The natural logarithmic function
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The Natural Logarithmic Function. Differentiation Integration. Properties of the Natural Log Function. If a and b are positive numbers and n is rational, then the following properties are true:. The Algebra of Logarithmic Expressions. The Derivative of the Natural Logarithmic Function.

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The Natural Logarithmic Function

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The natural logarithmic function

The Natural Logarithmic Function

Differentiation

Integration


Properties of the natural log function

Properties of the Natural Log Function

  • If a and b are positive numbers and n is rational, then the following properties are true:


The algebra of logarithmic expressions

The Algebra of Logarithmic Expressions


The derivative of the natural logarithmic function

The Derivative of the Natural Logarithmic Function

Let u be a differentiable function of x


Differentiation of logarithmic functions

Differentiation of Logarithmic Functions


Differentiation of logarithmic functions1

Differentiation of Logarithmic Functions


Differentiation of logarithmic functions2

Differentiation of Logarithmic Functions


Differentiation of logarithmic functions3

Differentiation of Logarithmic Functions


Differentiation of logarithmic functions4

Differentiation of Logarithmic Functions


Logarithmic properties as aids to differentiation

Logarithmic Properties as Aids to Differentiation

  • Differentiate:


Logarithmic properties as aids to differentiation1

Logarithmic Properties as Aids to Differentiation

  • Differentiate:


Logarithmic differentiation

Logarithmic Differentiation

  • Differentiate:

    This can get messy with the quotient or product and chain rules. So we will use ln rules to help simplify this and apply implicit differentiation and then we solve for y’…


Derivative involving absolute value

Derivative Involving Absolute Value

  • Recall that the ln function is undefined for negative numbers, so we often see expressions of the form ln|u|. So the following theorem states that we can differentiate functions of the form

    y= ln|u| as if the absolute value symbol is not even there.

  • If u is a differentiable function such that u≠0 then:


Derivative involving absolute value1

Derivative Involving Absolute Value

  • Differentiate:


Finding relative extrema

Finding Relative Extrema

  • Locate the relative extrema of

  • Differentiate:

  • Set = 0 to find critical points

    =0

    2x+2=0

    X=-1, Plug back into original to find y

    y=ln(1-2+3)=ln2 So, relative extrema is at (-1, ln2)


Homework

Homework

  • 5.1 Natural Logarithmic Functions and the Number e Derivative #19-35,47-65, 71,79,93-96


General power rule for integration

General Power Rule for Integration

  • Recall that it has an important disclaimer- it doesn’t apply when n = -1. So we can not integrate functions such as f(x)=1/x.

  • So we use the Second FTC to DEFINE such a function.


Integration formulas

Integration Formulas

  • Let u be a differentiable function of x


Using the log rule for integration

Using the Log Rule for Integration


Using the log rule with a change of variables

Using the Log Rule with a Change of Variables

Let u=4x-1, so du=4dx and dx=


Finding area with the log rule

Finding Area with the Log Rule

  • Find the area of the region bounded by the graph of y, the x-axis and the line x=3.


Recognizing quotient forms of the log rule

Recognizing Quotient Forms of the Log Rule


Definition

Definition

The natural logarithmic function is defined by

The domain of the natural logarithmic function is the set of all positive real numbers


U substitution and the log rule

u-Substitution and the Log Rule


Long division with integrals

Long Division With Integrals


How you know it s long division

How you know it’s long Division

  • If it is top heavy that means it is long division.

    • Example


Example 1

Example 1


Continue example 1

Continue Example 1


Example 2

Example 2


Continue example 2

Continue Example 2


Using long division before integrating

Using Long Division Before Integrating


Using a trigonometric identity

Using a Trigonometric Identity


Guidelines for integration

Guidelines for integration

  • Learn a basic list of integration formulas. (including those given in this section, you now have 12 formulas: the Power Rule, the Log Rule, and ten trigonometric rules. By the end of section 5.7 , this list will have expanded to 20 basic rules)

  • Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula.

  • If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative.

  • If you have access to computer software that will find antiderivatives symbolically, use it.


Integrals of the six basic trigonometric functions

Integrals of the Six Basic Trigonometric Functions


Homework1

Homework

  • 5.2 Log Rule for Integration and Integrals for Trig Functions (substitution)

    #1-39, 47-53,  67


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