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2011 Calculus BC free response #3a | AP Calculus BC | Khan Academy

Video vocabulary

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

distance

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US / 'dɪstəns /

UK / ˈdɪstəns /

A2
  • noun
  • Lack of friendliness toward another person
  • Amount of space between two places or things
  • verb
  • To make it clear that you are not involved

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

finally

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US / ˈfaɪnəli /

UK / ˈfaɪnəli /

A1
  • adverb
  • used especially at the beginning of a sentence to introduce the last point or idea
  • After a long time or some difficulty

hurt

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US / hɜ:t /

UK / hɚt /

A1
  • noun
  • Mental, emotional, or physical pain
  • verb
  • To cause pain, damage or injury

real

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US / rɪəl /

UK / ˈriəl, ril /

A1
  • adjecitve
  • Actually existing or happening, not imagined
  • Being important or special

region

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US / 'ri:dʒən /

UK / ˈridʒən /

A2
  • noun
  • Part of a country, of the world, area, etc.

rest

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US / rest /

UK / rɛst /

A1
  • noun
  • Time when one relaxes, sleeps, or is inactive
  • State in which there is a lack of motion
  • Part of something remaining after some was used
  • verb
  • To stop working or avoid activity to recover
  • (Of dead person) to lie in a grave
  • To lean, support or place something against
  • To not be tired because you had enough sleep

screen

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US / skri:n /

UK / skrin /

A2
  • noun
  • Wire net in a door or window to keep insects out
  • Displays in a theater on which movies are shown
  • Flat surface on a computer or TV showing images
  • verb
  • To hide or block something from being seen
  • To examine to find something, e.g. best candidate

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

estate

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US / ɪˈsteɪt /

UK / ɪˈstet /

A2
  • noun
  • A large piece of land, usually with a large house
  • Group of houses all built at the same time

along

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US / əˈlɒŋ /

UK / əˈlɔ:ŋ /

A1
  • adverb
  • As a companion; taken with someone
  • preposition
  • At a point on a line

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

height

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US / haɪt /

UK / haɪt /

A2
  • noun
  • Distance of something from the bottom to the top

put

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US / pʊt /

UK / pʊt /

A1
  • phrasal verb
  • to make a light stop shining by pressing or moving a switch
  • verb
  • To move or place a thing in a particular position
  • To rate something in importance or value
  • To write or say something in a certain manner

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

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

term

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US / tɜ:m /

UK / tɚm /

A1
  • noun
  • Conditions applying to an agreement, contract
  • Length of time something is expected to happen
  • Fixed period of weeks for learning at school
  • The (precise) name given to something
  • verb
  • To call; give a name to

three

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US / θri: /

UK / θri /

A1
  • number
  • Number 3

big

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US / bɪɡ /

UK / bɪɡ /

A1
  • adjecitve
  • Popular
  • Serious
  • Large

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

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

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

one

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US / wʌn /

UK / wʌn /

A1
  • pronoun
  • You; we; people in general; someone
  • The thing or person previously referred to
  • adjecitve
  • Used to refer to people generally
  • Being a strong example of (something mentioned)
  • number
  • Number 1

root

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US / ru:t /

UK / rut, rʊt /

A2
  • noun
  • Emotional connection with a place, history etc.
  • The origin or starting place of something
  • A number that is multiplied by itself
  • Part of plant growing underground, as to get water
  • Part of tooth, hair etc. from which the rest grows
  • Base part of a word to which affixes can be added
  • verb
  • To cause a plant to develop or grow roots
  • (Animals) to search for e.g. food in earth
  • To stop someone moving, as through fear

than

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US / ðən /

UK / ðən /

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

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

how

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US / haʊ /

UK / haʊ /

A1
  • adjecitve
  • To what degree? in what amount?
  • adverb
  • (Used to explain the way to do something)
  • exclamation
  • For what purpose or reason?
  • (Used to ask someone or something's condition)

