calculus

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A) How do you prove that if 0(<or=)x(<or=)10, then 0(<or=)sqrt(x+1)(<or=)10?

B) So once that is found, then how can you prove that if 0(<or=)u(<or=)v(<or=)10, then 0(<or=)sqrt(u+1)(<or=)sqrt(v+1)(<or=)10?

  • calculus -

    How do you prove that if 0(<or=)x(<or=)10, then 0(<or=)sqrt(x+1)(<or=)10?
    ===========================
    does the square root increase (is the derivative positive) as x goes from 0 to 10 ?
    If so the left side of the domain is minimum and the right side is maximum of the function and we only need to test the ends.

    d (x+1)^.5 / dx = .5 /sqrt(x+1)
    that is positive everywhere in the domain so all we have to prove is the end points.

    0 </= x </= 10

    if x = 0
    sqrt x+1 = sqrt 1 = 1
    if x = 10
    sqrt x+1 = sqrt 11 = 3.32

    so
    1 </ sqrt(x+1) </= 3.32

  • calculus -

    for part b again the derivative is positive throughout the domain so if v is right of u then sqrt (1+v) > sqrt(1+u)

  • calculus -

    thank you!

    Additionally,

    C) They give a recursively defined sequence: a_1=0.3; a_(n+1)=sqrt((a_n)+1)for n>1
    How do you find out the first five terms for it. then prove that this sequence converges. What is a specific theorem that will guarantee convergence, along with the algebraic results of parts A and B?

  • calculus -

    .3
    sqrt 1.3 = 1.14
    sqrt 2.14 = 1.46
    sqrt 2.46 = 1.57
    sqrt 2.57 = 1.60

    hmmm, not getting bigger very fast.
    let's see what happens to the derivative for large n
    .5/sqrt(x+1)
    ah ha, look at that. When n gets big, the derivative goes to zero. So the function stops changing.

  • calculus -

    But why would you look for the derivative to go to zero? Does it have to do anything with the theorem: If summation of a_n converges then limit_(n-->infinity) of a_n = 0. If so, what would the limit be approaching? 10 or infinity? But if not, then what theorem would we use? I know you explained about the larger n for the derivative, but I do not understand how that relates to one of the theorems.

  • calculus -

    But doesn't it converge to infinity and not 0?

  • calculus -

    we want it to converge to 0 right? But does it even converge if it goes to infinity, or is that divergence?

  • calculus -

    Do you do the limit on the derivative?

    Or is there another way to prove convergence with a theorem of some sort?

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