AP Calculus AB Chapter 5, Section 1

AP Calculus ABChapter 5, Section 1

Natural Logarithmic Functions: Differentiation

2013 - 2014

The Natural Logarithmic Function

• Evaluate

The Natural Logarithmic Function

• Definition of the Natural Logarithmic Function:

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

The Number e

• The number e is the base of ln. • e and ln are inverses of each other.• In the equation , the value of x to make this

statement true is e. • e is irrational and has the decimal

approximation

Definition of e

• The letter e denotes the positive real number such that

Let’s look at the graph• Set your window to [-1, 10] by [-5, 5]• Graph in and sketch below.

Theorem: Properties of the Natural Logarithmic Function

• The natural logarithmic function has the following properties:– The domain is (0, ∞) and the range is (-∞, ∞).– The function is continuous, increasing, and one-to-

one.– The graph is concave downward.

• Do you remember how we check for concavity?????

Theorem: Logarithmic Properties

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

Expanding Logarithmic Expressions

ln(𝑥2+3 )23√𝑥2+1

Condensing Logarithmic Expressions

The Derivative of the Natural Logarithmic Function

• Let u be a differential function of x

Differentiation of Logarithmic Functions

𝑑𝑑𝑥

[ ln(2 𝑥) ]

Differentiation of Logarithmic Functions

𝑑𝑑𝑥

[ ln √𝑥+1 ]

Differentiation of Logarithmic Functions

𝑦=ln (ln𝑥 ) , 𝑓𝑖𝑛𝑑 𝑦 ′

Differentiation of Logarithmic Functions

𝑑𝑑𝑥

[ (ln 𝑥 )3 ]

Derivative Involving Absolute Value

• If u is a differentiable function of x such that , then

Differentiation of Logarithmic Functions

𝑦=ln|cos 𝑥|, 𝑦 ′=¿

Logarithmic Properties as Aids to Differentiation

• Differentiate:

• Show that is a solution to the differential equation

Finding Relative Extrema

• Locate the relative extrema of

Ch 5.1 Homework

• Pg 329 – 330, #’s: 7 – 10, 15, 21, 27, 29, 33, 41, 49, 55, 61, 71, 75, 79