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mathguide_ak - Math Guide for Econ 11 Answers to Exercises...

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Math Guide for Econ 11 Answers to Exercises University of California, Los Angeles Exercise 1 dF dx 1. F ( x ) = x 2 dF dx = lim h ! 0 F ( x + h ) F ( x ) h = lim h ! 0 ( x + h ) 2 x 2 h = lim h ! 0 x 2 + 2 xh + h 2 x 2 h = lim h ! 0 2 xh + h 2 h = lim h ! 0 2 x + h = 2 x 2. F ( x ) = p x dF dx = lim h ! 0 F ( x + h ) F ( x ) h = lim h ! 0 p ( x + h ) p x h = lim h ! 0 p ( x + h ) p x h ± p ( x + h ) + p x p ( x + h ) + p x = lim h ! 0 x + h x h p ( x + h ) + p x ± = lim h ! 0 1 p ( x + h ) + p x = 1 2 p x Exercise 2 1. F ( x ) = 1 1+ x dF dx = 1 (1 + x ) 2 d 2 F dx 2 = 2 (1 + x ) 3 1
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2. F ( x ) = exp (1 + 2 x ) dF dx = 2 exp (1 + 2 x ) d 2 F dx 2 = 4 exp (1 + 2 x ) 3. F ( x ) = 1+ x 2 1+ x dF dx = 2 x (1 + x ) 1 + x 2 ± (1 + x ) 2 = x 2 + 2 x 2 (1 + x ) 2 d 2 F dx 2 = (2 x + 2) (1 + x ) 2 x 2 + 2 x 2 ± (1 + x ) 3 = 6 (1 + x ) 3 Exercise 3 Find the &rst- and second-order partial derivatives, and the cross- partial derivatives of 1. F ( x; y ) = xy @F @x = y @F @y = x @ 2 F @x 2 = 0 @ 2 F @y 2 = 0 @ 2 F @x@y = 1 @ 2 F @y@x = 1 2
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2. F ( x; y ) = x 2 y 3 @F @x = 2 xy 3 @F @y = 3 x 2 y 2 @ 2 F @x 2 = 2 y 3 @ 2 F @y 2 = 6 x 2 y @ 2 F @x@y = 6 xy 2 @ 2 F @y@x = 6 xy 2 3. F ( x; y ) = 2 xy x 2 2 y 2 3 x y @F @x = 2 y 2 x
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mathguide_ak - Math Guide for Econ 11 Answers to Exercises...

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