vertical

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US / ˈvɜ:tɪkl /

UK / ˈvɜ:rtɪkl /

B1
  • adjecitve
  • (Of a line, etc.) going up and down; not horizontal

year

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US / jə: /

UK / jɪr /

A1
  • noun
  • Unit of time equal to 12 months or 365 or 366 days
  • Used to refer to the age of a person

you

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US / jʊ /

UK / ju /

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

although

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US / ɔ:lˈðəu /

UK / ɔlˈðo /

A1
  • conjunction
  • Despite the fact that; however

quadrant

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US / ˈkwɒdrənt /

UK / ˈkwɑ:drənt /

C1
  • noun
  • Quarter of a circle or land area
  • An instrument used for measuring angles in the sky

below

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US / bɪ'ləʊ /

UK / bɪˈlo /

A1
  • preposition
  • In a lower position, place, or level

pythagorean

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US / paɪˌθægəˈri(ː)ən /

UK / ˌpɪθəˈgɔriən /

A1
  • adjecitve
  • Concerning Pythagoras or his geometry

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

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

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

segment

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US / ˈsegmənt /

UK / ˈsɛɡmənt /

B1
  • noun
  • Part divided from the other parts of something
  • verb
  • To separate something into different parts

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

formula

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US / 'fɔ:mjələ /

UK / ˈfɔrmjələ /

A2
  • noun
  • A mathematical or scientific rule
  • Plan, rule or method for doing or making something

measure

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US / ˈmeʒə(r) /

UK / ˈmɛʒɚ /

A1
  • noun
  • Plan to achieve a desired result
  • Tool used to calculate the size of something
  • verb
  • To determine the value or importance of something
  • To calculate size, weight or temperature of

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

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

save

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US / seɪv /

UK / sev /

A1
  • noun
  • Preventing a player in a game from scoring a goal
  • preposition
  • Except; not including
  • verb
  • To make your computer keep a file
  • To keep for future use
  • To make something unnecessary; prevent
  • To keep money so that you will have it later
  • To keep something for future use, such as money
  • To keep or rescue someone or something from harm
  • To prevent a player in a game from scoring a goal
  • To avoid the unnecessary waste of energy, etc.

same

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US / seɪm /

UK / sem /

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

but

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US / bət /

UK / bʌt,bət /

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

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

book

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US / bʊk /

UK / bʊk /

A1
  • noun
  • Thing made of pages with writing on that you read
  • verb
  • (Police) to officially record someone's crimes
  • To reserve (hotel room, etc.) in advance
  • To produce a red or yellow card to a soccer player

sum

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US / sʌm /

UK / sʌm /

A2
  • noun
  • Amount when all is added together; total
  • verb
  • To add two or more numbers together; total

want

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US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

first

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US / fɜ:st /

UK / fɚst /

A1
  • adjecitve
  • 1st; Coming before all others in time or place
  • noun
  • Something that has not happened before
  • Position of the winner of a competition

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

directly

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US / dəˈrektli /

UK / dɪˈrɛktli, daɪ- /

A2
  • adverb
  • In a frank and honest way
  • Right away; immediately

hard

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US / hɑ:d /

UK / hɑː(r)d /

A1
  • adjecitve
  • Difficult to do; difficult to understand
  • (Of facts) not able to be misunderstood; clear
  • (Of alcohol) strong
  • Involving or requiring lots of work, effort, care
  • Being full of pain, trouble, and problems
  • Strong
  • (Of edge) clear; sharp
  • Difficult to bend, break or cut; solid
  • adverb
  • With lots of force, power, or impact

enough

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US / ɪˈnʌf /

UK / ɪˈnʌf /

A1
  • adjecitve
  • Sufficient to answer the need
  • adverb
  • To a satisfactory degree
  • pronoun
  • Equal to what is needed; as much as required
  • Adequate number or amount

