Differential Calculus
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Transcript Differential Calculus
Engineering
Mathematics:
Differential Calculus
Contents
Concepts of Limits and Continuity
Derivatives of functions
Differentiation rules and Higher Derivatives
Applications
Differential Calculus
Concepts of Limits and
Continuity
The idea of limits
f(x) x
2
Consider a function
The function is well-defined for all real values of x
The following table shows some of the values:
x
2.9
2.99
2.99
3
3.001
3.01
3.1
F(x)
8.41
8.94
8.994
9
9.006
9.06
9.61
lim f ( x ) lim x 9
2
x 3
x 3
The idea of limits
Concept of Continuity
E.g. f ( x ) x 2 is continuous at x=3?
The following table shows some of the values:
x
2.9
2.99
2.99
3
3.001
3.01
3.1
F(x)
8.41
8.94
8.994
9
9.006
9.06
9.61
lim f ( x ) 9
lim f ( x ) 9
x 3
and
lim f ( x ) exists
x 3
as
lim f ( x ) 9 f ( 3 )
x 3
x 3
lim f ( x ) lim f ( x ) 9
x 3
x 3
=> f(x) is continuous at x=3!
Differential Calculus
Derivatives of functions
Derivative (導數)
y
Given y=f(x), if
variable x is given
an increment Dx
from x=x0, then y
would change to
f(x0+Dx)
Dy= f(x0+Dx) – f(x)
Dy/Dx is the slope
(斜率) of triangular
ABC
Y=f(x)
f(x0+Dx)
B
Dy
f(x0)
A
C
Dx
x0
x0+Dx
x
Derivative
What happen with Dy/Dx as Dx tends to 0?
It seems that Dy/Dx will be close to the slope of the curve
y=f(x) at x0.
We defined a new quantity as follows
dy
dx
df ( x )
dx
lim
Dx 0
Dy
Dx
lim
Dx 0
f ( x Dx ) f ( x )
Dx
If the limit exists, we called this new quantity as the
derivative (導數) of f(x).
The process of finding derivative of f(x) is called
differentiation (微分法).
Derivative
y
Y=f(x)
f(x0+Dx)
B
Dy
f(x0)
A
C
Dx
x0
x0+Dx
Derivative of f(x) at Xo = slope of f(x) at Xo
x
Differentiation from first principle
Find the derivative of y f ( x ) x 3 x with respect to (w.r.t.) x
2
dy
dx
lim
Dx 0
lim
Dy
Dx
f ( x Dx ) f ( x )
Dx
Dx 0
( x D x ) 3( x D x ) ( x 3 x )
2
lim
2
Dx
Dx 0
x 2 x( D x ) ( D x ) 3 x 3 D x x 3 x )
2
lim
2
2
Dx
Dx 0
2 x( D x ) ( D x ) 3D x
2
lim
Dx 0
Dx
lim ( 2 x D x 3 )
Dx 0
2x 3
To obtain the derivative of a function by its
definition is called
differentiation of the function from first principles
Differential Calculus
Differentiation rules and
Higher Derivatives
Fundamental formulas for
differentiation I
Let f(x) and g(x) be differentiable functions and c be a
constant.
