Abstract
In first-year calculus, constrained-optimization and related-rates word problems are two of the biggest stumbling blocks. In this note, I contrast the methods suggested in calculus textbooks for the solution of these two types of problems, and conclude that a different approach to constrained-optimization problems, similar to that widely used for related-rates problems, would be advantageous. Let us first consider related-rates problems. Traditional textbooks (see, for instance, Adams [1, p. 235]; Edwards and Penney [3, p. 193]; Finney, Weir, and Giordano [5, p. 209]; Johnston and Mathews [6, p. 316]; Stewart [8, p. 258], and Strauss et al. [9, p. 158]) introduce these shortly after implicit differentiation. These texts all suggest that implicit differentiation of the equation relating the rates should be an early step in the solution of such a problem. Nonetheless, many students, faced with a relatedrates problem, persistently avoid implicit differentiation by eliminating a variable. For instance:
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