Limits And Derivative

Done by ,
ASHAMS KURIAN
11 B
K.V.PATTOM

1. Done BY,
AchuthAn
xi B
k.v.pAttom

2. Limits and Derivatives

3. Concept of a Function

4. y is a function of x, and the
relation y = x
2
describes a function.
We notice that with such a
relation, every value of x
corresponds to one (and only one)
value of y.
y = x
2

5. Since the value of y depends on a
given value of x, we call y the
dependent variable and x the
independent variable and of the
function y = x
2
.

9. Notation for a Function : f(x)

20. The Idea of Limits

21. Consider the function
The Idea of
Limits
2
4
)(
2
-
-
=
x
x
xf
x1.91.991.9991.99992 2.00012.0012.012.1
f(x)

22. Consider the function
The Idea of
Limits
2
4
)(
2
-
-
=
x
x
xf
x1.91.991.9991.99992 2.00012.0012.012.1
f(x)3.93.993.9993.9999un-
defined
4.00014.0014.014.1

23. Consider the function
The Idea of
Limits
2)( +=xxg
x1.91.991.9991.99992 2.00012.0012.012.1
g(x)3.93.993.9993.99994 4.00014.0014.014.1
2)( +=xxg
x
y
O
2

24. If a function f(x) is a continuous at x
0
,
then . )()(lim
0
0
xfxf
xx
=
®
4)(lim
2
=
®
xf
x
4)(lim
2
=
®
xg
x
approaches to,
but not equal to

25. Consider the function
The Idea of
Limits
x
x
xh=)(
x-4-3-2 -1 0 1 2 34
g(x)

26. Consider the function
The Idea of
Limits
x
x
xh=)(
x-4-3-2 -1 0 1 2 34
h(x)-1-1-1 -1 un-
defined
1 2 34

27. 1)(lim
0
-=
-
®
xh
x
1)(lim
0
=
+
®
xh
x
)(lim
0
xh
x® does not
exist.

28. A function f(x) has limit l at x
0
if f(x)
can be made as close to l as we please
by taking x sufficiently close to (but
not equal to) x
0. We write
lxf
xx
=
®
)(lim
0

29. Theorems On Limits

30. Theorems On Limits

31. Theorems On Limits

32. Theorems On Limits

33. Limits at Infinity

34. Limits at Infinity
Consider
1
1
)(
2
+
=
x
xf

35. Generalized, if
±¥=
±¥®
)(limxf
x
then
0
)(
lim =
±¥® xf
k
x

36. Theorems of Limits at Infinity

37. Theorems of Limits at Infinity

38. Theorems of Limits at Infinity

39. Theorems of Limits at Infinity

40. Theorem
where θ is measured in radians.
All angles in calculus are
measured in radians.
1
sin
lim
0
=
®q
q
q

41. The Slope of the Tangent to a Curve

42. The Slope of the Tangent to a Curve
The slope of the tangent to a curve
y = f(x) with respect to x is defined
as
provided that the limit exists.
x
xfxxf
x
y
AT
xx D
-D+
=
D
D
=
®D®D
)()(
limlim of Slope
00

43. Increments
The increment
△
x of a variable is the
change in x from a fixed value x = x
0

to another value x = x
1
.

44. For any function y = f(x), if the variable x is
given an increment
△
x from x = x
0
, then the value
of y would change to f(x
0 +
△
x) accordingly.
Hence thee is a corresponding increment of y(
△
y)
such that △y = f(x
0
+
△
x) – f(x
0
).

45. Derivatives
(A) Definition of Derivative.
The derivative of a function y = f(x)
with respect to x is defined as
provided that the limit exists.
x
xfxxf
x
y
xx D
-D+
=
D
D
®D®D
)()(
limlim
00

46. The derivative of a function y = f(x)
with respect to x is usually denoted by
,
dx
dy
),(xf
dx
d
,'y ).('xf

47. The process of finding the
derivative of a function is called
differentiation. A function y = f(x)
is said to be differentiable with
respect to x at x = x
0
if the
derivative of the function with
respect to x exists at x = x
0
.

48. The value of the derivative of y = f(x)
with respect to x at x = x
0
is denoted
by or .
0xxdx
dy
=
)('
0xf

49. To obtain the derivative of a
function by its definition is called
differentiation of the function from
first principles.

50. •Let’s sketch the graph of the function f(x) = sin
x, it looks as if the graph of f’ may be the same
as the cosine curve.
DERIVATIVES OF TRIGONOMETRIC
FUNCTIONS
Figure 3.4.1, p. 149

51. •From the definition of a derivative, we have:
0 0
0
0
0
0 0 0
( ) ( ) sin( ) sin
'( ) lim lim
sin cos cos sinh sin
lim
sin cos sin cos sin
lim
cos 1 sin
lim sin cos
cos 1
limsin lim limcos lim
h h
h
h
h
h h h h
f x h f x x h x
f x
h h
x h x x
h
x h x x h
h h
h h
x x
h h
h
x x
h
® ®
®
®
®
® ® ® ®
+ - + -
= =
+ -
=
-é ù
= +
ê ú
ë û
é - ùæ ö æ ö
= +
ç ÷ ç ÷ê ú
è ø è øë û
-
= × + ×
0
sinh
h
DERIVS. OF TRIG. FUNCTIONS Equation 1

52. •Two of these four limits are easy to evaluate.
DERIVS. OF TRIG. FUNCTIONS
0 0 0 0
cos 1 sin
limsin lim limcos lim
h h h h
h h
x x
h h
® ® ® ®
-
× + ×

