Calculus

WUC 115/05WUC 115/05
UNIVERSITY MATHEMATHICSUNIVERSITY MATHEMATHICS
BB

Rules for differentiation:
1. The derivative of a constant function is zero.
example
2. The power rule:
example
[ ] 0=c
dx
d
[ ] 030 =
dx
d
[ ] 1−
= nn
nxx
dx
d
[ ] 3144
12)4(33 xxx
dx
d
== −
Unit 5: Differentiation
5.1 Concepts and Rules of Differentiation

3. The scalar multiple rule:
If f(x) is differentiable at x and c is any real number, then
example
4. The sum rule & difference rule:
example
[ ] [ ])()( xf
dx
d
cxcf
dx
d
= [ ] [ ] 3344
12)4(333 xxx
dx
d
x
dx
d
===
[ ] [ ] [ ])()()()( xg
dx
d
xf
dx
d
xgxf
dx
d
±=±
[ ] 12)5()()(5 22
−=+−=+− x
dx
d
x
dx
d
x
dx
d
xx
dx
d

Rules for differentiation:
5. The Product Rule:
Example: Differentiate
Solution:
[ ] )(')()().(')().( xgxfxgxfxgxf
dx
d
+=
)1)(13( 2
xxy +−=
[ ]
169
1366
)1)(13()6)(1(
)1()13()13()1()1)(13(
2
22
2
222
−+=
−++=
−++=
+−+−+=+−
xx
xxx
xxx
x
dx
d
xx
dx
d
xxx
dx
dy

6. The Quotient Rule: If
then
Example:
Differentiate with respect to x .
,0)(,
)(
)(
)( ≠= xh
xh
xg
xf
2
)]([
)()(')()('
)('
xh
xgxhxhxg
xf
×−×
=
23
12
)(
+
−
=
x
x
xf

7. The Chain Rule:
If is the composite function in which the
inside function g and the outside function f are
differentiable, then
Example:
Differentiate with respect to x .
))(( xgf
)(')).(('))](([ xgxgfxgf
dx
d
=
( )22
1−+= xxy

The derivative of an Exponential Function:
Example: Determine the derivative of
Solution:
5.1 Diff. of Exp. & Log. Function
( ) xx
ee
dx
d
=
x
xexf 3)( =
)3()(3)3()(' x
dx
d
ee
dx
d
xxe
dx
d
xf xxx
+==
)1(333 +=+= xeexe xxx

The derivative of a Natural Logarithm Function:
where x > 0
And by the chain rule:
where f (x) > 0
x
x
dx
d 1
)(ln =
)(
)('
)]([ln
xf
xf
xf
dx
d
=

Example:
Find the derivative of
Solution:
)52ln()( 2
−+= xxxf
52
22
)52(
)52(
)(' 22
2
−+
+
=
−+
−+
=
xx
x
xx
xx
dx
d
xf

Marginal functions in economics
Example: The total cost, C(x) associated with the producing
and marketing x units of MP4 player is given by
Find :
(a)The actual cost incurred for producing and marketing the
11th
MP4 player.
(b) Total cost when the output is 5 units.
(c) Average cost for an output of 10 units
(d) Marginal cost the output is 4 units.
5.2 Business Application
150201.05.0)( 23
+−−= xxxxC

Example: Suppose the relationship between the unit price p in
RM and the quantity demanded x of the laptop model S is
given by the equation
Find :
(a) the revenue function R(x)
(b) the marginal revenue foundation R’(x)
(c) Compute R’(200) .
30002.0)( +−= xxp

Example: For the above example , the revenue is function is
. Let the total cost of producing
x units of laptop model S be
Find :
(a) the profit function P(x)
(b) the marginal profit function P’(x)
(c) Compute P’(100)
xxx 30002.0)( 2
+−=
300000200)( += xxC

Unit 5 Integration
5.3 Integration as Anti-differentiation
Definition:
A function F is called an anti-derivative or an indefinite integral
of a function f(x) if the derivative F’(x) = f(x). We write
if
Example:
Since the derivative of x2
= 2x, or
therefore, x2
is an anti-derivative of 2x, or
∫= dxxfxF )()( )()(' xfxF =
xx
dx
d
2)( 2
=
∫ = 2
2 xxdx

Unit 5 Integration
x2
is not the only anti-derivative of 2x,
The derivative of x2
+ 5 is also 2x,
therefore x2
+ 5 is also anti-derivative of 2x.
Constant term is the only difference
In general,
it is always necessary to include the constant C
when writing a general anti-derivative:
x2
+ C

