Transcript Presentation1

Differentiation Recap
y
x
change in y
change in x
The first derivative gives the ratio for
which f(x) changes w.r.t. change in the x
value.
F(x) – Non linear
y
x
change in y
change in x
The closer the points are
together the more
accurate the
approximation of the
gradient
For a Non linear function, we can not take just ANY two points
If we wish to find the gradient at x=1 we must move the other point
at (x=3) closer and closer to the point at x=1
x=3 moved to
x=2
The
approximation
is move
accurate then
before
The blue line is the approximation to the tangent
x=2 moved to
x=1.5
Again the
approximation is
move accurate
then before
x=1.5 moved
to x=1.01
This red line through the
point is called the
TANGENT of f(x) at the
point x=1
The
approximation
is now very
accurate as
the points are
virtually
coincident
The tangent has the same
gradient as the curve f(x)
at the point in question
and touches f(x) at x=1
The tangent has the same
gradient as the curve f(x)
at the point in question
and touches f(x) at x=1
This red dashed line through the point
is called the Normal of f(x) at the point
x=1
It’s Perpendicular to the tangent
Definition of derivative
y
x
Limit
x  0
y
x
We call this limit
dy
dx
Some centuries ago Leibnitz and Isaac Newton
Both independently applied this limit to many different
functions
y
Limit
And noticed a general pattern.
x
x  0
This provided the basic rule of differentiation. And
made the process very easy for polynomial functions
The general rule is
f  x   kx n
d
 f  x   nkx n1
dx
Otherwise written as
y  kx n
dy
 nkx n 1
dx
Increasing functions
This function
is increasing
because
dy
0
dx
Always
+ve
Decreasing functions
This function is
increasing
because
dy
0
dx
always
-ve
Where the gradient is Zero
In the above graphs the gradient passes through zero at
x=0 we can write
These points of zero gradient are very important in
dy
mathematics applications as at these points the
0
dx x  0
rate of change of the function is zero
Consider this function
dy
0
dx
The gradient is
zero at these two
points
The tangents are
horizontal
And therefore with
dy
0
dx
The function does
not change value
when the gradient
is zero
we can find local
Maximum and Minimum values of functions
Importance
dy
0
dx
Will tell us when a function is at a local
maximum or minimum
Can be used to find the maximum
or minimum values in various
questions involving rates of change
But since BOTH Max and Min have
gradient = 0 we need a way of
distinguishing between the two
Function Max/Min
We could plot the function and look at the
graph
But an easier way is to consider the 2nd
derivative
d  dy 
d2y
 2


dx  dx 
dx
Consider the following function
f(x) = 2x3-4x-4
f’(x) = 6x2-4
f’’(x)=12x
f’(x)=12x
f’’(x)=6x2-4
f(x)=2x3-4x-4
f’(x)=0
f(x) decreasing
as f’(x)<0
f(x) increasing
as f’(x)>0
f ’’(x)<0
Concave up
f ’’(x)>0
Concave down
First derivative:
y
is positive
Curve is rising.
y
is negative
Curve is falling.
y
is zero
Possible local maximum or minimum.
Second derivative:
y  is positive
Curve is concave up. (MIN)
y  is negative
Curve is concave down. (MAX)
Multiple choice Test
Question
Show that the function
is a decreasing function
ANSWER
1 3
f ( x )   x  2 x 2  22 x
3
f ' ( x )  ( x  4 x  22)
2
f ' ( x )  ( x  4 x  22)
2
f ' ( x )  (( x  2)  4  22)
2
f ' ( x )  (( x  2)  18)
2
f ' ( x )  (  inside here )
.
y
A (1, 5)
C
R
O
B
D
x
The diagram above shows part of the curve C with equation y = 9 - 2x - 2
x
The point A(1, 5) lies on C and the curve crosses
the x-axis at B(b, 0), where b is a constant and b > 0.
,
(a)
Verify that b = 4.
(1)
The tangent to C at the point A cuts the x-axis at the point D, as shown in the
diagram above.
(b) Show that an equation of the tangent to C at A is y + x = 6.
(4)
(c) Find the coordinates of the point D.
(d) Find the Area of the ABD, assume it is a triangle
Answer
(a)
y = 9 – 2b2-
=0
=> b = 4
b
(c)
(d)
Let y = 0 and x = 6 so
D is (6, 0)
Area of shaded triangle is 4.5
Rates of Change
The following function f(s) = 3t is shown below
What does the gradient of the line represent ?
Rates of Change
The following function f(s) = 3t is shown below
What does the gradient of the line represent ?
dS
Velocity 
dt
The steeper the line the faster the velocity
The Black line is fastest as it arrives at B the quickest
The velocity is the Gradient
What about this graph?
What does the gradient represent
V Change in velocity

t
Change in time
What about this graph?
What does the gradient represent
accelerating
decelerating
V Change in velocity

t
Change in time
V
acceleration 
t
V
 0 
t
Constant
Velocity
Summary
We will need these formula
for some rates of change
Questions
s  f (t )
ds
Vel 
dt
dV
Acc 
dt
or
d
Acc 
dt
2
ds
d
s
 
 dt   dt 2
 