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    Assignment 13

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    prakashgupta12082000
    Maths assignment 13 is a very challenging assignment

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    Assignment 13

    Uploaded by

    prakashgupta12082000
    2 views1 page
    Maths assignment 13 is a very challenging assignment

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    © © All Rights Reserved

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    Maths assignment 13 is a very challenging assignment

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    2 views1 page

    Assignment 13

    Uploaded by

    prakashgupta12082000
    Maths assignment 13 is a very challenging assignment

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 1

    M102 - Mathematics II

    Assignment 13

    1. Let f (x) = cos (ax) for x ∈ R where a ̸= 0. Find f (n) (x) for n ∈ N, x ∈ R.

    2. Let g(x) = |x3 | for x ∈ R. Find g ′ (x) and g ′′ (x) for x ∈ R and g ′′′ (x) for x ̸= 0. Show
    that g ′′′ (0) does not exist.

    3. Find the nth derivative of the following functions:

    (a) x2 e3x sin 4x (c) x log x


    ln x sin x
    (b) (d)
    x+1 1 + x2

    4. If y = a cos log x + b sin log x, show that x2 y2n+2 + (2n + 1)xyn+1 + n(n + 1)yn = 0.
    √
    5. If y = log (x + 1 + x2 ). Prove that (1 + x2 )yn+2 + (2n + 1)xyn+1 + n2 yn = 0.

    6. 
     xn sin 1 if x ̸= 0
    Let f (x) = x
     0 if x = 0.
    Then f (k) (0) exists for all k < n, but f (n) (0) does not exist.

    7. Let m, n ∈ N. Consider the function


    (
    xn x > 0
    f (x) =
    xm x ≤ 0.

    Discuss its high-order differentiability.

    8. Determine whether or not x = 0 is a point of relative extremum of the following


    functions:

    (a) f (x) := x3 + 2 1
    (c) h(x) := sin x + x3
    6
    1
    (b) g(x) := sin x − x (d) k(x) := cos x − 1 + x2 .
    2

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