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Derivatives, Integrals, Limits and Convergence

Please see the attached file for the fully formatted problems.

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.

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Solution Summary

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

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