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Department
Mathematics
Course Code
MAT137Y1
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MAT137Y1b.doc
Page 1 of 38
Lecture #12 Thursday, October 16, 2003
DIFFERENTIATION
What is the slope of the secant line PQ?
(
)
(
)
( )
(
)
(
)
h
cfhcf
chc
cfhcf+
=
+
+ .
Idea: Get the slope of the tangent line as a limit of slopes of secant lines.
The slope of the tangent line at
c
x
=
ought to be
(
)
(
)
h
cfhcf
h
+
0
lim.
Example
(
)
xxf= at 0
=
x
This DOESNT have a well defined tangent line
Definition
f is differentiable at
c
x
=
if the limit
(
)
(
)
h
cfhcf
h
+
0
lim exists. If it does, we call it the derivative of f at
c and we denote it by
(
)
cf
.
Geometrically
(
)
cf
is the slope of the tangent line going through
(
)
(
)
cfc,.
What is the equation for tangent line?
(
)
(
)
(
)
cxcfcfy
=
P
c
Q
f
(
)
(
)
hcfhc++ ,
(
)
(
)
cfc,
c + h
www.notesolution.com
MAT137Y1b.doc
Page 2 of 38
Lecture #13 Tuesday, October 21, 2003
Example
For function
(
)
2
xxf=, the derivative of f at 2
=
c is
( )
(
)
(
)
( )
44lim
22
lim2
0
22
0
=+=
+
=
h
h
h
f
hh
.
The derivative of f is itself a function for
(
)
2
xxf= repeat the same calculation for any value of c.
( )
(
)
(
)
(
)
(
)
( )
chc
h
chchc
h
chc
h
cfhcf
cf
hhhh
22lim
2
limlimlim
0
222
0
22
00
=+=
++
=
+
=
+
=
At any fixed value of x,
(
)
xxf2=
.
Definition
The derivative of f is a function, denoted f
, and
( )
(
)
(
)
h
xfhxf
xf
h
+
=
0
lim, if it exists.
Terminology
To differentiate a function is to find the derivative.
Notice: The function f has to be defined in the interval
(
)
δ+δxx , in order for
(
)
xf
to be defined.
Example
Actually, even if f is continuous on
(
)
δ+δxx ,, it doesnt mean
(
)
xf
is defined.
Consider
(
)
0,== cxxf
Theorem
If f is differentiable at x, the f is continuous.
Being differentiable is better’ than being continuous.
Proof:
Because f is differentiable at x,
(
)
(
)
( )
xf
h
xfhxf
h
=
+
0
lim
( ) ( )( )
(
)
(
)
( )
00limlim
00
=
=
+
=+ xfh
h
xfhxf
xfhxf
hh
So f is continuous.
DIFFERENTIATION RULES
Building Blocks
If
(
)
cxf= (a constant function), then
(
)
0=
xf for all x.
If
(
)
xxf=, then
(
)
1=
xf for all x.
f(x)
x
www.notesolution.com
MAT137Y1b.doc
Page 3 of 38
Theorem: Sums and Scalar Multiples
Let f, g be differentiable at x and α a constant.
Then
(
)
gf+ and fα are differentiable, then
( ) ( ) ( ) ( )
xgxfxgf
+
=
+
( ) ( ) ( )
xfxf
α=
α
Proof:
( ) ( )
(
)
(
)
(
)
(
)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
xgxf
h
xghxg
h
xfhxf
h
xgxfhxghxf
h
xgfhxgf
xgf
h
h
h
+
=
+
+
+
=
+++
=
+
+
+
=
+
0
0
0
lim
lim
lim
Example
If
(
)
xxf10=,
(
)
10=
xf
Theorem: Differences and Linear Combinations
From the Sums and Scalar Multiples rule,
( ) ( ) ( ) ( )
xgxfxgf
=
( ) ( ) ( ) ( ) ( )
xfxfxfxfff nnnn
α++
α+
α=
α++α+α...... 22112211
Theorem: Product Rule
If f and g are differentiable at x, then gf is differentiable and
( ) ( ) ( ) ( ) ( ) ( )
xgxfxgxfxgf
+
=
Proof:
( ) ( )
(
)
(
)
(
)
(
)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
xgxfxgxf
gf
h
xghxg
xfhxg
h
xfhxf
h
xgxfhxgxfhxgxfhxghxf
h
xgxfhxghxf
h
xgfhxgf
xgf
hh
h
h
h
+
=
+
++
+
=
+++++
=
++
=
+
=
.continuous are they able,differenti are , Because
limlim
lim
lim
lim
00
0
0
0
Theorem: Power Rule
Using the Product Rule, we derive the Power Rule.
