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    Derivatives, Integrals, Limits and Convergence

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    Question 1
    Differentiate the function f(x) =

    (a)
    xlnx - x

    (b)
    x5lnx

    (c)
    (lnx)2

    (d)
    1-x
    ________________________________________lnx
    Question 2
    Figure 2.1
    ?(x) = ln ^/¯ (9-x2)
    ________________________________________(4+x2)

    Apply laws of logarithms to simplify the function (Figure 2.1). Then find its derivative.
    Question 3
    Find
    limx→3 2x4-3x3-81
    ________________________________________x5-10x3+27
    Apply l'Hopital's rule as many times as
    necessary, verifying your results after each application.
    Question 4
    Find
    limx→∞ ( 3x-2
    ________________________________________3x+2 )x
    Question 5
    Given
    ?(x)=log10x
    find ?'(x).
    Question 6
    Evaluate
    ∫ log3x
    ________________________________________2x dx
    Question 7
    Evaluate:
    ∫ sinh6 x cosh xd x
    Question 8
    Given
    ?(x)= csch-1 1
    ________________________________________x2
    find
    ?'(x)
    .
    Question 9
    Evaluate
    ∫ (^/¯x+4)3
    ________________________________________3^/¯x dx
    Question 10
    Evaluate
    ∫ x2sin2x dx.
    Question 11
    Figure 11.1
    0<x< &#61552;
    ________________________________________2

    R is bounded below by the x-axis and above by the curve
    y = 2cosx, Figure 11.1. Find the volume of the solid generated by
    revolving R around the y-axis by the method of cylindrical shells.
    Question 12
    Evaluate
    &#8747; sin5xdx
    Question 13
    Evaluate
    &#8747; 3x+3
    ________________________________________x3-1 dx

    Question 14
    Use trigonometric substitution to evaluate
    &#8747; 1
    ________________________________________^/¯1+x2 dx
    Question 15
    Figure 15.1
    y= 1
    ________________________________________x2+4x+5

    R is the region that lies between the curve (Figure 15.1) and the
    x-axis from x = -3 to x = -1. Find:
    (a) the area of R,
    (b) the volume of the solid generated by revolving R around the y-axis.
    (c) the volume of the solid generated by revolving R round the x-axis.
    Question 16
    Figure 16.1
    &#8734;&#8747;-&#8734; 1
    ________________________________________1+x2 dx

    Determine whether Figure 16.1 converges or diverges. If it converges,
    evaluate the integral.

    © BrainMass Inc. brainmass.com March 4, 2021, 8:19 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/derivatives-integrals-limits-and-convergence-159988

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    Solution Summary

    A variety of calculus problems are solved. The solution is detailed and well presented.

    $2.49

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