Anonymous Christian

  • Facebook
  • Google
  • SoundCloud
  • STAY CONNECTED

  • HOME
  • ABOUT US
  • BLOG
  • E-BOOKNEW
    • Other New Christian Books
    • Parable of the 5 Wise and 5 Foolish Virgins by Lord Jesus Christ with Church Fathers in First Christianity and Protestant Reformers Martin Luther & John Wesley
    • Almsgiving and Atonement Mystery – Essays in First Christianity Book
    • Lake of Fire – Hope for the Wicked One Day? – Essays in First Christianity Book
    • First Christianity – Thirty Sixty and Hundredfold Mystery Book
    • Book of 1Enoch – Christ Centered Universalism Mystery – Essays in First Christianity
    • Lost Orthodoxy (Paternal Light – Heart of the Gospel through the Eyes of the Kings of Christian Theology) St. Justin Martyr and St. Irenaeous of Lyons
    • Saviour of The World Book
  • AUDIO
  • LYRICS
  • GALLERY
    • PHOTO
    • VIDEO
  • CONTACT US

OUR BLOG

  • Calculus help

    Anonymous Christian
    Sep 5, 2014
    MATH
    Comments Off on Calculus help

    Question

    a.) Suppose that we want to approximate sqrt(x) for x near 81…?

     …Find the linear approximation to sqrt(x) at x=81. L(x)=? 
    b.) Use the linear approximation at x=81 that you found in part a to approximate the square root of 76.
    Answer
    Let f(x) = x^(1/2). The idea.

    Whats “linear approximation”? Basically, you interpolate linearly. How? As follows :
    [ f(y) -f(x) ] / ( y – x ) = ‘the gradient value at y ; the basis for linear approximation used’. [ an approximate equation ] … ( 1)

    Now, in our case, y =81 because we are using ’81’ to ‘approximate’ the ‘square root of ’76’. Thus, x = 76.

    We can readily ‘compute’ the following :
    f(x) = f(76) =76^(1/2).
    f(y) = f(81) = 81^(1/2) = 9.
    x = 76 & y = 81.
    ( y – x ) = 81 – 76 = 5.

    The trick is that ‘calculus’ helps us with the ”the gradient value at y ; the basis for linear approximation used’ part”.
    Indeed,
    ”the gradient value at y” = f ‘(y) or value of ‘dy/dx’ at y.
    So,
    f ‘(y)
    = dy/dx
    = (1/2 ) x^(1/2 -1)
    =(1/2) x^( -1/2 )

    Thus,
    ”the gradient value at y = 81 the basis for linear approximation used’ part”
    is =(1/2) 81^( -1/2 )
    = (1/2) * ( 1/9)
    = 1/18

    Hence, ‘plugging’ all those values computed into the (1)…we get,

    [ f(y) -f(x) ] / ( y – x ) = 1/18
    [ 9 – 76^(1/2) ] / 5 = 1/18
    76^(1/2) = 9 – [ 5/18 ] = 8.722 ( approximately)

    Note that The Linear approximation is NOT THE EXACT value. It is derived based on the assumption that since the points are close together, hence their gradient of the curve sqrt(x) between these two points is approximately linear! If you didnt understand what I said in this paragraph… its ok. Because you dont need to know it but its good to know it.

    Also, L(x) = f(x) above. The reason they introduce the term ‘L(x)’ is because our L(x) is NOT the exact value of f(x) but is an approximation for f(x) using the Linear Interpolation Idea. However, we have “loosely” used ‘L(x)’ to ‘equal’ f(x) in the above because it does NOT thwart our purpose.

    I just wanted to clarify as much things as possible so that you may now not just know “how to do it” but “understand the idea behind this method and how it related to calculus as well”.

    If you noticed, you could have “approximated” sqrt 76 by the method above without a calculator since we are just dividing and multiplying integers and fractions! The method above is “one way” where one can “estimate” the “square root” of a “integer” without using a calculator… Go ahead and impress your friends!

anonymous-christian

Jonathan Ramachandran is founder at AnonymousChristian.org which serves as a Christian Blog discussing the Greater Hope of Biblical Christian in Christ Alone. He is also an IMO (International Math Coach) and a featuring Guitarist for many bands.

Advertisement

Recent Posts

  • Dialogues – FAQ and Doctrine to Possibilities
  • Bible and Church Fathers Quotes for Charity Doctrine
  • Water of Life – Two References 1000 years Apart in Chiliasm Timeline and meaning of Unsaved with the Shepherd of Hermas, St. Papias and St. Irenaeous
  • Mystery of the Marriage of Christ and His Church through the Eyes of the Direct Disciples of the Apostles only
  • Earning Salvation Fallacy and God’s Commands by Faith Mystery to the Endless Beauty of His Written Word

Search

Advertisement

Categories

  • ages of Ages (21)
  • AIMO (4)
  • AMC (1)
  • APMO (3)
  • Apostolic Fathers (6)
  • ASMO (2)
  • Book of Enoch (6)
  • Calculus (2)
  • Charity (14)
  • Christian Leaders (45)
  • Earning Salvation Fallacy (3)
  • Fatherless and Orphans (13)
  • First Christianity (53)
  • Foreigner (1)
  • Forever (10)
  • Free Choice (17)
  • Gehenna (20)
  • GMC (2)
  • Godliness (2)
  • Gospel Hope for Non Christians (75)
  • Hate and Love (9)
  • Head Covering (5)
  • Heavens Definition in First Christianity (18)
  • Holy Bible Aionion Edition (12)
  • Hundredfold Mystery (4)
  • ICAS (1)
  • IJMO (1)
  • IMC (1)
  • IMO (11)
  • Jewish Hell (11)
  • Justice Human Law (3)
  • KBAT/HOTS (1)
  • KMC (3)
  • Lake of Fire (20)
  • LGBT (2)
  • Marriage (8)
  • Math (174)
  • Math Olympiad (154)
  • Math Olympiad Classes (11)
  • Math Olympiad Primary (11)
  • Math Olympiad Secondary (10)
  • Millennial Reign (8)
  • Money Matters (11)
  • Music (1)
  • Non Elect Salvation (6)
  • Old Testament (6)
  • OMK (14)
  • Parent Children Duty (2)
  • PMONK (4)
  • Poem (1)
  • Preterism (1)
  • Probability (3)
  • Purgatorial Hell Universalism (33)
  • Puzzle (12)
  • Rapture (3)
  • Reason (1,799)
  • Recapitulation (7)
  • Repentance Afterlife (12)
  • Sabbath and Tithes (5)
  • Sadhu Sundar Singh (2)
  • Saints (5)
  • SASMO (4)
  • Shadow of Death (10)
  • Shepherd of Hermas (12)
  • Sibyline Oracles (9)
  • Singapore Maths (3)
  • Slavery (1)
  • SMOPS (2)
  • Spirit World Visions (9)
  • Statistics (1)
  • Testimony (6)
  • Theology (1,665)
  • Tree of Life (3)
  • Trinity (4)
  • UKMT (1)
  • Uncategorized (7)
  • Unconditional Love (14)
  • Understanding Evil (5)
  • Videos (1)
  • Water of Life (4)
  • Wicked Spirit Saved (14)
  • Women (5)
  • Zionism (6)
© Copyright 2014 - 2022 Anonymous Christian | All Rights Reserved.