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Understanding Natural Exponential Functions in Calculus II

Dive into the concepts and properties of natural exponential functions in this engaging section of Calculus II. This resource covers key definitions, limits at infinities, and the relationships between exponential functions and their inverses. Students will find examples and exercises designed to reinforce understanding and application of the material. With 33 problems to solve, this assignment encourages early preparation and thorough review of previous sections for a comprehensive learning experience.

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Understanding Natural Exponential Functions in Calculus II

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  1. MAT 1235Calculus II Section 6.3* The Natural Exponential Functions http://myhome.spu.edu/lauw

  2. Homework and … • WebAssign HW 6.3* • 33 problems -- most of them are straight forward • Due Wednesday • Be sure to start early

  3. Preview

  4. Recall • 6.1 • 6.2*

  5. Definition (6.2*) • Let be the number such that

  6. Definition (6.2*) • Let be the number such that • ………

  7. Definition (Inverse of ) Let be the inverse function of In particular, we have

  8. Then,… , i.e. , i.e. , where is an rational number, we get

  9. Then,… , i.e. , i.e. , where is an rational number, we get

  10. So,… • and give the same values for rational numbers • However, is undefined for irrational number (not yet!)

  11. It makes sense to define…

  12. Properties

  13. Properties (limits at infinities)

  14. Properties (limits at infinities)

  15. Example 1 (a)

  16. Example 1 (a)

  17. Example 1 (b)

  18. Example 1 (c)

  19. Remarks • ∞ is not a number, it is a concept. • In particular, you should not write: • Again, you can blame whoever told you that this is ok. But it is your responsibility to do the right thing from now on.

  20. Derivatives & Antiderivatives

  21. Derivatives & Antiderivatives

  22. Example 2

  23. Example 3

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