Unit 5 Integration
Integration:
If then, integrating f(x) produces
the anti-derivative F(x) + C.
We write:
[ ] )()( xfxF
dx
d
=
∫ += CxFdxxf )()(

Unit 5 Integration
Basic Integration Rules:
Rule 1: The indefinite integral of a constant
Example:
∫ += Ckxkdx
∫ += Cxdx 99

Unit 5 Integration
Rule 2: The power rule
Example:
)( 1
1
1 1
−≠+
+
=∫
+
nCx
n
dxx nn
∫ +=+
+
= +
CxCxdxx 5144
5
1
14
1

Unit 5 Integration
Rule 3: The constant multiple rule
(h = constant)
Example:
Try Activity 5.1 (Page 8)
∫ ∫= dxxfhdxxhf )()(
∫ ∫ +





== Cxxdxxdx 2
2
1
333

Unit 5 Integration
5.3 Indefinite Integrals
Rule 4: The sum rule
Example:
[ ]∫ ∫∫ ±=± dxxgdxxfdxxgxf )()()()(
( )
Cxxx
Cx
xx
dxxdxdxxdxxx
+−+=
+−





+





=
−+=−+∫ ∫ ∫ ∫
23
2
2
6
3
3
263263
23
23
22

Unit 5 Integration
Rule 5: The indefinite integral of the exponential
function
(k = constant)
Example:
∫ += Ce
k
dxe kxkx 1
∫ += −−
Cedxe xx 8585
5
1

Unit 5 Integration
Rule 6: The indefinite integral of
Example:
x
xxf
11
== −
)(
∫ ∫ ≠+==−
0
11
xCxdx
x
dxx ln
∫ ∫ ≠+==−
)(ln 05
1
55 1
xCxdx
x
dxx

Unit 5 Integration
5.3 Definite Integrals
Theorem:
Let f be continuous on [a, b]. If G is any anti-derivative
for f on [a, b], then
Example: Evaluate .
Solution: Since is an antiderivative of f(x) = x, thus
∫ −=
b
a
aGbGdxxf )()()(
∫
2
1
dxx
2
)(
2
x
xF =
2
3
)14(
2
1
2
2
1
2
2
1
=−==∫
x
dxx

Unit 5 Integration
Evaluating the definite Integral:
Let f and g be continuous function, then
( c = constant)
[ ]
( )cbadxxfdxxfdxxf
dxxgdxxfdxxgxf
dxxfcdxxcf
dxxfdxxf
dxxf
c
b
b
a
c
a
b
a
b
a
b
a
b
a
b
a
b
a
a
b
a
a
<<+=
±=±
=
−=
=
∫∫∫
∫∫∫
∫ ∫
∫ ∫
∫
)()()(.
)()()()(.
)()(.
)()(.
)(.
5
4
3
2
01

Unit 5 Integration
Example: Given that and
Find: (a) (b)
Solution:
(a) (b)
5)(
7
4
−=∫ dxxf 8)(
7
4
=∫ dxxg
∫ +
7
4
)]()([ dxxgxf ∫ +−
7
4
]3)()([ dxxgxf
3
85
)()(
)]()([
7
4
7
4
7
4
=
+−=
+=
+
∫∫
∫
dxxgdxxf
dxxgxf
4
)47(385
3)()(
]3)()([
7
4
7
4
7
4
7
4
−=
−+−−=
+−=
+−
∫∫∫
∫
dxdxxgdxxf
dxxgxf

Unit 5 Integration
Example: Given that and
Find
Solution:
8)(
5
0
=∫ dxxf 3)(
2
0
=∫ dxxf
∫
5
2
)( dxxf
∫
∫
∫ ∫∫
=−=
=+
=+
5
2
5
2
5
2
5
0
2
0
538)(
8)(3
)()()(
dxxf
dxxf
dxxfdxxfdxxf

Example: Evaluate
Solution:
Unit 5 Integration
( ) .4
3
0
2
dxx∫ −
( )
3
)00()912(
3
44
3
0
3
3
0
2
=
−−−=






−=−∫
x
xdxx

Calculus

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Calculus

Similar to Calculus (20)

More from Sheila Shamuganathan

More from Sheila Shamuganathan (11)

Recently uploaded

Recently uploaded (20)

Calculus