ’(x)>0, and f ’(x) < 0.
IB Subject Guide
•
•
•
•
•
•
Gradients of curves for given values of x.
Values of x where f ′(x) is given.
Equation of the tangent at a given point.
Increasing and decreasing functions.
Graphical interpretation of f′(x)>0, f′(x)=0, f′(x)<0.
Values of x where the gradient of a curve is 0 (zero): solution
of f′(x) =0.
• Local maximum and minimum points.
GDC
• Equations of tangents in the calculator:
•
•
•
•
put function in [Y=]
graph
[2nd] [prgm] Draw 5: Tangent(
type the value where you want your tangent
Section 19G – The Second Derivative
Example 1
Find f ’’(x) given that
3
f ( x)  x 
x
3
Section 19HI – Curve Properties
• Increasing function
– An increase in x produces
an increase in y
• Decreasing function
– An increase in x produces
a decrease in y.
Consider y = x3 – 3x + 4
• What is happening to
the slopes of the
tangent lines?
• Where f(x) is increasing,
f ’(x) is _____.
• Where f(x) is decreasing
f ’(x) is _____.
• Where f(x) is at a
maximum or minimum,
f ’(x) is _____.
Understanding:
• Derivative = slope of tangent line.
• If the tangent line has a negative slope, then the derivative
is negative.
– This happens where the function, f(x), is decreasing.
• If the tangent line has a positive slope, then the derivative
is positive.
– This happens where the function, f(x), is increasing.
• If the tangent line is horizontal, then the derivative is zero.
– This happens where the function, f(x), is at a maximum or
minimum.
IB Example 1
Given the graph of f (x) state:
a) the intervals from A to L in which f (x) is increasing.
b) b) the intervals from A to L in which f (x) is decreasing.
f(x )
D
y = f(x)
C
A
E
L
B
K
H
F
G
x
IB Example 2
The function f(x) is given by the formula
f(x) = 2x3 – 5x2 + 7x – l
a) Evaluate f (1).
b) Calculate f '(x).
c) Evaluate f '(2).
d) State whether the function f (x) is
increasing or decreasing at x = 2.
Consider y = x3 – 3x + 4
• What is the tangent line at the maximum and the minimum?
In order to find the x-coordinate of any maximum or
minimum points, solve the equation f ’(x) = 0
Example 3
The function f(x) is defined as
1 3 1 2
f ( x)  x  x  12 x  4
3
2
Determine the x-coordinates of the points where
the graph has a gradient of zero.
Example 4
3
2
f
(
x
)

x

3
x
 9x 
The function f(x) is defined as
Determine the x-coordinates of the points where
the graph has a gradient of zero.
IB Example 5
Consider the function f(x)=2x3 – 3x2 – 12x + 5
a) (i) Find f ‘(x).
(ii) Find the gradient of the curve f(x) when x = 3.
b) Find the x-coordinates of the points on the curve where
the gradient is equal to –12.
c) (i) Calculate the x-coordinates of the local maximum
and minimum points.
(ii) Hence find the coordinates of the local minimum.
d) For what values of x is the value of f(x) increasing?
Homework
•
•
•
•
•
Worksheet S-46a
Worksheet S-46b
Pg 624 #1bce
Pg 626 #3ace
Worksheet S-47 #1-2