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Unformatted text preview: Chapter 5 What the Derivative tells us about a function 5.1 A zero of a function is a place where f ( x ) = 0. (a) Find the zeros, local maxima, and minima of the polynomial y = f ( x ) = x 3 3 x (b) Find the local minima and maxima of the polynomial y = f ( x ) = (2 / 3) x 3 3 x 2 + 4 x . (c) A point of inflection is a point at which the second derivative changes sign. Determine whether each of the polynomials given in parts (a) and (b) have an inflection point. 5.2 Find the absolute maximum and minimum values on the given interval: (a) y = 2 x 2 on 3 ≤ x ≤ 3 (b) y = ( x 5) 2 on 0 ≤ x ≤ 6 (c) y = x 2 x 6 on 1 ≤ x ≤ 3 (d) y = 1 x + x on 4 ≤ x ≤  1 2 . 5.3 Sketch the graph of x 4 x 2 + 1 in the range 3 to 3. Find its minimum value. 5.4 Identify all the critical points of the following function. y = x 3 27 v.2005.1  September 4, 2009 1 Math 102 Problems Chapter 5 5.5 Consider the function g ( x ) = x 4 2 x 3 + x 2 . Determine locations of critical points and inflection points. 5.6 Consider the polynomial y = x 3 + 3 x 2 + ax + 1. Show that when a > 3 this polynomial has no critical points. 5.7 Find the values of a , b , and c if the parabola y = ax 2 + bx + c is tangent to the line y = 2 x + 3 at (2 , 1) and has a critical point when x = 3. 5.8 The position of a particle is given by the function y = f ( t ) = t 3 + 3 t 2 ....
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 Winter '09
 Allard
 Critical Point, Derivative, Michaelis Menten kinetics

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