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EE101Calculus and Analytical

Geometry 2

ngcy@ucsiuniversity.edu.my

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Natural Logarithm

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Definition of the natural logarithm

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Derivatives of Natural Logarithms

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The Derivative of y = ln x

ln = 1

.

= 1

= 1

= l n

Differentiation of ln ax = 1/x was proven

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Trigonometry Function

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Example 1 & 2

Example 1

(a)ln 6

(b)ln 4ln 5

(c)ln

Example 2

(a) ln 4 + ln s

(b) ln+

(c) ln sec x

(d) ln 1

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u)u

1(IntegralThe

= +

1 C ; n 1

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Integration Table

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Example 5

() tan

() tan 2 /

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Exponential Functio

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The inverse of ln x and the number of e

The graph of ln-1xis the gr

ln x reflected across the line

= 1

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The function y = ex

=. ;= 1

;

/ =

= = =

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Exercise

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Example 6

Find y if ln y = 3t + 5

Example 7

Find kif e2k= 10

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Example 8:

dxe

2ln

0

x3

dxe x4

Example 9

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axand loga x

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The Function ax

Since a = e ln afor any positive number a, we can as (e ln a) x = e x ln a.

We therefore make the following definition.

DEFINITIONFor any numbers a> 0 and x,

ax= e x ln a

Laws of exponents

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Laws of exponents

For a> 0, and any xand y:

x

x

y

x

x

yx

yx

ab

aa

a

b

a

a

aa

)(

.

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Example 10:

xdxcos2

xsin

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Example 11(a):

d2

1

0

Example 11(b):

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Example 11(b):

2

D fi iti

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Definition

For any positive nu

Log ax

= inver

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We can derive the Equation (5) from Equation

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Example

2 = ln 2

ln 10 0.69315

2.302590.30103

Apply the rules of

=ln ln

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The Derivative of logaU

To find the derivative of a base alogarithm, we first conver

natural logarithm. If u is a positive differentiable function o

log =

ln ln =

1ln .

ln =

1ln .

1

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Example 12

1x3logdxd 10

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Example 13

log

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Example 14

ln2 log dx

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Example 15

log 10

/

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EXTRA EXERCISES

Q1

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Solve the Equations forx

3 = 5 3. 10

ln 4

=1 log 100

ANS: (a) x = 2 and

Q1

Q2

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Find the derivative of y

= l o g 73 2

ANS:1

3 2

Q2

l h i l iQ3

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Evaluate the integrals in:

13

1 l n

log

() log

ANS: (a) 22 l n 3

ANS: (b) 32 760

ANS: (c) ln 4

: lnln

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