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Cameos for Calculus: Visualization in the First-Year Course by Roger B. Nelsen

By Roger B. Nelsen

A thespian or cinematographer could outline a cameo as a short visual appeal of a recognized determine, whereas a gemologist or lapidary may outline it as a priceless or semiprecious stone. This ebook provides fifty brief improvements or vitamins (the Cameos) for the first-year calculus direction within which a geometrical determine in brief seems to be. the various Cameos illustrate mainstream issues corresponding to the by-product, combinatorial formulation used to compute Riemann sums, or the geometry in the back of many geometric sequence. different Cameos current themes obtainable to scholars on the calculus point yet now not frequently encountered within the path, comparable to the Cauchy-Schwarz inequality, the mathematics mean-geometric suggest inequality, and the Euler-Mascheroni constant.

There are fifty Cameos within the publication, grouped into 5 sections: half I Limits and Differentiation; half II Integration; half III limitless sequence; half IV extra themes, and half V Appendix: a few Precalculus subject matters. a few of the Cameos comprise workouts, so recommendations to all of the workouts follows half V. The e-book concludes with References and an Index.

Many of the Cameos are tailored from articles released in journals of the MAA, reminiscent of The American Mathematical Monthly, Mathematics Magazine, and The collage arithmetic Journal. a few come from different mathematical journals, and a few have been created for this ebook. by means of collecting the Cameos right into a e-book we are hoping that they're going to be extra available to lecturers of calculus, either to be used within the school room and as supplementary explorations for students.

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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.

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