# 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< 

________________________________________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

∫ sin5xdx

Question 13

Evaluate

∫ 3x+3

________________________________________x3-1 dx

Question 14

Use trigonometric substitution to evaluate

∫ 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

∞∫-∞ 1

________________________________________1+x2 dx

Determine whether Figure 16.1 converges or diverges. If it converges,

evaluate the integral.

#### Solution Summary

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