For Z
>
nn ,0, if
(
)
n
xxf=, then
(
)
1
=
n
nxxf.
Proof (by induction):
True for 1=k
Assume
(
)
1
=
kk kxx . Prove for 1+k:
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Description
MAT137Y1b.doc Lecture #12 Thursday, October 16, 2003 D IFFERENTIATION (c, fc )) P f Q (c + h, (c+ h)) c c + h What is the slope of the secant line PQ ?f ( + ) f( ) = f(c+h )f c( ). (c + c h Idea: Get the slope of the tangent line as a limit of slopes of secant lines. The slope of the tangent line at x = c ought to be limf(c + ) f c ). h0 h Example f(x)= x at x = 0 This DOESNT have a well defined tangent line Definition f(c +h ) f( ) f is differentiable at x = c if the lih0 lim h exists. If it does, we call it the derivative of f at c and we denote it by fc ). Geometrically f (c) is the slope of the tangent line going throu(c, fc ). What is the equation for tangent line? f(c)= f (c)x c ) Page 1 of 38 www.notesolution.com MAT137Y1b.doc Lecture #13 Tuesday, October 21, 2003 Example 2 2 2 (2+ h ) 2 For function f ( ) x , the derivative of f atc =2 is f(2 = h0 h = h0 4+ h = )4 . 2 The derivative of f is itself a function (x) = x repeat the same calculation for any value of c. 2 2 2 2 2 f c )= lim f c + h) f c)= lim (c+ h) ()c = lim c + 2ch+ h c = lim(2c + h =)2c h0 h h0 h h0 h h0 At any fixed value of x, f(x) = 2x . Definition f x+ h) f x ) The derivative of fs a function, denoted f , and f x )= lim , if it exists. h0 h Terminology To differentiate a function is to find the derivative. Notice: The function f has to be defined in the interval, x+ ) in order for f(x) to be defined. Example Actually, even if f is continuous o(x , x+ ), it doesnt mean f(x) f (x) is defined. Consider f(x)= x ,c =0 x Theorem If f is differentiable at x, the f is continuous. Being differentiable is better than being continuous. Proof: f x+ h) f x ) Because f is differentiable ath0 h = f (x ) lim ( (x+ h )f x = lim) f(x+ h) f ( )h = f(x 0 = 0 h0 h0 h So f is continuous. D IFFERENTIATION R ULES Building Blocks If f(x)= c (a constant function), then f(x)= 0 for all x. If f (x)= x , then f(x)= 1 for all x. Page 2 of 38 www.notesolution.com MAT137Y1b.doc Theorem: Sums and Scalar Multiples Let f, g be differentiable at x and a constant. Then (f + g) and f are differentiable, then (f + g )x = ) x + g( ) ( ) (f )x(= f x ( ) Proof: (f +g x( +h )( f+ g x ) (f + g )x(= lim h0 h f x +h +)g x+ h f x) g x( ) ( ) = h0 h f(x + h )f x (g)x + h )g x ( ) = h0 h + h = f (x ) g x ( ) Example If f (x)= 10x , f x )=10 Theorem: Differences and Linear Combinations From the Sums and Scalar Multiples rule, (f g )x = ) x g( ) ( ) (1 1+ 2 +2..+ f n n x = 1) 1 x + 2f 2 x(+...+ n f n x ( ) ( ) Theorem: Product Rule If f and g are differentiable at x, then f g differentiable and(f g x = )f x (x + f ( g x ( ) ( ) Proof: (f )x(=)lim (fg x( + h)( fg x( ) h0 h = lim f x + h x +(h f x x ( ) ( ) h0 h f x + h x +(h f x x + h + f x x)+ h (f x x ) ( ) ( ) = lim h0 h f x + h )f x ( ) g x + h )g x ( ) = h0 h g x + h +)h0 f x ( ) h Because f , g aredifferentiable, they arecontinuous. = f (x g x(+)f x g (x) ( ) Theorem: Power Rule Using the Product Rule, we derive the Power Rule. For n > 0,nZ , if f( )= x , then f x( )nx n1. Proof (by induction): True for k =1 Assume (x) = kxk1. Prove for k +1: Page 3 of 38 www.notesolution.com
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