find

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US / faɪnd /

UK / faɪnd /

A1
  • noun
  • Object discovered by chance
  • Things that have been discovered from searching
  • verb
  • To become aware of something that is happening
  • To discover or meet by chance
  • To decide if someone is guilty of a crime, or not
  • To learn through study, research, or effort
  • To discover something by looking for it

expand

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US / ɪk'spænd /

UK / ɪkˈspænd /

A2
  • verb
  • To make something larger in size, number, amount

great

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US / ɡreɪt /

UK / ɡret /

A1
  • adjecitve
  • Very large in size
  • Very important
  • Very good; fantastic; wonderful
  • adverb
  • Very good; better than before
  • noun
  • Successful and well-admired person

derive

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US / dɪˈraɪv /

UK / dɪˈraɪv /

B1
  • verb
  • To come from another thing

respect

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US / rɪ'spekt /

UK / rɪˈspɛkt /

A2
  • noun
  • Behavior that follows the established rules
  • Admiration for someone
  • Regard or admiration for someone or something
  • verb
  • To follow the established rules
  • To think very highly of another person

then

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US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

essentially

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US / ɪˈsenʃəli /

UK / ɪˈsenʃəli /

A2
  • adverb
  • Basically; (said when stating the basic facts)

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

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)

even

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US / 'i:vn /

UK / ˈivən /

A1
  • adjecitve
  • Being equal in influence, power, possibility etc.
  • Being divided equally or fairly
  • (Of teams/contest) with similar chances of success
  • Smooth, level, or regular
  • Can be divided by 2; ending in 2, 4, 6, 8, or 0
  • At a regular, equally steady manner
  • Remaining steady and unchanging
  • adverb
  • (When this is surprising) as well; too
  • (Used to compare two things/stress the difference)
  • verb
  • To make something level; to flatten
  • To make something equal

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

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

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

arc

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US / ɑ:k /

UK / ɑrk /

B2
  • noun
  • Smoothly curving line or movement
  • verb
  • To bend in a smooth curve

square

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US / skweə(r) /

UK / skwɛr /

A2
  • adjecitve
  • Being fair or honest
  • Multiply a number by itself e.g. 2x2 or 6x6
  • noun
  • Shape with 4 equal length sides and 4 equal angles
  • The result of multiplying Length by the width
  • Open public area in the middle of the town
  • verb
  • To make something balance or be equal
  • To multiply a number by itself, e.g. 2x2 or 6x8
  • To straighten the edge to form a four-sided shape

plus

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US / plʌs /

UK / plʌs /

A2
  • noun
  • A benefit or advantage
  • preposition
  • In addition; add
  • In addition to; Indicates a number above
  • Having a value greater than zero

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

integral

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US / ˈɪntɪgrəl /

UK / ˈɪntɪɡrəl, ɪnˈtɛɡrəl /

B1
  • adjecitve
  • Being an essential part of something; necessary

use

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US / ju:s /

UK / ju:s /

A1
  • noun
  • Ability to employ something
  • Need to enable it to work
  • Permission to do something with something
  • Act of spending money
  • Operation of a machine for its function
  • verb
  • To have; to eat, drink, consume etc.
  • To need to operate or work
  • To operate a machine, e.g. a computer
  • To treat a person badly
  • To do something with, for a task or purpose

write

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US / raɪt /

UK / raɪt /

A1
  • verb
  • To compose letters and words on paper or a screen
  • To invent or create a computer program
  • To create a song or piece of music

here

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US / hɪə(r) /

UK / hɪr /

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

small

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US / smɔ:l /

UK / smɔl /

A1
  • adjecitve
  • Little in size; not big
  • (Of letters) not written in capital letters
  • Being insignificant or minor
  • (Of a business) with few people or things; local
  • (Of voice) being soft and quiet
  • (Of a child) being young

