2.1RATES OF CHANGE AND LIMITS LIMITS. AVERAGE SPEED=DISTANCE/TIME A rock breaks loose from the top of a tall cliff. What is the average speed during the.

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