differentiation recap the first derivative gives the ratio for which f(x) changes w.r.t. change in...
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Differentiation Recap
xinchange
yinchange
x
y
The first derivative gives the ratio for which f(x) changes w.r.t. change in the x value.
F(x) – Non linearxinchange
yinchange
x
y
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
The closer the points are together the more accurate the approximation of the gradient
x=3 moved to x=2
The approximation is move accurate then before
The blue line is the approximation to the tangent
x=1.5 moved to x=1.01
The approximation is now very accurate as the points are virtually coincidentThis red line through the
point is called the TANGENT of f(x) at the point 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
The tangent has the same gradient as the curve f(x) at the point in question and touches f(x) at x=1
0
xx
yLimit
Some centuries ago Leibnitz and Isaac Newton Both independently applied this limit to many different functions
And noticed a general pattern.
This provided the basic rule of differentiation. And made the process very easy for polynomial functions
The general rule is
1
n
n
nkxxfdx
d
kxxf
1
n
n
nkxdx
dy
kxy
Otherwise written as
Where the gradient is Zero
In the above graphs the gradient passes through zero at x=0 we can write
00
xdx
dyThese points of zero gradient are very important in mathematics applications as at these points the rate of change of the function is zero
Consider this function
The gradient is zero at these two points
The tangents are horizontal
The function does not change value when the gradient is zero
And therefore with we can find local
Maximum and Minimum values of functions
0dx
dy
0dx
dy
Importance
0dx
dy 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/MinWe could plot the function and look at the graph
But an easier way is to consider the 2nd derivative
2
2
dx
yd
dx
dy
dx
d
Consider the following function f(x) = 2x3-4x-4
f’(x) = 6x2-4 f’’(x)=12x
f’(x)=12xf’’(x)=6x2-4
f(x)=2x3-4x-4
f’(x)=0
f(x) decreasingas f’(x)<0
f(x) increasingas f’(x)>0
f ’’(x)<0Concave up
f ’’(x)>0Concave 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)
QuestionShow that the function is a decreasing function xxxxf 222
3
1)( 23
ANSWER
)224()(' 2 xxxf
)224)2(()(' 2 xxf
)18)2(()(' 2 xxf
)()(' hereinsidexf
)224()(' 2 xxxf
x
y
O B D
R
C
A (1 , 5 )
x
2
.
The diagram above shows part of the curve C with equation y = 9 - 2x -
,
(a) Verify that b = 4. (1) (1)The tangent to C at the point A cuts the x-axis at the point D, as shown in thediagram 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
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.
Answer(a) y = 9 – 2b - = 0 => b = 4
b
2
(c) Let y = 0 and x = 6 so D is (6, 0)
(d) Area of shaded triangle is 4.5
Rates of ChangeThe following function f(s) = 3t is shown below
What does the gradient of the line represent ?
Rates of ChangeThe following function f(s) = 3t is shown below
What does the gradient of the line represent ?
The steeper the line the faster the velocityThe 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
timeinChange
velocityinChange
t
V
t
Vonaccelerati
decelerating
accelerating
0t
V ConstantVelocity
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