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

infinitesimally

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US / ˌɪnfɪnɪ'tesɪməlɪ /

UK / ˌɪnfɪnɪ'tesɪməlɪ /

A1
  • adverb
  • Infinitely or immeasurably small

integrate

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US / ˈɪntɪgreɪt /

UK / ˈɪntɪˌɡret /

A2
  • verb
  • To combine together; make into one thing

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

should

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US / ʃʊd /

UK / ʃʊd /

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

probably

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US / ˈprɒbəbli /

UK / ˈprɑbəbli /

A1
  • adverb
  • That is likely to happen or be true

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

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

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

super

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US / ˈsu:pə(r) /

UK / ˈsupɚ /

A2
  • adjecitve
  • Of the highest quality; great; marvelous

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

derivative

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US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

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

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

old

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US / əʊld /

UK / old /

A1
  • adjecitve
  • Having existed for a long time, not young or new
  • Age of a person since birth

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

show

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US / ʃəʊ /

UK / ʃo /

A1
  • noun
  • False display of love, emotion or action
  • An event for displaying or promoting goods
  • Play, opera or theater production
  • Presentation that is broadcast on TV or radio
  • verb
  • To be easily seen or displayed
  • To display your emotions or feelings
  • To let someone see something; display
  • To explain or teach how something is done
  • To lead someone somewhere; point them to something
  • To prove something to be true, correct
  • To reveal or allow something to be seen

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

zoom

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US / zu:m /

UK / zum /

B1
  • noun
  • Camera lens used to magnify images
  • verb
  • To move along or go somewhere very quickly

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

between

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US / bɪˈtwi:n /

UK / bɪˈtwin /

A1
  • preposition
  • From one person, thing, or place, to another
  • In the space that separates two objects
  • Among two or more people who share something

approximate

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US / ə'prɒksɪmət /

UK / əˈprɑksəmɪt /

B2
  • adjecitve
  • About; not exact but close
  • verb
  • To be similar to, but not exactly the same

curve

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US / kə:v /

UK / kɚv /

A2
  • noun
  • Smooth rounded line that is not straight
  • verb
  • To make something bend into a rounded shape

they

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US / ðeɪ /

UK / ðe /

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

reference

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US / 'refrəns /

UK / ˈrɛfərəns, ˈrɛfrəns /

A2
  • noun
  • A letter of recommendation, e.g. for a job
  • Note in a book, etc. giving source of information
  • A location on a map
  • verb
  • To cite a piece of research in speech or writing

very

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US / 'verɪ /

UK / ˈvɛri /

A1
  • adjecitve
  • Concerning a large degree or amount
  • adverb
  • A lot; much; many
  • Used to stress or emphasize a noun

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

expression

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US / ɪk'spreʃn /

UK / ɪkˈsprɛʃən /

A2
  • noun
  • Act of making your thoughts and feelings known
  • Group of words that have a specific meaning

length

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US / leŋθ /

UK / leŋθ /

A2
  • noun
  • Distance from one end of a pool to the other
  • Measurement of distance or of time

draw

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US / drɔ: /

UK / drɔ /

A1
  • noun
  • Something that attracts people to visit a place
  • A lottery or prize
  • Result of a game, contest where the score is equal
  • verb
  • To attract attention to someone or something
  • To get closer to or approach something or someone
  • To influence a person's involvement in something
  • To move an object by pulling
  • To create an image using pen or pencil and paper
  • To take one thing out of a container, etc.
  • (Of 2 teams) to finish a game with the same score

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

part

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US / pɑ:t /

UK / pɑ:rt /

A1
  • adjecitve
  • To some degree; half; not fully
  • noun
  • Division of a book
  • Ratio of something, e.g. 3 of gin, 1 of tonic
  • A line in a person's hair, made with a comb
  • Character an actor plays on stage, movie, or TV
  • Some, but not all of a specific thing
  • Role in causing something to happen
  • verb
  • To make a line in a person's hair, by using a comb
  • To move two things apart creating a space
  • To leave someone, or to break off a relationship

