5.1 The Natural Logarithm - Battaly · 5.1 The Natural Logarithm Introduction Calculus Home Page Class Notes: Prof. G. Battaly, Westchester Community College, NY

Natural Logarithm: Definition and Derivative

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Background

5.1 The Natural Logarithm Introduction

Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY

Homework Part 1 Homework Part 2

Background: Integrate the following.

Study 5.1; p. 331 # 7, 9, 11, 15, 1927, 3143, 49, 51, 5763, 6973, 77, 80, 83


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Background

5.1 The Natural Logarithm



Background: Integrate the following.


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Intro




Background Summary


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What about?


What about ?




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Think Antiderivative


What about ?



Remember the 2nd FTC?

Think: What function has a derivative = 1/x?


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2nd FTC



4.4 FTC, Part 2: MVT, Ave. Value, 2nd FTC From Calc 1, Section 4.4

2nd FTCIf f is continuous on an open interval I containing a,

then, for every x on the interval:

What about ?

http://www.battaly.com/calc/classnotes/4_4FTC_MVT_AV_2ndFTC.pdf


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Need a function




OR:

So, to find we need to define afunction that has a derivative = 1/x.

What about ?


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Definition of ln x




Definition of the natural logarithm

What is its domain?What is its range?

What does graph look like?

Need to investigate y = 1/x


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explore




If x > 1, ln x ? 0If 0 < x < 1, ln x ? 0

Because of Vertical Asymptote,limit domain to x > 0

x

web; change f to 1/xthen change b

http://www.battaly.com/calc/geogebra/integral/


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plot




0.5

Investigation:Let x = 0.5, 1, 1.5, ....Use calculator to find the integral of y=1/x from 1 to 0.5, from 1 to 1.5, etc.Plot points (0.5, ln 0.5), (1, ln 1), (1.5, ln 1.5), etc.


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sketch ln x




0.5

Result is y = ln x


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properties of ln x




Properties of y = ln x

1. Domain: x > 0 Range: all y

2. continuous, increasing over all domain

3. concave down


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example



Remember Properties of Logarithms ...

1. ln (1) = 0 2. ln (e) = 1

3. ln (a b) = ln(a) + ln(b)

4. ln a = ln(a) ln(b) b

5. ln an = n ln(a)


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example





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example





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example




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example




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example



Write as the natural log of a single expression.


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example



Write as the natural log of a single expression.


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limits

X is approaching 6 from the left

(6x) approaches 0 from the right




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limits

X is approaching 6 from the left

(6x) approaches 0 from the right




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derivative of ln x



Derivative of the Natural Logarithm: Use the 2nd FTC

d (ln x) = dx 1

By definition:

d (ln x) = 1 dx x

d (ln u) = 1 du = u' dx u dx u


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example




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example



OR:


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example




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example




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example




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example




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eq of tangent line




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eq of tangent line

Point slope form of straight line




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ln(absolute value)




The Derivative of the Natural Logarithm of the Absolute Value

since:

and:

Note: u ≠0


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Jan 286:01 PM



HW problems


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example




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example




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example




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example



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5.1 The Natural Logarithm - Battaly · 5.1 The Natural Logarithm Introduction Calculus Home Page Class Notes: Prof. G. Battaly, Westchester Community College, NY