d( c )
0
dx
d ( f ( x ) g ( x ))
df ( x )
dx
d ( cf ( x ))
dx
c
dx
df ( x )
dx
dg ( x )
dx
n
d( x )
dx
nx
n 1
for any real number n
Examples
3 x 4 x 1 and
Differentiate
w.r.t. x
y 3x 4x 1
2
d( 3x )
dx
3
d( 4x )
dx
2
3
d ( 1 )
dy
dx
dx
dx
d( x )
4
d( x )
dx
3( 2 x ) 4 0
6x 4
3
2
y 5x 8x 6x 9
2
dy
5x 8x 6x 9
2
dx
0
2
3
5
d (x )
2
(8)
dx
d (x )
dx
5 (3 x ) 8 ( 2 x ) 6 0
2
15 x 16 x 6
2
6
d ( x)
dx
d (9 )
dx
Table of derivative (1)
df ( x )
Function f ( x )
Constant k
x
kx
kx
n
ln x
ln kx
Derivative
0
1
k
knx
1
x
1
x
n 1
dx
Table of Derivatives (2)
df ( x )
Function
e
f (x)
kx
Derivative
ke
kx
sin kx
k cos kx
sin( kx )
cos kx
k co s( kx )
cos( kx )
dx
k sin kx
k sin( kx )
Angles in
radians
Differential Calculus
Product Rule, Quotient Rule
and Chain Rule
- The product rule and the quotient rule
Form of product function
Form of quotient function
e.g.1
y( x ) u ( x )v ( x )
y( x )
u( x )
v( x )
y ( x ) x cos x
2
This is a _________ function with u ( x )
and v ( x )
e x
x
e.g.2
y( x )
ln x
This is a _________ function with
and
t 1
2
e.g.3
y(t )
cos t
e.g.4
f ( t ) ( t 6 ) cos 2 t
e.g.5
f ( x ) (3 x 7 )e
2
2 x
u( x )
v( x )
The product rule
Consider the function
y( x ) u ( x )v ( x )
dy
Using the product rule,
dx
du
vu
dx
u ' v uv '
dy
e.g.1 Find
Solution:
dx
where y x 2 cos x
dv
dx
e.g.2 Find y ' where y ( x )
Solution:
df
e.g.3 Find f '
dt
Solution:
where
xe
2x
f ( t ) t ln t
3
The quotient rule
Consider the function
u( x )
y( x ) .
v( x )
Applying the quotient rule,
dy
dx
e.g.1 Find y ' where y
cos x
x
Solution:
v
du
u
dx
v
2
dv
dx vu ' uv '
2
v
e x
x
e.g.2 Find y ' where y ( x )
ln x
Solution:
e.g.3 Find
Solution:
dy
y '
dt
t 1
2
where y ( t )
cos 3 t
More Example (1)
y
Differentiate
f ( x ) 2 3x
dy
( 2 3x )
d( 2 3x )
( 2 3x )
dx
2
( 2 3 x )( 3 ) ( 2 3 x )( 3 )
( 2 3x )
12
( 2 3x )
2
d( 2 3x )
dx
( 2 3x )
dx
2 3x
w.r.t. x
g( x ) 2 3x
2 3x
2
More Example (2)
Differentiate
dy
y ( 2 x 4 x 1 )( 3 x 2 x 5 )
2
d( 3x 2x 5 )
2
( 2x 4x 1)
2
dx
d( 2x 4x 1)
2
( 3x 2x 5 )
2
dx
( 2 x 4 x 1 )( 3
2
dx
2
2
dx
dx
dx
dx
d( 5 )
) ( 3 x 2 x 5 )( 2
2
dx
( 2 x 4 x 1 )( 3( 2 x ) 2 ) ( 3 x 2 x 5 )( 2 ( 2 x ) 4 )
2
2
( 2 x 4 x 1 )( 6 x 2 ) ( 3 x 2 x 5 )( 4 x 4 )
2
2
24 x 24 x 2 x 22
3
2
w.r.t. x
2
dx
2
dx
4
dx
dx
d ( 1 )
dx
)
Fundamental formulas for
differentiation II
Let f(x) and g(x) be differentiable functions
d ( f ( x ) g ( x ))
f ( x )
dx
d ( f ( x ) / g ( x ))
dx
d ( g ( x ))
g( x )
d ( f ( x ))
dx
g( x )
df ( x )
dx
f(x)
dx
2
( g ( x ))
dg ( x )
dx
Fundamental formulas for
differentiation III
d cos( x )
sin( x )
d sin( x )
dx
dx
d tan( x )
1
dx
(cos( x ))
d sec( x )
2
(sec( x ))
d cot( x )
2
d csc( x )
sec( x ) tan( x )
1
dx
dx
de
cos( x )
(sin( x ))
2
(csc( x ))
csc( x ) cot( x )
dx
x
e
x
where
dx
d ln( x )
dx
e 1 x
x
x
2
2!
