quiz3-10.40-soln. - f& x ±&&& x& 2 ± 2...

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MATH 101-B Fall 2004 Quiz 3 ( 10:40-12:30 Group) Oct. 15, 2004 Time: 15 minutes Name: Student No: Follow the directions. No work ° No credit!! Problem (8 ± 2 pts.) a. Write a possible polynomial function for both graphs below. Explain your answers. 4 3 2 1 0 -1 -2 -3 -4 -5 -6 200 175 150 125 100 75 50 25 0 -25 -50 -75 -100 -125 -150 -175 -200 x y 4 3 2 1 0 -1 -2 -3 -4 -5 -6 50 40 30 20 10 0 -10 -20 -30 -40 -50 x y Solution. In the first picture, there are five x ° intercepts. Therefore, if we denote the polynomial function whose graph is this one by f ° x ± , it has at least degree 5. The intercepts x ° ° 1 and x ° 2 come from factors with even exponent since at these points the graph just touches the x ° axis. The other intercepts correspond to factors with odd exponent. Hence, f ° x ± could be of the form ° x ° 2 ± 2 x ° x ± 1 ± 2 ° x ± 3 ±° x ± 4 ± . However, the function above behaves like x 5 polynomial as x ² ³ ± , whereas the graph’s end points behaviour is like ° x 5 . Hence, the correct answer is
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Unformatted text preview: f & x ± & & & x & 2 ± 2 x & x ± 1 ± 2 & x ± 3 ±& x ± 4 ± . With a similar argument, a possible polynomial function for the second graph is g & x ± & & x & 2 ± x & x ± 1 ±& x ± 3 ±& x ± 4 ± . b. Assume that any exponential function a x (for a ´ 1) dominates (i.e. larger than) a power function x n ( n is an integer) for large x values (i.e. as x ² ± ). Use this to explain why any such exponential function dominates any polynomial function for large x values. Solution. We know that a polynomial function of degree n behaves like the power function x n for large x values. So, if a x (for a ´ 1) dominates x n for large x values, then it also dominates a polynomial function (of degree n ) as x ² ± ....
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