By L. V. Tarasov

An interactive textbook of Calculus provided as a discussion among the writer and the reader.

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1a). 1. The derivative of the arctangent The arctangent is an odd function (since the tangent is), so its derivative is even. Hence _ we need only find d arctan t=dt for t 0. t C t /2 for t > 0 (the case where t < 0 is similar). t C t / arctan t . t C t / t ; or arctan t 17 18 CAMEO 8. t C t / t arctan t D 1 ; 1 C t2 and similarly for the limit as t ! 0 , So the ordinary limit as t ! 0 exists, and we have d 1 arctan t D : dt 1 C t2 The derivatives of other inverse trigonometric functions now follow using the chain rule and identities: arcsin t D arctan p 1 t t2 ; arccos t D arcsin t; 2 1 arcsec t D arccos ; etc.

1) follows. 1. 2. The AM-GM inequality p for two numbers. a C b/=2 and the geometric mean ab for positive numbers a and b in the preceding Cameo. 2) p with equality if and only if a D b. 1 C x/=2 x for p x > 0. 2. 1 C x/=2 23 24 CAMEO 11. 1 C x/=2 only if x D 1/. 2. 1 C x/=2 x and multiply both sides by a. 1. 2). 3. 2. 2), expand and simplify the inequality p . a b/2 0 (which is obviously true since squares are never negative). 3. 2) are equivalent. 4. x 3 C 1=x 3 / for x > 0. x C 1=x/3 and b D x 3 C 1=x 3 .

115, and C n2 /”, College Mathematics D. 12 C 22 C Journal, 22 (1991), p. 124. CAMEO 16. -K. Siu, “Proof without words: Sum of squares,” Mathematics Magazine, 57 (1984), p. 92. 5: A. n C 1//2 =4” Mathematics Magazine, 62 (1989), p. 259, and W. Lushbaugh, Mathematical Gazette, 49 (1965), p. 200. 6: S. Golomb, “A geometric proof of a famous identity,” Mathematical Gazette, 49 (1965), pp. 198–200. CAMEO 17 Summation by parts Summation by parts is a deceptively simple yet remarkably powerful method for computing certain sums in calculus, and can be used in higher level courses as well.