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

know

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US / nəʊ /

UK / noʊ /

A1
  • verb
  • To be familiar with a person or place, thing
  • To feel sure or be convinced about something
  • To discover or be aware of something
  • To be able to distinguish
  • To have knowledge of things

with

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US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

problem

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US / ˈprɒbləm /

UK / ˈprɑbləm /

A1
  • adjecitve
  • Causing trouble
  • noun
  • Something difficult to deal with or causes trouble
  • Question to show understanding of a math concept

prime

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US / praɪm /

UK / praɪm /

A2
  • adjecitve
  • Concerning the highest in quality or value
  • Concerning the highest level of achievement
  • Most important, chief importance in value
  • verb
  • To get a person ready for; prepare someone
  • To coat with paint to prepare it for the main coat

theorem

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US / ˈθɪərəm /

UK / ˈθiərəm, ˈθɪrəm /

B2
  • noun
  • A statement that can be proved

radical

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US / ˈrædɪkl /

UK / ˈrædɪkəl,'rædɪkḷ /

B1
  • adjecitve
  • Very new and different from what is traditional
  • Concerning something's most basic part or form
  • noun
  • Person who believes in extreme political change

see

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US / si: /

UK / si /

A1
  • verb
  • To go on a date with someone
  • To predict whether something is, or will be true
  • To understand what someone says
  • To use your eyes to look at something
  • To visit a place often for sightseeing or pleasure
  • To visit or meet with someone
  • To find out by waiting or looking
  • To watch (a game, movie or TV show)

waste

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US / weɪst /

UK / west /

A2
  • noun
  • Materials left over after a process has finished
  • verb
  • To kill or severely harm someone
  • To use valuable things ineffectively

ask

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US / ɑ:sk /

UK / æsk /

A1
  • verb
  • To say to someone that you want something

exact

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US / ɪɡ'zækt /

UK / ɪɡˈzækt /

A2
  • adjecitve
  • Completely correct; accurate; specific
  • verb
  • To get something, sometimes using force

piece

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US / pi:s /

UK / pis /

A1
  • noun
  • A counter in a board game
  • Small part of something larger
  • verb
  • To put items together to assemble something

yourself

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US / jɔ:ˈself /

UK / jʊrˈsɛlf, jɔr-, jor-, jə- /

A1
  • pronoun
  • Reflexive form of 'you', used for an emphasis

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

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

coordinate

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US / kəʊ'ɔ:dɪneɪt /

UK / ko'ɔrdəˈnet /

B1
  • noun
  • Point on a map defining a location e.g. 135N, 34W
  • verb
  • To match styles or colors that go well together
  • To cause body parts to move smoothly together
  • To make plans to meet a shared goal
  • To bring order and organization to 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)

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

perimeter

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US / pəˈrɪmɪtə(r) /

UK / pəˈrɪmɪtɚ /

B2
  • noun
  • Outline or border; outer edge

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

define

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US / dɪ'faɪn /

UK / dɪˈfaɪn /

A2
  • verb
  • To explain the meaning of words
  • To set or mark the limits of something

those

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US / ðəʊz /

UK / ðoz /

A1
  • determiner
  • The people or things we are talking about
  • pronoun
  • The objects far from the speaker

