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Solutions 5.pdf

Solutions 5.pdf - Math for Econ I Homework Assignment...

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Math for Econ I, Homework Assignment Solutions 5 Instructor: Antonio De Rosa Due: Thursday, October 19th, during Lecture New York University 1. (3 pts) Suppose f ( x ) = ax 2 + bx + c , where a, b, c are constants. For what value (or values) of x is f 0 ( x ) = 0? Solution : f 0 ( x ) = 2 ax + b , so f 0 ( x ) = 0 when x = - b 2 a . 2. (6 pts) If f (3) = 1, f 0 (3) = 3, g (3) = 5, and g 0 (3) = 4, find: (a) (3 pts) ( f · g ) 0 (3). Note: this is the derivative of ( f times g ), not the derivative of the composition ( f g ). (b) (3 pts) f g 0 (3). Solution : (a) ( f · g ) 0 = f 0 · g + f · g 0 , so ( f · g ) 0 (3) = f 0 (3) g (3) + f (3) g 0 (3) = 3 · 5 + 1 · 4 = 19 . (b) By the quotient rule, f g 0 (3) = g (3) f 0 (3) - f (3) g 0 (3) ( g (3)) 2 = 5 · 3 - 1 · 4 5 2 = 11 25 . 3. (4 pts) The derivative g 0 of some function g is graphed below. On what intervals is g increasing? [ - 15 , - 5] and [0 , 5] (it’s alright if students exclude the endpoints) For what x -value(s) is the tangent line for g horizontal? x = - 15 , - 5 , 0 , 5 , 15. 4. (4 pts) Let f ( x ) = | x | . Graph f 0 ( x ). (note: f 0 (0) will be undefined! Drawing the picture of f should help you graph the derivative: | x | is a piecewise function consisting of two lines.) Solution : The student does not need to do all of the following computations, but he or she should at least justify
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