Calculus: Analytic Geometry and Calculus, with Vectors

(^466) Trigonometric functions and show that 0 is a maximum when x = ab. Remark: Addicts of elementary plane geometry can be ple ...

8.3 Inverse trigonometric junctions 467 10 Letting N be a positive integer (which we shall not call "arbitrary"), manufacture an ...

468 Trigonometric functions This and the sandwich theorem imply that 0 as n and the defini. tion involving (5.622) shows that (8 ...

8.4 Integration by trigonometric and other substitutions 469 17 Try to understand and even prove the statement that each of the ...

(^470) Trigonometric functions a sense in which this whole business is utterly silly, but it really is a convenience to imagine ...

8.4 Integration by trigonometric and other substitutions 471 The required conclusion (8.441) then follows from the fact that a a ...

472 Trigonometric functions careless about quadrants in which angles lie. There is a "theory of analytic functions" that guarant ...

8.4 Integration by trigonometric and other substitutions 473 we look at itand generate the idea that we should try setting x = a ...

474 t Trigonometric functions It is always a good idea to display u and v' in one lineand u' and v in a lower line and to know t ...

8.4 Integration by trigonometric andother substitutions 475 Problems 8.49 1 By making an appropriate figure and trigonometric su ...

(^476) Trigonometric functions 4 In Problem 2 of Problems 4.39, we called attention to the important non- elementary beta integr ...

8.5 Integration by substitutingz = tan 012 477 7 It is easy to evaluate the integral Kl = fx sin x dx with the aid of the formul ...

478 Trigonometric functions of quotients of polynomials in z. We suppose that -7r < 0 < 7r, that -n/2 < 0/2 < a/2, a ...

8.5 Integration by substituting z = tan 0/2 479 where Pl and Ql are polynomials in z. Several examples appear in the problems at ...

9 Exponential and logarithmic functions 9.1 Exponentials and logarithms At least a modicum of basic information about exponentia ...

9.1 Exponentials and logarithms 481 physical theory of hydrogen, is neither so brief nor unimportant that it is unworthy of seve ...

482 Exponential and logarithmic functions Theorem 9.15 .ds a function of the "rational variable" x, the function having values a ...

9.1 Exponentials and logarithms 483 very substantial,and we proceed to use it to define and study ax when x is real. Let x be a ...

484 Exponential and logarithmic functions the procedure by letting r, s, t, and u be rational numbers for which r x5sandt<y&l ...

9.1 Exponentials and logarithms 485 and putting x = b in the first gives the second. Putting b = e and a= 10 gives the formula ( ...