2 . 1 RAT E S O F C H A N G E A N D L I M I T S
LIMITS
AVERAGE SPEED=DISTANCE/TIME
• A rock breaks loose from the top of a tall cliff. What is the average speed during the first 2 seconds of fall?• Need to know:
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• Find the average speed from 2 seconds to slightly past 2 seconds…
INSTANTANEOUS RATE OF CHANGE
𝛥𝑦𝛥𝑥
=𝑓 (2+h )− 𝑓 (2 )
(2+h )−2
¿16 (2+h )2−16 (2 )2
2+h−2
¿16 (4+4 h+h2 )−16 (4 )
h
¿ 64+64 h+16h2−64
h
¿h (64+16h )
h=64+16h
lim𝑡→0
(64+16h )=64
TYPES OF LIMITS
• Graph.
DEFINITION OF A LIMIT
• Let a function “f” be defined on an open interval containing “a” except possibly itself. The statement “limit as x approaches a in the function is equal to k” means that f(x) is approaching the y-value of “k” as x gets sufficiently close to “a”.
BEHAVIOR OF A LIMIT
• Show graphically, numerically, and analytically• Find f(2) and f(4)• Use substitution first:• Indeterminate form: • Factor and simplify• Graph equation (show removable discontinuity)• Show table (count by .001)• Show analytically by substituting x=4 into simplified equation
PROPERTIES OF LIMITS
lim𝑥→𝑎
𝑐 (constant )=𝑐
lim𝑥→𝑎
𝑥=𝑥
¿
USE SUBSTITUTION FIRST!!!USE FACTORING SECOND
SIMPLIFY THIRD
1. lim𝑥→1
4 𝑥3−3 𝑥2+5𝑥+72.lim𝑥→ 9
√𝑥−3𝑥−9
(hint : let u =√𝑥 )
3.lim𝑥→5
1𝑥−15
𝑥−54.lim𝑥→ 2
𝑥2+2𝑥+4
𝑥+2
DISCOVER
• Show graph…show numerically…compare answers
USE PRODUCT RULE
• Homework: 1-9
LEFT AND RIGHT LIMITS
• means the limit is approaching the x-value “c” from the left
• means the limit is approaching the x-value “c” from the right
• means the limit is approaching the x-value “c” from the left and the right
• only exists if =
FIND THE LIMITS
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Homework : Interact Math.com lesson 2.1