1
x
where
x
3
3!
x
4
4!
x
5
5!
ln( e ) x
x
ln(x) is called natural logarithm (自然對數)
2
Differentiation of composite functions
y
x 1
2
To differentiate
w.r.t. x, we may have problems
as we don’t have a formula to do so.
The problem can be simplified by considering composite
function:
u x 1
2
Let
so
y
and
u
y
x 1
2
we know derivative of y w.r.t. u (by formula):
1
dy
du
d
u
du
du
du
2
1
2
1
u2
1
1
2
u
1
2
but still don’t know
dy
dx
Chain Rule (鏈式法則)
Chain Rule states that :
given y=g(u), and u=f(x)
dy
dy
dx
So our problem
dy
x 1
d
dx
dx
1
2
u
du
y g( u )
2
dy
du
dx
and
u
u f ( x ) x 1
2
du
dy
dx
du
( 2 x ) x( x 1 )
2
2
du
dx
d
u
du
1
1
2
du
1
u
1
2
2
d x 1
2
dx
2x
Example 1
Differentiate y (cos( x )) 3
w.r.t. x
Simplify y by letting u cos( x ) so now
By chain rule
dy
dy du
dx
dy
dy
dx
du
du
du
dy
du
du
dx
3u
dx
du
2
3
du
3
yu
dx
d (cos( x ))
sin( x )
dx
3 u ( sin( x )) 3 sin( x )(cos( x ))
2
2
Example 2
2
Differentiate y e w.r.t. x
x
u
Simplify y by letting u x so now y e
By chain rule
dy
dy du
2
dx
dy
du
de
du
u
e
du
u
du
dy
dx
dx
dx
dy
du
du
dx
dx
2
2x
dx
e 2 x 2 xe
u
x
2
Example 3
Differentiate
y ln( 2 x 3 )
2
w.r.t. x
Simplify y by letting u 2 x 3 so now y ln( u )
By chain rule
dy
dy du
2
dx
dy
du
d ln( u )
du
dx
1
du
du
u
dx
dy
dy
dx
du
du
dx
d( 2x 3 )
2
4x
dx
1
u
4x
4x
2x 3
2
Higher Derivatives (高階導數)
If the derivatives of y=f(x) is differentiable function of x, its
derivative is called the second derivative (二階導數) of y=f(x)
2
d
y or f ’’(x). That is
and is denoted by
dx
2
2
f ''( x) f
(2)
( x)
d y
dx
d
2
(
dy
)
dx dx
3
Similarly, the third derivative =
f
(3)
( x)
d y
dx
n
the n-th derivative = f
(n)
(x)
d y
dx
n
d
dx
(
d
3
dx
n 1
dx
d
y
n 1
)
2
(
d y
dx
2
)
Example
2
Find
dy
4
dx
2
dx
dx
4
dx
dx
d y
2
3
dy d y d y
, 2 , 3
dx dx
dx
d
12
dx
dx
(
dx
dy
3
3
4
2
8
dx
) 16
dx
dx
d y
dx
3
y 4 x x 12 x 8 x 6
if
3
3
dx
dx
2
(
d y
dx
2
dx
dx
d
dx
) 48
3
d( 6 )
16 x 3 x 24 x 8
3
2
dx
2
24
dx
dx
2
dx
dx
2
dx
6
dx
dx
d(8 )
48 x 6 x 24
2
dx
d ( 24 )
dx
96 x 6
Differential Calculus
Applications
Slope of a curve
Recall that the derivative of a curve evaluate at a point is
the slope of the curve at that point.
