By Roger B. Nelsen
A thespian or cinematographer may outline a cameo as a quick visual appeal of a identified determine, whereas a gemologist or lapidary may perhaps outline it as a useful or semiprecious stone. This booklet provides fifty brief improvements or vitamins (the Cameos) for the first-year calculus path during which a geometrical determine in short appears to be like. the various Cameos illustrate mainstream themes akin to the by-product, combinatorial formulation used to compute Riemann sums, or the geometry in the back of many geometric sequence. different Cameos current subject matters available to scholars on the calculus point yet no longer frequently encountered within the path, corresponding to the Cauchy-Schwarz inequality, the mathematics mean-geometric suggest inequality, and the Euler-Mascheroni constant.
There are fifty Cameos within the booklet, grouped into 5 sections: half I Limits and Differentiation; half II Integration; half III limitless sequence; half IV extra subject matters, and half V Appendix: a few Precalculus subject matters. some of the Cameos comprise routines, so strategies to all of the routines follows half V. The booklet concludes with References and an Index.
Many of the Cameos are tailored from articles released in journals of the MAA, akin to 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 e-book. by means of collecting the Cameos right into a publication we are hoping that they are going to be extra available to academics of calculus, either to be used within the school room and as supplementary explorations for students.
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Additional resources for Cameos for Calculus: Visualization in the First-Year Course
35 CAMEO 15. 2. 3) we see how a box with sides x; y, and z fits inside the union of three right pyramids whose bases are squares with sides x; y, and z and whose altitudes are also x; y, and z, respectively. 2). 3. 2) algebraically. 2) for three numbers is a powerful problem-solving tool. We illustrate its use with four examples—the first and fourth can be solved with single-variable calculus, but the second and third require multivariable calculus. 3. 3 of the Native American encampment at the Pan-American Exposition in Buffalo, NY in 1901.
Let x = length, y = width, and z = height in inches, with volume V = 560 in3 . 2xy C 5xz C 7yz/ for some constant c. 70V 2 /1=3 D 3 2xy 5xz 7yz Ä 3 3c with equality if and only if 2xy D 5xz D 7yz, or x D 7z=2 and y D 5z=2. Hence V D 560 D 72 z 52 z z D 35 z 3 so that z = 4 in, x = 14 in, and y = 10 in. 6 are most likely not designed to minimize the amount of plastic used for their construction. 6. 6. If the costs of metal, cardboard, and plastic are m, c, and p cents per in2 respectively, what should the ratio of the height h to the base radius r be in order to minimize the construction cost for a can with volume V ?
2xy C 5xz C 7yz/ for some constant c. 70V 2 /1=3 D 3 2xy 5xz 7yz Ä 3 3c with equality if and only if 2xy D 5xz D 7yz, or x D 7z=2 and y D 5z=2. Hence V D 560 D 72 z 52 z z D 35 z 3 so that z = 4 in, x = 14 in, and y = 10 in. 6 are most likely not designed to minimize the amount of plastic used for their construction. 6. 6. If the costs of metal, cardboard, and plastic are m, c, and p cents per in2 respectively, what should the ratio of the height h to the base radius r be in order to minimize the construction cost for a can with volume V ?