College physics textbooks (algebra based) tend to shy away from topics that are usually thought to require calculus. I suspect that most students are just as happy to avoid these topics. Occasionally, I encounter students who are not so easily satisfied, and have found it useful to maintain a storehouse of non-calculus solutions for some common problems. One such type of problem is finding maximums and minimums of functions. A number of authors have presented solutions using, among other techniques, inequalities, geometry, finding the apex of a parabola, or forcing the discriminant of a quadratic equation to be zero; here, I discuss several examples of finding maximums and minimums using trigonometry instead of calculus.
References
1.
Harald
Helfgott
and Michael
Helfgott
, “Maxima and minima before calculus
,” Pro Mathematica
12
, 135
–158
(1998
). Examples 1, 6.1, and 6.4 are readily solved with the method described here. Examples 3 and 4 are solvable using a triple angle identity.2.
Raymond A.
Serway
and Chris
Vuille
, College Physics
Vol. 1
, 11th ed. (Cengage Learning
, Boston
, 2018
), p. 115
.3.
William J.
Leonard
, “Dragging a box: The representation of constraints and the constraint of representations
,” Phys. Teach.
39
, 412
(Oct.
2001
). This solution makes use of a clever geometric construction. Although phi was initially introduced in the present work as an alternate variable for μK without physical significance, further reflection during revision revealed that phi is the real angle Leonard’s combined contact force makes with the horizontal surface.4.
Se-yuen
Mak
, “Extreme values problems in mechanics without calculus
,” Am. J. Phys.
55
, 929
–931
(Oct.
1987
).5.
Paul A.
Tipler
, Physics for Scientists and Engineers
, Vol. 2
, 3rd ed. (Worth Publishers
, New York
, 1991
), p. 729
.6.
John S.
Thomsen
, “Maxima and minima without calculus
,” Am. J. Phys.
52
, 883
–881
(Oct.
1984
).7.
8.
David
Halliday
and Robert
Resnick
, Fundamentals of Physics
, 3rd ed. (Wiley
, New York
, 1988
), p. 328
.© 2019 American Association of Physics Teachers.
2019
American Association of Physics Teachers
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