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    Question Bank for Calculus

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    baladurgaprasad2

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    © All Rights Reserved
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    Question Bank for Calculus

    Uploaded by

    baladurgaprasad2
    7 views2 pages

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    © All Rights Reserved
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    Download as docx, pdf, or txt
    7 views2 pages

    Question Bank for Calculus

    Uploaded by

    baladurgaprasad2

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
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    QUESTION BANK FOR CALCULUS

    1.Verify Rolle’s theorem for the function f(x)= (x−a)m ( x−b )n where m, n are positive integers in [a,
    mb+ na
    b]. ANS (c= )
    m+ n

    2. Verify Rolle’s theorem for f(x) = log ( x 2+ ab


    x (a+b) )
    in [a, b], a>0, b>0. ANS (c=√ ab )

    3. Verify Rolle’s theorem for f(x) = |x| in [-1, 1]. ANS (LHD≠ RHD f(x) is not derivable at x=0 .Hence
    Rolle’s theorem is not applicable.

    sin x −x π
    4. Verify Rolle’s theorem for f(x) = x or e sin x in [0, π ¿ . ANS(c= )
    e 4
    5. Verify Rolle’s theorem can be applied to the following functions in the intervals cited.

    (i) f(x) = tan x in [0, π ¿. ANS (Rolle’s not applicable)

    1
    (ii) f(x) = 2 in [-1,1]. ANS (Rolle’s not applicable)
    x
    (iii) f(x) = x 3 in [1,3]. ANS (Rolle’s not applicable)

    6. Verify Rolle’s theorem for(i) f(x)= 2 x 3+ x 2-4x-2 in [√ 3, -√ 3]

    (ii)f(x)= 2 x 3+ x 2-4x-2 in [√ 2 , -√ 2 ¿ .

    7. Verify Lagrange’s mean value theorem for

    1+ √37
    (i) f(x) = x3-x2-5x+3 in [0,4].ANS (c= )
    4
    (ii) f(x) = log ex in [1,e]. ANS (c=e-1)

    π
    (iii) f(x)= cos x in [0, ]. ANS(c=sin-12 π )
    2
    b−a -1 -1
    b−a
    8. If a<b, prove that 2 < tan b -tan a < 2 using Lagrange’s mean value theorem .
    (1+b ) (1+a )
    Deduce the following :

    π 3 4 π 1
    (i) + < tan-1 < +
    4 25 3 4 6
    5π+4 π+2
    (ii) < tan-12 < .
    20 4
    π 1 π 1
    9. Prove that (i) + < sin-1( 3/5 ) < + .
    6 5 √3 6 8

    (ii)
    π
    -
    1
    3 5 √3
    > cos-1 ()
    3 π 1
    > - .
    5 3 8
    10. Show that for any x>0, 1+x< ex< 1+xex.

    11. Find c of Cauchy’s mean value theorem for

    1
    (i) f(x) = √ x and g(x) = in [a, b]. ANS(c=√ ab )
    √x
    a+b
    (ii) f(x)= ex and g(x)=e-x. on [a, b] .ANS (c= ) and f(x)= ex and g(x)=e-x on [3,7]ANS (c=5)
    2
    π π
    (iii) f(x) =sin x and g(x)= cos x on [0, ]. ANS (c= )
    2 4
    log b−log a
    12. If f(x) = log x and g(x) =x2 in [a, b] with b>a>1, using Cauchy’s theorem prove that =
    b−a
    2
    (a+ b)/2 c .
    13. Verify Cauchy’s mean value theorem for the functions f(x) and f’(x) in (1,e) given f(x) =logx.

    e
    ANS (c= )
    e−1
    14. Obtain the Taylor’s series expansion for

    (i) f(x)=ex about x=-1.

    π
    (ii) f(x) = sin2x in powers of x -
    4
    π
    (iii) sin x in powers of x -
    4
    5
    15. Verify Taylor’s theorem for f(x)= (1−x ) 2 with Lagrange’s form of remainder upto 2 terms in the
    interval [0.1].
    5
    16. Verify Taylor’s theorem for f(x)= (1−x ) 2 with Lagrange’s form of remainder upto 3 terms in the
    interval [0.1].

    17. Obtain Maclaurin’s series expansion of the following function

    (i) ex (ii) cos x (iii) sin x (iv) cos hx (v) sin hx (vi) log e(1+x)
    −1 3
    sin x x
    18. Show that = x+4 +……….
    √1−x 2 3!
    x
    x x2 x 4 e 1 x x3
    19.Show t5hat log(1+x) = log2 + + - + ……. Hence deduce that x = + - +……….
    2 8 192 e + 1 2 4 48

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