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

memorize

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US / ˈmeməraɪz /

UK / ˈmɛməˌraɪz /

A2
  • verb
  • To study something so that you remember it
  • Subtitles section
  • Problem three.
  • Let f of x is equal to e to the 2x.
  • Let R be the region in the first quadrant bounded by the graph of f, the coordinate axes, and the vertical line, x is equal to k.
  • So they drew that right over here.
  • And it's in the figure above in the actual AP exam.
  • But I put it below here so I didn't have to waste the screen real estate above that.
  • And you see right here, this is our f of x, this is the line x is equal to k, and this region R is between those, and it's in the first quadrant right over here.
  • So this right over here is our region R.
  • So what are they asking us to do?
  • So k is greater than 0, the region R shown the figure above.
  • And this is the figure.
  • Part a.
  • Write, but do not evaluate an expression involving an integral that gives the perimeter of R in terms of k.
  • So for the perimeter of R, we're going to have to find the length of the sides of R, and the hardest of these is to find the length of the actual curve between this point and this point right over here.
  • And I like to always rederive the arc length formula, although if you're about to take the AP exam, it probably doesn't hurt to have it memorized just so you could save some time.
  • But you should always know how it's derived so that when you're 35 years old and you don't have a calculus book as a reference, you can come up with it yourself.
  • And so you know where the formula comes from.
  • So let's rederive it.
  • So let's say we were to zoom in on a little bit of the curve right over there.
  • So let's say let's zoom in on that part of the curve right over there.
  • And what we could do is, same way we derived an arc length when the function was defined parametrically, which we did, I think, in part one of this AP exam.
  • This time it's not defined parametrically, so we can do it in terms of just x and y.
  • So if you viewed this distance right over here, this is going to be a very small change in x.
  • Let's call that dx.
  • And then this right over here is going to be a very small change in y, dy.
  • And we know that dy, the derivative of y with respect to x, or dy/dx, this is equal to f prime of x.
  • Or we could say that if we multiply both sides by dx, we know that dy is equal to f prime of x dx.
  • So this is equal to f prime of x dx.
  • And then for a very small change in x and very small change in y, we can approximate this curve length with the Pythagorean theorem.
  • And if we get small enough, it'll probably directly kind of measure that curve length.
  • So we use the Pythagorean theorem right here.
  • This length is going to be the square root of dx squared plus dy squared.
  • And if we write dy this way, that arc length right over here, this little small arc length, is going to be equal to, right over here, the square root of dx squared plus-- this is dy-- so plus f prime of x squared times dx squared.
  • All I did is I squared this. dy squared is this thing right over here.
  • And let me expand the radical sign a little bit.
  • And just the same exact way we did it when we had the function defined parametrically, we can factor out a dx squared here.
  • So this is going to be equal to the square root of-- if you factor out a dx squared of this, you get a 1.
  • You factor out a dx squared out of this part, plus f prime of x squared.
  • And we factor a dx squared out of the radical sign, it just becomes a dx.
  • And so that would be the length of this little small arc here, and it wouldn't even be this big.
  • It would be a super infinitesimally small piece of arc.
  • We could call it d arc right over there.
  • And essentially want to sum up all of these little segments.
  • So we want to sum them up from x is equal to 0, all the way to x is equal to k, because we're integrating with respect to x right over here.
  • So this arc length right over here-- so let me write what they want us to figure out; we're trying to figure out the perimeter of R-- is equal to first, this part.
  • So this is the integral from 0 to k. x is equal to 0 to x is equal to k of the square root of 1 plus-- and f prime of x.
  • f of x is e to the 2x.
  • So f prime of x is equal to the derivative of this with respect to x, which is 2e to the 2x.
  • The derivative of 2x is 2.
  • Derivative of e to the 2x with respect to 2x is e to the 2x.
  • So 2e to the 2x, we want to square that, because this is f prime of x, but we want to square f prime of x.
  • So that's 4e to the 4x.
  • And of course, we have a dx.
  • So this is the length of the arc, and that's the hard part.
  • And now we just have to get the rest of the perimeter.
  • So you have this little segment right over here.
  • That's length 1.
  • We're going from 1 to 0 or 0 to 1.
  • So plus 1.
  • Then you have this part right along the x-axis.
  • Well, that's going to be of length k.
  • Plus k.
  • And then finally, you have this height right over here.
  • And that's going to be of length of f of k, or e to the 2k.
  • Plus e to the 2k.
  • And we're done.
  • We found the perimeter of R, and we don't have to evaluate it.
  • So we're done with this part.