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Finding relative extrema (first derivative test) | AP Calculus AB | Khan Academy

Video vocabulary

into

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US / 'ɪntə /

UK / ˈɪntu /

A1
  • preposition
  • Moving or going inside something
  • In the same direction as; in the direction of
  • So as to become; taking the state of

other

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US / 'ʌðə(r) /

UK / ˈʌðər /

A1
  • adjecitve
  • A thing different from that mentioned
  • determiner
  • (Something) else; not the first (one)
  • pronoun
  • Being the one and only thing of that category

than

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US / ðən /

UK / ðən /

A1
  • conjunction
  • Used to compare two things that are not equal

let

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US / let /

UK / lɛt /

A1
  • verb
  • To allow someone to do something
  • Introducing a suggestion to do something together
  • To rent a house, etc. to others

the

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US / ðə /

UK / ðə /

A1
  • article
  • Used to refer to something already mentioned
  • Used to show there is only one of something
  • determiner
  • Used to describe all of a family

less

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US / les /

UK / lɛs /

A1
  • adjecitve
  • Smaller in amount or degree
  • adverb
  • To a smaller degree
  • noun
  • A smaller amount
  • preposition
  • With the subtraction of

this

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US / ðɪs /

UK / ðɪs /

A1
  • adjecitve
  • Used to indicate something already been discussed
  • Used to indicate something happening around now
  • adverb
  • An indication by a physical motion or gesture
  • A specific amount of
  • determiner
  • Person, thing, or idea near you
  • pronoun
  • Person, thing that has already been discussed

order

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US / 'ɔ:də(r) /

UK / ˈɔrdɚ /

A1
  • noun
  • Religious or social group, as of nuns
  • Instructions to others to do what you want
  • State in which things are neat and organized
  • What you ask to buy, e.g. food in restaurant
  • Social or political control of an area
  • verb
  • To put things in a sequence, e.g. a, b, c
  • To give instructions to others to do what you want
  • To ask for something you want to buy, e.g. a meal

make

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US / meɪk /

UK / mek /

A1
  • noun
  • Product made by a particular company; brand
  • verb
  • To arrange or prepare something e.g. dinner
  • To get to (a place) on time
  • To gain the status of (teacher, lawyer, etc.)
  • To create something by putting things together
  • To earn a certain amount of money at a job
  • To cause or force a person to do something
  • To cause something to happen or be formed

critical

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US / ˈkrɪtɪkl /

UK / ˈkrɪtɪkəl /

A2
  • adjecitve
  • Making a negative judgment of something
  • Being important or serious; vital; dangerous
  • Most important part

identify

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US / aɪ'dentɪfaɪ /

UK / aɪˈdɛntəˌfaɪ /

B1
  • verb
  • To indicate who or what someone or something is

maximum

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US / ˈmæksɪməm /

UK / ˈmæksəməm /

A2
  • noun
  • Largest or highest number or amount of something

you

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US / jʊ /

UK / ju /

A1
  • pronoun
  • Person someone is speaking or writing to
  • Person or people in general

meet

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US / mi:t /

UK / mit /

A1
  • noun
  • Sporting competition e.g. swimming
  • verb
  • To provide something that is necessary
  • To come together at a certain time or place

come

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US / kʌm /

UK / kʌm /

A1
  • verb
  • To arrive at a place
  • (Of mail) to be delivered
  • To move toward someone; go with someone
  • To reach a certain point or state

local

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US / ˈləʊkl /

UK / ˈloʊkl /

A2
  • adjecitve
  • Relating to a particular area, city, or town
  • noun
  • Bus or train, that stops at most stations
  • People who live in the area being discussed
  • The pub you most frequently go to

particular

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US / pəˈtɪkjələ(r) /

UK / pɚˈtɪkjəlɚ, pəˈtɪk- /

B2
  • adjecitve
  • One specific (one)
  • Having firm ideas about what is desirable
  • Being special or more than is typical

think

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US / θɪŋk /

UK / θɪŋk /

A1
  • verb
  • To have an idea about something without certainty
  • To have an idea, opinion or belief about something

from

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US / frəm /

UK / frʌm, frɑm,frəm /

A1
  • preposition
  • Starting at a particular place, time or level

out

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US / aʊt /

UK / aʊt /

A1
  • adverb
  • No longer stylish (e.g. fashion)
  • Away from home, business, or the usual place
  • In a direction/movement away from inside or center
  • noun
  • Act when a batter cannot continue playing
  • verb
  • To tell people a secret

over

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US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

another

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US / əˈnʌðə(r) /

UK / əˈnʌðɚ /

A1
  • adjecitve
  • One more, but not this
  • determiner
  • One more added
  • pronoun
  • One more (thing)

criterion

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US / kraɪ'tɪərɪən /

UK / kraɪˈtɪriən /

B1
  • noun
  • Standard used to make judgments or decisions

keep

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US / ki:p /

UK / kip /

A1
  • noun
  • Large, strong tower in the middle of a castle
  • Money spent on you so you can live in a place
  • verb
  • To continue doing something; not to stop
  • To stop balls going in a goal in sports
  • To not give back or return something
  • (Of food) to not spoil or not rot
  • To take possession of something to own, e.g. a pet
  • To have and continue to hold something
  • To do something that you have promised or agreed
  • To stay in a certain place or condition
  • To maintain the rules

call

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US / kɔ:l /

UK / kɔl /

A1
  • noun
  • A order or request for action
  • The sound an animal makes, often when in danger
  • A correct prediction, e.g. a decision in a game
  • A short visit to a place or person
  • A decision by a referee or umpire in a game
  • verb
  • To make a request or order for action
  • (Animal) to make sounds to warn of danger
  • To announce the date of an election
  • To make a judgment about something
  • To give someone or something a name
  • To guess in advance the result or outcome
  • To visit a place or person for a short time
  • To phone someone
  • To speak, announce or shout in a loud voice

cross

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US / krɒs /

UK / krɔs, krɑs /

A2
  • adjecitve
  • Angry and upset
  • From one side of something to the other
  • noun
  • (Of animals) mixture of breeds in one animal
  • Person's name
  • verb
  • To put one thing over another e.g. arms
  • To breed one type of animal or plant with another
  • To not to agree with someone; oppose
  • To go from one side of something to the other
  • To meet at one point

far

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US / fɑ:(r) /

UK / fɑr /

A1
  • adjecitve
  • A long way away
  • Being a long way away
  • adverb
  • To a large degree or extent

where

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US / hwɛə /

UK / hwɛr, wɛr /

A1
  • conjunction
  • At or in the place which
  • In whatever place
  • pronoun
  • Word you use to ask the location of something

and

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US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

both

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US / bəuθ /

UK / boθ /

A1
  • determiner
  • Used to refer to two things at the same time
  • pronoun
  • Used to refer to two things at the same time

change

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US / tʃeɪndʒ /

UK / tʃendʒ /

A1
  • noun
  • Exchange of one set of clothes for another
  • Money in the form of coins instead of paper
  • Money returned after giving too much
  • Act of making or becoming something else
  • verb
  • To exchange one set of clothes for another
  • To exchange one kind of money for another
  • To replace something with another thing
  • To make or become something else
  • To go from one train, bus, etc. and go to another

approach

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US / ə'prəʊtʃ /

UK / əˈprəʊtʃ /

A2
  • noun
  • Means of reaching a place, often a road or path
  • Request of someone with a specific goal in mind
  • Specific way to handle a project, task, problem
  • verb
  • To get close to reaching something or somewhere
  • To request someone to do something specific

take

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US / teɪk /

UK / tek /

A1
  • noun
  • Filming of a single scene for TV or the movies
  • verb
  • To accept something that is offered
  • To use a method of traveling, e.g. a bus or train
  • To use a camera to make (a photo)
  • To pick up something and go away with it
  • To study a subject in school
  • To attempt to pass (an exam)
  • To require a certain amount of time, money, space

derivative

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US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

B1
  • adjecitve
  • Not from original ideas; from somewhere else

look

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US / lʊk /

UK / lʊk /

A1
  • noun
  • Person's beauty or physical attractiveness
  • Particular style of dress, clothes, or appearance
  • The way someone, something appears to be
  • Act of searching for or examining something
  • verb
  • To appear to be; seem

sub

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US / sʌb /

UK / sʌb /

B2
  • prefix
  • A prefix meaning 'under' or 'instead of'

pretty

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US / 'prɪtɪ /

UK / ˈprɪti /

A1
  • adjecitve
  • Being attractive to the eye in a simple way
  • adverb
  • Very; to a great degree

figure

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US / ˈfiɡə /

UK / ˈfɪɡjɚ /

A1
  • noun
  • Your body shape
  • Numbers in a calculation
  • Doll-like thing meant to represent a person
  • Picture or diagram giving information in a text
  • Person who is very important or famous
  • Shape of a person seen indistinctly or in outline
  • Amount that is expressed in numbers
  • verb
  • To appear in a game, play or event
  • To calculate how much something will cost
  • To understand after thinking; work out

way

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US / weɪ /

UK / we /

A1
  • adverb
  • Used to emphasize the amount of something
  • noun
  • Particular aspect of something being thought about
  • One of various choices that could be made
  • Particular existing condition or state
  • (Often long) distance or length of time
  • How something is done
  • Route to go to a place
  • Small street
  • Usual behavior, emotions or habits of someone

true

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US / tru: /

UK / tru: /

A1
  • adjecitve
  • Agreeing with the facts; not false; real or actual

past

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US / pɑ:st /

UK / pæst /

A2
  • adjecitve
  • No longer current or new
  • adverb
  • So as to go beyond, or further than something
  • noun
  • Time that existed before now, e.g. long ago
  • preposition
  • After a particular time
  • At a later point in distance; beyond

somehow

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US / ˈsʌmhaʊ /

UK / ˈsʌmˌhaʊ /

A1
  • adverb
  • In a way that is not known or certain

exactly

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US / ɪgˈzæktli /

UK / ɪɡˈzæktli /

A1
  • adverb
  • No more and no less than; precisely

either

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US / ˈaɪðə(r) /

UK / ˈiðɚ, ˈaɪðɚ /

A1
  • adjecitve
  • One of two (used when there is a choice of two)
  • adverb
  • (After negative statements) likewise, in addition
  • conjunction
  • Used as the first of two connected things
  • determiner
  • Used as the first of two connected things
  • pronoun
  • The one or the other

here

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US / hɪə(r) /

UK / hɪr /

A1
  • adverb
  • In this place; at this location, position or time

try

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US / traɪ /

UK / traɪ /

A1
  • noun
  • Act of putting the ball over the rugby goal line
  • verb
  • To make an effort, to attempt to do something
  • To examine someone or something in a court of law
  • To test to see if it is good or you like it
  • To make a person go through hardship; test
  • To examine something to see if it works

once

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US / wʌns /

UK / wʌns /

A1
  • adverb
  • One time; one instance
  • At a time in the past
  • conjunction
  • When; as soon as

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

A1
  • adverb
  • Used to add new (often different) information

after

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US / 'ɑ:ftə(r) /

UK / ˈæftɚ /

A1
  • adverb
  • Later than another time, or behind something
  • conjunction
  • Following something before
  • preposition
  • When an event happened or time has passed

point

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US / pɔɪnt /

UK / pɔɪnt /

A1
  • noun
  • An item to be discussed
  • Small spot or dot
  • Fact or detail
  • Feature of something
  • The size of the text on a page or screen
  • A certain time or stage in a process
  • Certain position in time or space
  • Idea or fact you try to convince people about
  • Measurement of scores in a game, sport
  • Sharp end of something, e.g. a pen or pin
  • verb
  • To face a certain direction, e.g. north
  • To indicate something with your finger to others

sure

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US / ʃʊə(r) /

UK / ʃʊr /

A1
  • adjecitve
  • Having no doubts about something
  • Have confidence in your own words or actions
  • Firmly established or held; being firm or stable
  • Being correct or true
  • adverb
  • What you say to agree willingly

switch

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US / swɪtʃ /

UK / swɪtʃ /

A2
  • noun
  • A change or shift from one method to another
  • Thing you move to turn electrical items on or off
  • Thin flexible piece of wood or stick
  • verb
  • To exchange or replace something for another thing

can

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US / kæn /

UK / kən,kæn /

A1
  • noun
  • A metal container for storing food
  • other
  • Able to; have the skill to; have the time to
  • verb
  • To have permission to
  • To be possible to do
  • To preserve food by sealing it in a metal container

minimum

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US / ˈmɪnɪməm /

UK / ˈmɪnəməm /

B1
  • noun
  • Smallest degree or level that is acceptable

whether

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US / ˈweðə(r) /

UK / ˈhwɛðɚ, ˈwɛð- /

A1
  • conjunction
  • If something will happen or not

become

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US / bɪ'kʌm /

UK / bɪˈkʌm /

A1
  • verb
  • To begin to be; grow to be; develop into
  • To be suitable for a particular job or role

naught

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US / nɔ:t /

UK / nɔt /

B2
  • noun
  • Quantity of no importance, nonexistence

min

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US / mɪn /

UK / mɪn /

A1
  • noun
  • Abbreviation for minute

same

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US / seɪm /

UK / sem /

A1
  • adjecitve
  • Being equal in quality and amount
  • Identical; not different
  • Not changing; constant

not

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US / nɒt /

UK / nɑt /

A1
  • adverb
  • Word indicating the negative
  • (Used to form the negative of verbs)

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

whole

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US / həul /

UK / hol /

A1
  • adjecitve
  • Complete or full; all of
  • Being in a good condition
  • noun
  • Complete amount or sum; total

but

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US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

especially

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US / ɪˈspeʃəli /

UK / ɛˈspɛʃəli, ɪˈspɛʃ- /

A1
  • adverb
  • More than usual; extremely

good

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US / ɡʊd /

UK / ɡʊd /

A1
  • adjecitve
  • Proper, appropriate or right
  • (Of an amount) enough; plenty
  • Excellent; high quality
  • Acting well or properly; of moral character
  • In a positive or happy emotional state
  • noun
  • Advantage or benefit

involve

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US / ɪn'vɒlv /

UK / ɪnˈvɑlv /

A2
  • verb
  • To have or be included as a part of something

would

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US / wʊd /

UK / wʊd /

A1
  • other
  • Used to indicate something imagined
  • Used to ask or request something politely
  • Verb used to indicate possibility, request etc.
  • Used to indicate a preference
  • Used to indicate a reaction
  • Indicates something that occurred repeatedly
  • Used to indicate willingness or ability to do

video

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US / 'vɪdɪəʊ /

UK / ˈvɪdiˌo /

A2
  • adjecitve
  • Of a moving film, with sound
  • noun
  • Recording capturing action with sound
  • verb
  • To make a moving film of, with sound

equal

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US / 'i:kwəl /

UK / ˈikwəl /

A1
  • adjecitve
  • Same in shape, size, or number
  • verb
  • To add up or be the same as

now

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US / naʊ /

UK / naʊ /

A1
  • adverb
  • At the present time or moment
  • Right away or in the next possible moment
  • Used to refer to something that just happened
  • Time between the present and a moment in the past
  • conjunction
  • Resulting from or because of something

just

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US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

definitely

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US / ˈdefɪnətli /

UK / ˈdefɪnətli /

A2
  • adverb
  • Without question; beyond doubt

slope

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US / sləʊp /

UK / sloʊp /

B1
  • noun
  • Angle of a decline, e.g. where a road dips down
  • Surface on which people can ski
  • verb
  • To have one end that is not as high as the other

necessarily

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US / ˌnesəˈserəli /

UK / ˌnɛsɪˈsɛrəli, -ˈsɛr- /

A2
  • adverb
  • In a way that is needed/required/is unavoidable

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something

time

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US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

start

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US / stɑ:t /

UK / stɑrt /

A1
  • noun
  • First time or place that a thing exists; beginning
  • First opportunity to achieve something, e.g. a job
  • Sudden action or movement because you are scared
  • other
  • Beginning of something in place or time
  • verb
  • To do, be or happen for the first time; begin
  • To turn something on

well

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US / wel /

UK / wɛl /

A1
  • adjecitve
  • Being in a fortunate, good condition or state
  • Healthy and in good condition
  • adverb
  • In an acceptable, proper, or successful manner
  • In a complete, total, or full manner
  • A great and considerable degree
  • In a successful way, in a good or satisfactory way
  • exclamation
  • Used to begin or continue a particular statement
  • Used to show you want to start saying something
  • Used to express that you are waiting for a reply
  • You say this to change the topic of a conversation
  • Used to indicate you are making a final remark
  • noun
  • Long deep hole in the ground to get water
  • verb
  • To cause your eyes to fill with tears

undefined

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US / ˌʌndɪˈfaɪnd /

UK / ˌʌndɪˈfaɪnd /

B1
  • adjecitve
  • Not precisely limited or described

max

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US / mæks /

UK / mæks /

A2
  • noun
  • Person's name
  • Short for 'maximum'

green

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US / gri:n /

UK / ɡrin /

A1
  • adjecitve
  • Color of young leaves
  • Covered with vegetation, e.g. grass
  • Concerned about protecting the environment
  • The color of young leaves
  • Lacking in experience/knowledge due to being young
  • Being not fully grown or ripe, as in fruit
  • noun
  • Large grassy area
  • used as a part of a name of a place (UK)

visually

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US / 'vɪʒʊəlɪ /

UK / ˈvɪʒʊəlɪ /

B1
  • adverb
  • With respect to vision

positive

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US / ˈpɒzətɪv /

UK / ˈpɑzɪtɪv /

A2
  • adjecitve
  • Showing agreement or support for something
  • Being sure about something; knowing the truth
  • Having the charge produced by electrons
  • Being good or useful
  • Showing that a chemical is present in a test

could

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US / kəd /

UK / kʊd /

A1
  • other
  • Past form of 'can' to mean have ability
  • Used to ask for something politely, or to offer

increase

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US / ɪn'kri:s /

UK / ɪnˈkris /

A1
  • noun
  • Fact of increasing; amount something grows by
  • verb
  • To make or become larger in size or amount

have

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US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

last

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US / lɑ:st /

UK / læst /

A1
  • adjecitve
  • Most recent; the latest
  • The one before
  • adverb
  • Final; with nothing following; after all others
  • Most recent; the latest
  • determiner
  • The one at the end
  • verb
  • To remain or continue for a certain period of time

explore

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US / ɪk'splɔ:(r) /

UK / ɪkˈsplɔr /

A2
  • verb
  • To examine something in detail to learn about it
  • To travel to a place to discover more about it

prime

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US / praɪm /

UK / praɪm /

A2
  • adjecitve
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A2
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  • Subtitles section
  • In the last video we saw that if a function takes on a minimum or maximum value, min max value for our function at x equals a, then a is a critical point.
  • But then we saw that the other way around isn't necessarily true. x equal a being a critical point does not necessarily mean that the function takes on a minimum or maximum value at that point.
  • So what we're going to try to do this video is try to come up with some criteria, especially involving the derivative of the function around x equals a, to figure out if it is a minimum or a maximum point.
  • So let's look at what we saw in the last video.
  • We saw that this point right over here is where the function takes on a maximum value.
  • So this critical point in particular was x naught.
  • What made it a critical point was that the derivative is 0.
  • You have a critical point where either the derivative is 0 or the derivative is undefined.
  • So this is a critical point.
  • And let's explore what the derivative is doing as we approach that point.
  • So in order for this to be a maximum point, the function is increasing as we approach it.
  • The function is increasing is another way of saying that the slope is positive.
  • The slope is changing but it stays positive the whole time which means that the function is increasing.
  • And the slope being positive is another way of saying that the derivative is greater than 0 as we approach that point.
  • Now what happens as we pass that point?
  • Right at that point the slope is 0.
  • But then as we past that point, what has to happen in order for that to be a maximum point?
  • Well the value of the function has to go down.
  • If the value of the function is going down, that means the slope is negative.
  • And that's so another way of saying that the derivative is negative.
  • So that seems like a pretty good criteria for identifying whether a critical point is a maximum point.
  • So let's say that we have critical point a.
  • We are at a maximum point if f prime of x switches signs from positive to negative as we cross x equals a.
  • That's exactly what happened right over here.
  • Let's make sure it happened at our other maximum point right over here.
  • So right over here, as we approach that point the function is increasing.
  • The function increasing means that the slope is positive.
  • It's a different positive slope.
  • The slope is changing, it's actually getting more, and more, and more steep.
  • Or more, and more, and more positive.
  • But is definitely positive.
  • So it's positive going into that point.
  • And then it becomes negative after we cross that point.
  • The slope was undefined right at the point.
  • But it did switch signs from positive to negative as we crossed that critical point.
  • So these both meet our criteria for being a maximum point.
  • So, so far our criteria seems pretty good.
  • Now let's make sure that somehow this point right over here, which we identified in the last video as a critical point, let's make-- and I think we called this x0, this was x1, this was x2.
  • So this is x3.
  • Let's make sure that this doesn't somehow meet the criteria because we see visually that this is not a maximum point.
  • So as we approach this, our slope is negative.
  • And then as we cross it, our slope is still negative, we're still decreasing.
  • So we haven't switched signs.
  • So this does not meet our criteria, which is good.
  • Now let's come up with the criteria for a minimum point.
  • And I think you could see where this is likely to go.
  • Well we identified in the last video that this right over here is a minimum point.
  • We can see that.
  • It's a local minimum just by looking at it.
  • And what's the slope of doing as we approach it?
  • So the function is decreasing, the slope is negative as we approach it.
  • f prime of x is less than 0 as we approach that point.
  • And then right after we cross it-- this wouldn't be a minimum point if the function were to keep decreasing somehow.
  • The function needs to increase now.
  • So let me do that same green.
  • So right after that, the function starts increasing again.
  • f prime of x is greater than 0.
  • So this seems like pretty good criteria for a minimum point.
  • f prime of x switches signs from negative to positive as we cross a.
  • If we have some critical point a, the function takes on a minimum value at a if the derivative of our function switches signs from negative to positive as we cross a, from negative to positive.
  • Now once again, this point right over here, this critical point x sub 3 does not meet that criteria.
  • We go from negative to 0 right at that point then go to negative again.
  • So this is not a minimum or a maximum point.