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Unformatted text preview: (8/19/08) Math 20C. Lecture Examples. Sections 14.2 and 14.3. Limits and partial derivatives Example 1 What is lim ( x,y ) (3 , 2) ( x 2 + y 2 )? Answer: lim ( x,y ) (3 , 2) ( x 2 + y 2 ) = 13 Example 2 Find the x- and y-derivatives of f ( x,y ) = x 3 y- x 2 y 5 + x . Answer: f x = 3 x 2 y- 2 xy 5 + 1 f y = x 3- 5 x 2 y 4 Example 3 What are g x (2 , 5) and g y (2 , 5) for g ( x,y ) = x 2 e 3 y ? Answer: g x (2 , 5) = 4 e 15 g y (2 , 5) = 12 e 15 Example 4 The volume of a right circular cylinder of radius r and height h is equal to the product V ( r,h ) = r 2 h of its height h and the area r 2 of its base (Figure 1). What are (a) the rate of change of the volume with respect to the radius and (b) the rate of change of the volume with respect to the height and what are their geometric significance? r h [Area of base ] = r 2 [Volume] = r 2 h [Lateral surface area] = 2 rh FIGURE 1 Answer: (a) V r = 2 rh is the area of the lateral surface (the sides) of the cylinder. (b) V h = r 2 is the area of the base. Lecture notes to accompany Sections 14.2 and 14.3 of Calculus, Early Transcendentals by Rogawski. 1 Math 20C. Lecture Examples. (8/19/08) Sections 14.2 and 14.3, p. 2 Partial derivatives as slopes of tangent lines When we hold y equal to a constant y = y , z = f ( x,y ) becomes the function...
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