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**Additional info for Analysis and partial differential equations. Dedicated to Mischa Cotlar**

**Example text**

1. , finding the distance from a given point to a given curve. The basic procedure in solving such problems is to: (1) Set up a diagram including the essentials of the given problem. (2) Find a formula (try to make it explicit) for the appropriate function f(x) to be extremized, through relationships suggested from the diagram. 34 35 Volumes of Solids with Known Cross-Sections (3) Determine the domain of f(x) from the data of the given problem. :::; b. (4) Find all extremal points wheref'(x) = 0, a < x < b.

Chapter II Geometry In this chapter we consider three types of applied one-variable calculus problems which are basically geometrical in character: maxima and minima (extrema) associated with geometrical configurations, related rates, and volumes of solids with similar cross-sections. ) Extremal and related rates problems involve applications of the differential calculus (derivatives) whereas volume problems involve the integral calculus (integrals). In all such problems the student should first construct a diagram including the essentials of the given problem.

From the above information one can construct Fig. L17e showing the sign and "direction" of h (x). , f(x) is the derivative of the integral h(x). Proceeding from this fact, make the following observations. Slope of h(x). - 00 < x < - 2: h(x) is increasing (since f(x) > 0). h'( - 3) == + 3. x = -2: h'( -2) = 0 ~ zero slope for h(x). - 2 < x < 0: h (x) is decreasing. In this interval the slope has its minimum value of -! at x = - L x = 0: h'(O) = o. 0< x < 2: h(x) is increasing. In this interval the slope has its maximum value of!