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## DIFFERENTIAL CALCULUS

by: Mrs. Dedric Little

24

0

3

# DIFFERENTIAL CALCULUS MTH 251H

Mrs. Dedric Little
OSU
GPA 3.79

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 3 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 251H at Oregon State University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/224448/mth-251h-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15
Review Exponential and Logarithmic Functions Whose smarter you or your calculator What is a anyway How about loga 90 Your calculator seems to know and so should you Here is a summary of some facts you must have at your nger tips Exponential Functions some things old and some things new A general exponential function has the form fxaquot fooltacltoo where a gt 0 is a xed real number called its base Laws of Exponents for ab gt 0 and ac and y any real numbers 1 axy away axiy 0 aim 7 ay a any 7 an 11 0 1 a i a a i a 7 a abx ambquot lt7 b b a for ac Irrational ln algebra you learned what amquot means for any rational number exponent m n What did you learn Once amquot is de ned for rational exponents some form of a limiting argument that transcends algebra is needed to give meaning to the expression a for 90 an irrational number What is an irrational number One avenue is to use the following de nition De nition For so irrational a lim aT 7 rational 7 41 where the limit is taken through rational numbers 7 that converge to x Properties of a for a gt 1 The function f a has domain the interval 7007 00 has range the interval 0 00 is increasing and continuous a a 00 as so a 00 a a 0 as so a 700 Stop Sketch a graph of a that has these properties Compare with calculator generated graphs for 2quot 7139 and 10quot Logaritth Fix a base a gt 0 Let x be a positive real number You may have learned the following de nition the logarithm of so with base a denoted logaac is the exponent to which a must be raised to equal 90 aloga an This amounts to saying that logax is the inverse function of the exponential functions That is each of these functions reverses the action of the other alogaquot so for any so gt 0 loga a for any real number x Equivalently yaquot ltgt 9010ng The laws of exponents imply the following laws for logarithms Laws of Logaritth for a gt 0 u and 1 any positive real numbers and p any real number ix u loga m loga u logav log 7 logau 7 log L a 7 11 leg up ploga u log 1 0 Properties of logaac for a gt 1 The function f log 90 has domain the interval 0 00 has range the interval 700 00 is increasing and continuous logaac a 700 as so a 0 logaxaoo asxaoo For the record there is a special base 6 an irrational in fact a transcendental number whose decimal expansion begins 6 2718281828459045235360287 that is especially useful in calculus It so useful that the exponential function e is programmed into every scienti c calculator Stay tuned

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