53. •Since we regard x as a constant
when computing a limit as h → 0,
we have:
DERIVS. OF TRIG. FUNCTIONS

lim
h®0
sinx=sinx
lim
h®0
cosx=cosx

54. •The limit of (sin h)/h is not so obvious.
•In Example 3 in Section 2.2, we made
the guess—on the basis of numerical and
graphical evidence—that:
0
sin
lim 1
q
q
q
®
= =
DERIVS. OF TRIG. FUNCTIONS Equation 2

55. •We can deduce the value of the remaining
limit in Equation 1 as follows.
0
0
2
0
cos 1
lim
cos 1 cos 1
lim
cos 1
cos 1
lim
(cos 1)
q
q
q
q
q
q q
q q
q
q q
®
®
®
-
- +æ ö
= ×
ç ÷
+è ø
-
=
+
DERIVS. OF TRIG. FUNCTIONS

56. 2
0
0
0 0
0
sin
lim
(cos 1)
sin sin
lim
cos 1
sin sin 0
lim lim 1 0
cos 1 1 1
cos 1
lim 0
q
q
q q
q
q
q q
q q
q q
q q
q q
q
q
®
®
® ®
®
-
=
+
æ ö
=- ×
ç ÷
+è ø
æ ö
=- × =- × =
ç ÷
+ + è ø
-
=
DERIVS. OF TRIG. FUNCTIONS Equation 3

57. •If we put the limits (2) and (3) in (1),
we get:
•So, we have proved the formula for sine,
0 0 0 0
cos 1 sin
'( ) limsin lim limcos lim
(sin ) 0 (cos )1
cos
h h h h
h h
f x x x
h h
x x
x
® ® ® ®
-
= × + ×
= × + ×
=
DERIVS. OF TRIG. FUNCTIONS Formula 4
(sin ) cos
d
x x
dx
=

58. •Differentiate y = x
2
sin x.
–Using the Product Rule and Formula 4,
we have:
2 2
2
(sin ) sin ( )
cos 2 sin
dy d d
x x x x
dx dx dx
x x x x
= +
= +
Example 1DERIVS. OF TRIG. FUNCTIONS
Figure 3.4.3, p. 151

59. •Using the same methods as in
the proof of Formula 4, we can prove:
(cos ) sin
d
x x
dx
=-
Formula 5DERIV. OF COSINE FUNCTION

60. 2
2
2 2
2
2 2
2
sin
(tan )
cos
cos (sin ) sin (cos )
cos
cos cos sin ( sin )
cos
cos sin 1
sec
cos cos
(tan ) sec
d d x
x
dx dx x
d d
x x x x
dx dx
x
x x x x
x
x x
x
x x
d
x x
dx
æ ö
=
ç ÷
è ø
-
=
× - -
=
+
= = =
=
DERIV. OF TANGENT FUNCTION Formula 6

61. •We have collected all the differentiation
formulas for trigonometric functions here.
–Remember, they are valid only when x is measured
in radians.
2 2
(sin ) cos (csc ) csc cot
(cos ) sin (sec ) sec tan
(tan ) sec (cot ) csc
d d
x x x x x
dx dx
d d
x x x x x
dx dx
d d
x x x x
dx dx
= =-
=- =
= =-
DERIVS. OF TRIG. FUNCTIONS

62. •Differentiate
•For what values of x does the graph of f have
a horizontal tangent?
sec
( )
1 tan
x
f x
x
=
+
Example 2DERIVS. OF TRIG. FUNCTIONS

63. •The Quotient Rule gives:
2
2
2
2 2
2
2
(1 tan ) (sec ) sec (1 tan )
'( )
(1 tan )
(1 tan )sec tan sec sec
(1 tan )
sec (tan tan sec )
(1 tan )
sec (tan 1)
(1 tan )
d d
x x x x
dx dx
f x
x
x x x x x
x
x x x x
x
x x
x
+ - +
=
+
+ - ×
=
+
+ -
=
+
-
=
+
Example 2Solution:
tan2 x + 1 =
sec2 x

64. •Find the 27th derivative of cos x.
–The first few derivatives of f(x) = cos x
are as follows:
(4)
(5)
'( ) sin
''( ) cos
'''( ) sin
( ) cos
( ) sin
f x x
f x x
f x x
f x x
f x x
=-
=-
=
=
=-
Example 4DERIVS. OF TRIG. FUNCTIONS

65. –We see that the successive derivatives occur
in a cycle of length 4 and, in particular,
f
(n)
(x) = cos x whenever n is a multiple of 4.
–Therefore, f
(24)
(x) = cos x
–Differentiating three more times,
we have:
f
(27)
(x) = sin x
Example 4Solution:

66. •Find
–In order to apply Equation 2, we first rewrite
the function by multiplying and dividing by 7:
0
sin7
lim
4
x
x
x
®
sin7 7 sin7
4 4 7
x x
x x
æ ö
=
ç ÷
è ø
Example 5DERIVS. OF TRIG. FUNCTIONS

67. •If we let θ = 7x, then θ → 0 as x → 0.
So, by Equation 2, we have:
0 0
0
sin7 7 sin7
lim lim
4 4 7
7 sin
lim
4
7 7
1
4 4
x x
x x
x x
q
q
q
® ®
®
æ ö
=
ç ÷
è ø
æ ö
=
ç ÷
è ø
= × =
Example 5Solution:

68. •Calculate .
–We divide the numerator and denominator by x:
by the continuity of
cosine and Eqn. 2
0
lim cot
x
x x
®
Example 6DERIVS. OF TRIG. FUNCTIONS
0 0 0
0
0
cos cos
lim cot lim lim
sinsin
limcos
cos0
sin 1
lim
1
x x x
x
x
x x x
x x
xx
x
x
x
x
® ® ®
®
®
= =
= =
=

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