Y=f(x)
f(x0+Dx)
B
Dy
f(x0)
A
Derivative of f(x) at Xo
= slope of f(x) at Xo
C
Dx
x0
x0+Dx
x
Slope of a curve
Find the slope of y=2x+3
at x=0
To find the slope of a
curve, we have to
compute the derivative of
y and then evaluate at a
point
dy
dx
d( 2x 3 )
2
dx
The slope of y at x=0
equals 2
(y=mx+c now m=2)
Slope of a curve
Find the slope of y x 1
at x=0, 2, -2
2
dy
dx
d( x 1)
2
2x
dx
The slope of y = 2x
The slope of y (at x=0)
= 2(0) = 0
The slope of y (at x=-2)
=2(-2) = -4
The slope of y (at x=2)
=2(2) = 4
X=-2
X=2
X=0
Local maximum and minimum point
dy
0
For a continuous function, the point at which
dx
is called a stationary point.
This gives the point local maximum or local minimum of
the curve
D
B
X1
C
A
X2
First derivative test (Max pt.)
Given a continuous function y=f(x)
If dy/dx = 0 at x=xo & dy/dx changes from +ve to –ve through x0,
x=x0 is a local maximum point
local maximum point
x=x0
First derivative test (Min pt.)
Given a continuous function y=f(x)
If dy/dx = 0 at x=xo & dy/dx changes from -ve to +ve through x0,
x=x0 is a local minimum point
x=x0
local minimum point
Example 1
Determine the position of any local maximum and
minimum of the function y x 2 1
First, find all stationary point (i.e. find x such that dy/dx = 0)
dy
d( x 1)
2
dx
when
dx
x 0,
dy
dx
2 x , so
2x 0
dy
0
dx
when
x 0,
when x=0
dy
2x 0
dx
By first derivative test x=0 is a local minimum point
Example 2
Find the local maximum and minimum of y x 3 6 x 2 9 x 2
Find all stationary points first:
dy
y x 6x 9x 2
3
2
3 x 12 x 9 3( x 1 )( x 3 )
2
dx
dy
0,
x 1
when
x3
and
dx
x
0
x<1
x=1
1<x<3
x=3
x>3
dy/dx
+
+
0
-
0
+
Example 2 (con’d)
By first derivative test,
x=1 is the local maximum (+ve -> 0 -> -ve)
x=3 is the local minimum (-ve -> 0 -> +ve)
Second derivative test
Second derivative test states:
There is a local maximum point in y=f(x) at x=x0,
2
dy
d
y < 0 at x=x0.
if
0 at x=x0 and
dx
dx
There is a local minimum point in y=f(x) at x=x0,
2
dy
d
y > 0 at x=x0.
if
0 at x=x0 and
dx
2
d y
2
dx
2
If dy/dx = 0 and
2 =0 both at x=x0, the second derivative
dx
test fails and we must return to the first derivative test.
Example
Find the local maximum and minimum point of
y x 6x 9x 2
3
2
Solution
Find all stationary points first:
y x 6x 9x 2
3
2
dy
3 x 12 x 9 3( x 1 )( x 3 )
2
dx
dy
0,
then
x 1
or
2
d y
x3
dx
dx
2
6 x 12 x
2
d y
dx
2
6 ( 1 ) 12 6 0
2
x 1
d y
dx
6 ( 3 ) 12 6 0
2
x3
By second derivative test, x=1 is max and x=3 is min.
So, (1,2) is max. point and (3,-2) in min. point.
Practical Examples 1
e.g.1 A rectangular block, with square base of side x
mm, has a total surface area of 150 mm2. Show that
the volume of the block is given by V 1 ( 75 x x 3 ) .
2
Hence find the maximum volume of the block.
Solution:
Practical Examples 2
A window frame is made in the shape of a rectangle with
a semicircle on top. Given that the area is to be 8 ,
8
show that the perimeter of the frame is P r ( 2 ).
r
2
Find the minimum cost of producing the frame if 1 metre
costs $75.
Solution: