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Maximizing function at value

Video vocabulary

max

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US / mæks /

UK / mæks /

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

necessarily

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US / ˌnesəˈserəli /

UK / ˌnɛsɪˈsɛrəli, -ˈsɛr- /

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

visually

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US / 'vɪʒʊəlɪ /

UK / ˈvɪʒʊəlɪ /

B1
  • adverb
  • With respect to vision

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

grow

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US / ɡrəʊ /

UK / ɡro /

A1
  • verb
  • To get bigger and more mature; make plants do this
  • To improve; to develop better skills over time
  • To develop and become bigger or taller over time

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

never

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US / 'nevə(r) /

UK / ˈnɛvɚ /

A1
  • adverb
  • Not ever; not at any time

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

pause

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US / pɔ:z /

UK / pɔz /

A2
  • noun
  • A temporary break before continuing
  • Function on a device to stop and restart it
  • Sign to hold a musical note a bit longer
  • verb
  • To stop doing for a while before continuing

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)

line

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US / laɪn /

UK / laɪn /

A1
  • noun
  • Border that divides two places or regions
  • Long crease in the skin, often from age
  • Mark that is long, straight and very thin
  • Row of words, letters or numbers that are written
  • Telephone connection
  • People waiting one behind the other; queue
  • Long thin wire, e.g. for electrical signal
  • verb
  • To put material along the inside of something

graph

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US / græf /

UK / ɡræf /

A2
  • noun
  • Image expressing numerical data in lines or shapes
  • verb
  • To show data using images

lower

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US / 'ləʊə(r) /

UK / ˈlaʊɚ, laʊr /

A2
  • adjecitve
  • In a position more near the bottom
  • verb
  • To move something downward to the ground

through

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US / θru: /

UK / θru /

A2
  • adjecitve
  • having finished using or doing something
  • adverb
  • From the beginning of something until the end
  • preposition
  • Over, in, across an entire thing or place
  • Allowing you to pass between, or to
  • By a particular way or process; by the use of

axe

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US / æks /

UK / æks /

B1
  • noun
  • Tool with a sharp metal edge used to chop wood
  • verb
  • To make someone lose their job

for

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US / fə(r) /

UK / fɔr,fə /

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

limit

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US / 'lɪmɪt /

UK / ˈlɪmɪt /

A2
  • noun
  • Point beyond which it is not possible to go
  • verb
  • To stop or prevent an increase past a point

large

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US / lɑ:dʒ /

UK / lɑ:rdʒ /

A1
  • adjecitve
  • Big; of great size; broad, tall, wide, or fat

anything

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US / ˈeniθɪŋ /

UK / ˈɛniˌθɪŋ /

A1
  • pronoun
  • A thing of any kind

minus

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US / ˈmaɪnəs /

UK / ˈmaɪnəs /

A2
  • adjecitve
  • Being close to the bottom of a letter grade
  • noun
  • Negative quality; disadvantage
  • Sign that indicates subtraction
  • preposition
  • Being less than zero in number
  • Used when subtracting one number from another
  • verb
  • To take one number away from another; subtract

help

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US / help /

UK / hɛlp /

A1
  • noun
  • Something given to enable a person to do something
  • Assistance given to someone in need
  • Someone who makes something easier or possible
  • verb
  • To act to enable a person to do something; assist
  • To make something better or less difficult

inaudible

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US / ɪnˈɔ:dəbl /

UK / ɪnˈɔdəbəl /

B2
  • adjecitve
  • Impossible to hear; imperceptible by the ear

any

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US / 'enɪ /

UK / ˈɛni /

A1
  • adjecitve
  • One, or some, no matter which
  • pronoun
  • One (thing) of many; some

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

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

invoke

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US / ɪn'vəʊk /

UK / ɪnˈvok /

B1
  • verb
  • To appeal for help/protection to higher authority
  • To use a law, concept, or right to achieve a goal

something

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US / ˈsʌmθɪŋ /

UK / ˈsʌmθɪŋ /

A1
  • adverb
  • In some degree or measure
  • noun
  • a person or thing of some value or consequence
  • pronoun
  • Thing that is not yet known or named
  • Approximately

two

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US / tu: /

UK / tu /

A1
  • number
  • 2

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

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

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

too

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US / tu: /

UK / tu /

A1
  • adverb
  • In addition, also; with emphasis
  • Say that something is more than you want

always

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US / 'ɔ:lweɪz /

UK / ˈɔlwez,-wɪz,-wiz /

A1
  • adverb
  • For a very long time
  • At all times; in every situation
  • For ever

should

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US / ʃʊd /

UK / ʃʊd /

A1
  • other
  • Used to indicate what is proper or reasonable

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

obviously

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US / ˈɒbviəsli /

UK / ˈɑːbviəsli /

A2
  • adverb
  • In a way that is obvious/easy to see or understand

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

here

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US / hɪə(r) /

UK / hɪr /

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

exactly

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US / ɪgˈzæktli /

UK / ɪɡˈzæktli /

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

will

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US / wɪl /

UK / wɪl /

A1
  • noun
  • Desire or choice of someone
  • Document saying who gets your money when you die
  • Strong desire to do something; determination
  • Desire to do something; strong wish to do
  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it

side

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US / saɪd /

UK / saɪd /

A1
  • adjecitve
  • In addition to the central or most important thing
  • Next to, or very close to, the right or left of
  • noun
  • Position or opinion that is opposite to another
  • Right or left part of a person's body
  • Feature of someone's personality
  • Straight edge of an object
  • One of two or more groups fighting each other
  • verb
  • To decide to agree with one point, not the other

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

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

encourage

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US / ɪn'kʌrɪdʒ /

UK / ɛnˈkɚrɪdʒ, -ˈkʌr- /

A2
  • verb
  • To increase the chance of something happening
  • To make someone more determined or confident

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

multiply

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US / ˈmʌltɪplaɪ /

UK / ˈmʌltəˌplaɪ /

B1
  • verb
  • To increase in number by reproducing/having babies
  • To increase something by a large quantity
  • To have increased by a large quantity
  • To calculate two numbers e.g. 2 x 4

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

they

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US / ðeɪ /

UK / ðe /

A1
  • pronoun
  • Two or more people, animals, or things

possible

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US / ˈpɒsəbl /

UK / ˈpɑsəbəl /

A1
  • adjecitve
  • Being able or appropriate for a task or purpose
  • Having a chance of happening, or being true

travel

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US / ˈtrævl /

UK / ˈtrævəl /

A1
  • noun
  • Going to other places, esp. distant places
  • verb
  • To go to a place that is far away

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

bit

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US / bɪt /

UK / bɪt /

A1
  • noun
  • Device put in a horse's mouth to control it
  • Small piece of something
  • verb
  • (E.g. of fish) to take bait and be caught

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

fast

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US / fɑ:st /

UK / fæst /

A1
  • adjecitve
  • In a way that is difficult to move or change
  • Moving or able to move quickly
  • Showing time that is later than the actual time
  • adverb
  • With a great deal of speed
  • noun
  • Period of time when a person chooses not to eat
  • verb
  • To refuse food for a long time

work

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US / wɜ:k /

UK / wɚk /

A1
  • adjecitve
  • Activity completed in order to make money
  • noun
  • The product of some artistic or literary endeavor
  • Everything created by an author, artist, musician
  • Activity you do in order to make money
  • Place of employment
  • Final product or result of a some labor or effort
  • verb
  • To bring into a specific state of success
  • To be functioning properly, e.g. a car
  • To do your job in your company or workplace
  • To make an effort to do something
  • To operate a machine or device

all

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US / ɔ:l /

UK / ɔl /

A1
  • adverb
  • Completely; totally
  • determiner
  • Being every one of something
  • pronoun
  • 100% of something; the complete amount

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

comma

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US / ˈkɒmə /

UK / ˈkɑ:mə /

B2
  • noun
  • Mark showing a pause or list in a sentence

little

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US / 'lɪtl /

UK / ˈlɪtl /

A1
  • adjecitve
  • Being short in length of time
  • Not great in size, amount, or degree; small
  • Being very young or younger than others
  • adverb
  • By an amount or degree that is not very much
  • Not often; rarely
  • noun
  • A person's name

you

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US / jʊ /

UK / ju /

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

rate

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US / reɪt /

UK / ret /

A1
  • noun
  • Current price or cost of something
  • Speed or frequency of events over time
  • verb
  • To assess something or consider its qualities

although

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US / ɔ:lˈðəu /

UK / ɔlˈðo /

A1
  • conjunction
  • Despite the fact that; however

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

than

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US / ðən /

UK / ðən /

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

might

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US / maɪt /

UK / maɪt /

A1
  • other
  • Used to politely request an action
  • Used to express that something could happen
  • Power, strength, force or influence

kind

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US / kaɪnd /

UK / kaɪnd /

A1
  • adjecitve
  • In a caring and helpful manner
  • friendly and considerate
  • noun
  • One type of thing

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

really

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US / ˈri:əli /

UK / ˈriəˌli, ˈrili /

A1
  • adverb
  • Used to stress something that is said; definitely
  • In a true or accurate manner
  • Very

visualize

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US / ˈvɪʒuəlaɪz /

UK / ˈvɪʒuəˌlaɪz /

B1
  • verb
  • To create a picture in your mind of something

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

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

give

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US / ɡɪv /

UK / ɡɪv /

A1
  • noun
  • Degree of flexibility in something, a material
  • verb
  • To hand over or present something to someone
  • To cause someone to have or experience something

scale

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US / skeɪl /

UK / skel /

A2
  • noun
  • Size, level, or amount when compared
  • Small hard plates that cover the body of fish
  • Device that is used to weigh a person or thing
  • Range of numbers from the lowest to the highest
  • Dimensions or size of something
  • verb
  • To change the size of but keep the proportions
  • To climb something large (e.g. a mountain)
  • To remove the scales of a fish

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

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

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

same

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US / seɪm /

UK / sem /

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

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

but

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US / bət /

UK / bʌt,bət /

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

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

not

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US / nɒt /

UK / nɑt /

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

what

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US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

get

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US / ɡet /

UK / ɡɛt /

A1
  • verb
  • To become affected by illness or disease
  • To begin to feel or understand an emotion or idea
  • To send or bring someone or something home, etc.
  • To go somewhere to obtain something
  • To (cause to) do a particular thing
  • To obtain, receive or be given something
  • To currently have
  • To prepare for use
  • To understand something being said or read

like

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US / laɪk /

UK / laɪk /

A1
  • adjecitve
  • Being equal in status or kind to something else
  • Similar; the same
  • adverb
  • For example
  • preposition
  • Used as a filler; used to ad emphasis
  • In a way similar to the way something is done
  • Such as; for example
  • verb
  • To want to
  • To find something pleasing; to prefer something

about

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US / ə'baʊt /

UK / əˈbaʊt /

A1
  • adverb
  • To be ready to; going to
  • Nearly; approximately; roughly
  • Near; next to; close to
  • In every direction; in the area of
  • preposition
  • Concerning or of (a topic)

right

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US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
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essentially

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instantaneous

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part

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theorem

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prime

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differentiable

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know

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  • Subtitles section
  • Let f be a differentiable function for all x.
  • If f of negative 2 is equal to 3 and f prime of x is less than or equal to 7 for all x, then what is the largest possible value of f of 10?
  • And so I encourage you to think about this on your own, pause the video, try to figure out the largest possible value for f of 10.
  • And then we'll work through it together.
  • So I'm assuming you've given a go at it.
  • So let's visualize this.
  • So let me draw some axes here.
  • So let's say that's my x-axis.
  • That's my x-axis right over there.
  • And this right over here is my y-axis.
  • That's my y-axis [INAUDIBLE] I'll graph y equals f of x.
  • And they tell us f of negative 2 is equal to 3.
  • And the two axes aren't going to be drawn to scale.
  • So let's say this is negative 2.
  • And this right over here is the point negative 2 comma 3.
  • And they tell us that f prime of x is a less than or equal to 7, that the instantaneous slope is always less than or equal to 7.
  • So really, the way to get the largest possible value of f-- we don't have to necessarily invoke the mean value theorem, although the mean value theorem will help us know for sure-- is to say well, look, the largest possible value of f of 10 is essentially if we max this thing out.
  • If we assume that the instantaneous rate of change just stays at the ceiling right at 7.
  • So if we assumed that our function, the fastest growing function here would be a line that has a slope exactly equal to 7.
  • So the slope of 7 would look-- and obviously, I'm not drawing this to scale.
  • Visually, this looks more like a slope of 1, but we'll just assume this is a slope of 7 because it's not at the same-- the x and y are not at the same scale.
  • So slope is equal to 7.
  • And so if our slope is equal to 7, where do we get to when x is equal to 10?
  • When x is equal to 10, which is right over here, well what's our change in x?
  • So what's our change in x?
  • Let's just think about it this way.
  • Our change in y over change in x is going to be what?
  • Well our change in y is going to be f of 10 minus f of 2.
  • f of 2 is 3, so minus 3, over our change in x.
  • Our change in x is 10 minus negative 2.
  • 10 minus negative 2 is going to be equal to 7.
  • This is the way to max out what our value of f of 10 might be.
  • If at any point the slope were anything less than that, because remember, the instantaneous rate of change can never be more than that.
  • So if we start off even a little bit lower, than the best we can do is get to that.
  • Remember, we can't do something like that.
  • That would get us too steep.
  • So it has to be like that.
  • And then we would get to a lower f of 10.
  • Every time you have a slightly lower rate of change, then it kind of limits what happens to you.
  • So remember, our slope can never be more than 7.
  • So this part should be parallel.
  • So this should be parallel to that right over there.
  • This should be parallel.
  • But we can never have a higher slope than that.
  • So the way to max it out is to actually have a slope of 7.
  • And so what is f of 10 going to be?
  • So let's see, 10 minus negative 2, that is 12.
  • Multiply both sides by 12, you get 84.
  • So f of 10 minus 3 is going to be equal to 84.
  • Or f of 10 is going to be equal to 87.
  • So if you have a slope of 7, the whole way, you travel 12.
  • That means you're going to increase by 84.
  • If you started at 3, you increase by 84, you're going to get to 87.