1. Show that
where
,
,
. Hence determine the
th derivatives of the functions
,
, and show in particular that
.
2. Trace the curve , where and are positive. Show that has an infinity of maxima whose values form a geometrical progression and which lie on the curve
3. Integrals containing the exponential function. Prove that
[Denoting the two integrals by
,
, and integrating by parts, we obtain
Solve these equations for
and
.]
4. Prove that the successive areas bounded by the curve of Ex. 2 and the positive half of the axis of form a geometrical progression, and that their sum is
5. Prove that if then
6. If then . [Integrate by parts. It follows that can be calculated for all positive integral values of .]
7. Prove that, if is a positive integer, then and
8. Show how to find the integral of any rational function of . [Put , when , , and the integral is transformed into that of a rational function of .]
9. Integrate distinguishing the cases in which is and is not equal to .
10. Prove that we can integrate any function of the form , where denotes a polynomial. [This follows from the fact that can be expressed as the sum of a number of terms of the type , where is a positive integer.]
11. Show how to integrate any function of the form
12. Prove that , where and is greater than the greatest root of the denominator of , is convergent. [This follows from the fact that tends to infinity more rapidly than any power of .]
13. Prove that , where , is convergent for all values of , and that the same is true of , where is any positive integer.
14. Draw the graphs of , , , , , , and , determining any maxima and minima of the functions and any points of inflexion on their graphs.
15. Show that the equation , where and are positive, has two real roots, one, or none, according as , , or . [The tangent to the curve at the point is which passes through the origin if , so that the line touches the curve at the point . The result now becomes obvious when we draw the line . The reader should discuss the cases in which or or both are negative.]
16. Show that the equation has no real root except , and that has three real roots.
17. Draw the graphs of the functions
18. Determine roughly the positions of the real roots of the equations
19. The hyperbolic functions. The hyperbolic functions , , … are defined by the equations Draw the graphs of these functions.
Establish the formulae
21. Verify that these formulae may be deduced from the corresponding formulae in and , by writing for and for .
[It follows that the same is true of all the formulae involving
and
which are deduced from the corresponding elementary properties of
and
. The reason of this analogy will appear in
Ch. X.]
22. Express and in terms (a) of (b) of . Discuss any ambiguities of sign that may occur.
23. Prove that
[All these formulae may of course be transformed into formulae in integration.]
24. Prove that and .
25. Prove that if then , if then , and if then . Account for the ambiguity of sign in the first case.
26. We shall denote the functions inverse to , , by , , . Show that is defined only when , and is in general two-valued, while is defined for all real values of , and when , and both of the two latter functions are one-valued. Sketch the graphs of the functions.
27. Show that if and is positive, and , then
28. Prove that if then , and is equal to or to , according as or .
29. Prove that if then is equal to or to , according as is less than or greater than . [The results of Exs. 28 and 29 furnish us with an alternative method of writing a good many of the formulae of Ch.VI.]
30. Prove that
31. Prove that
32. Solve the equation , where , showing that it has no real roots if , while if it has two, one, or no real roots according as and are both positive, of opposite signs, or both negative. Discuss the case in which .
33. Solve the simultaneous equations , .
34. as . [For , and . Cf. Ex. XXVII. 11.] Show also that the function has a maximum when , and draw the graph of the function for positive values of .
35. as .
36. If , where , as , then . [For , and so (Ch.IV, Misc. Ex. 27).]
37. as .
[If
then
. Now use Ex. 36.]
38. as .
39. Discuss the approximate solution of the equation .
[It is easy to see by general graphical considerations that the equation has two positive roots, one a little greater than
and one very large, and one negative root a little greater than
. To determine roughly the size of the large positive root we may proceed as follows. If
then
roughly, since
and
are approximate values of
and
respectively. It is easy to see from these equations that the ratios
and
do not differ greatly from unity, and that
gives a tolerable approximation to the root, the error involved being roughly measured by
or
or
, which is less than
. The approximations are of course very rough, but suffice to give us a good idea of the scale of magnitude of the root.]
40. Discuss similarly the equations