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Understanding Natural Logarithmic Functions and Their Differentiation in AP Calculus AB

This section delves into Natural Logarithmic Functions, focusing on their properties and differentiation techniques. We explore the definition of the natural logarithmic function, its domain, and range, emphasizing its behavior on graphs. Key theorems outline the critical properties of logarithms, including concavity and continuity. We'll also examine the base 'e' and its role as an inverse of ln. Finally, the section guides how to apply logarithmic properties to differentiate functions effectively, aiding students in mastering calculus concepts.

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Understanding Natural Logarithmic Functions and Their Differentiation in AP Calculus AB

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  1. AP Calculus ABChapter 5, Section 1 Natural Logarithmic Functions: Differentiation 2013 - 2014

  2. The Natural Logarithmic Function • This about what you know about the integral power rule. • One important disclaimer: it doesn’t apply when n = -1. • What would happen if n = -1??

  3. 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.

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

  5. 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?????

  6. Theorem: Logarithmic Properties • If a and b are positive numbers and n is rational, then the following properties are true:

  7. Expanding Logarithmic Expressions

  8. 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

  9. Definition of e • The letter e denotes the positive real number such that

  10. Evaluating Natural Logarithmic Expressions

  11. The Derivative of the Natural Logarithmic Function • Let u be a differential function of x

  12. Differentiation of Logarithmic Functions

  13. Differentiation of Logarithmic Functions

  14. Differentiation of Logarithmic Functions

  15. Differentiation of Logarithmic Functions

  16. Logarithmic Properties as Aids to Differentiation • Differentiate:

  17. Logarithmic Properties as Aids to Differentiation • Differentiate:

  18. Logarithmic Differentiation • Find the derivative of

  19. Derivative Involving Absolute Value • If u is a differentiable function of x such that , then

  20. Derivative Involving Absolute Value • Find the derivative of

  21. Finding Relative Extrema • Locate the relative extrema of

  22. Ch 5.1 Homework • Pg 329 – 330, #’s: 7 – 10, 15, 21, 27, 29, 33, 41, 49, 55, 61, 71, 75, 79

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