Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 43. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Polynomial Functions and Their Graphs 401 9. Find a possible equation to fit each polynomial graph below. Notice that a is a negative number. 10. Find a polynomial to fit the graph below. 11. Each of the graphs on the following page is the graph of a polynomial P x . For each graph do the following. a Determine whether the degree of P x is even or odd. b Despite the fact that you have just categorized each of the polynomials as being of either odd or even degree none of the polynomials graphed are even functions and none are odd functions. Explain. c Determine whether the leading coefficient is positive or negative. d Determine a good lower bound for the degree of the polynomial. Explain your reasoning. For example the last graph on the right has one turning point so it must be of degree 2 or more. It is not a parabola since it has a point of inflection therefore we know the degree is higher than 2. It cannot be a polynomial of degree 3 because for x large enough P x is positive. Therefore it must be a polynomial of degree 4 or more. 402 CHAPTER 11 A Portrait of Polynomials and Rational Functions 12. a Suppose P x is a polynomial of degree 5. Which of the statements that follow must necessarily be true If a statement is not necessarily true provide a counterexample an example for which the statement is false . i. P x has at least one zero. ii. P x has no more than four zeros. iii. The graph of P x has at least one turning point. iv. The graph of P x has at most four turning points. b Suppose P x is a polynomial of degree 5 with its natural domain -œ œ . If P n 0 and P n 5 then which one of the following statements is true Explain your answer. i. P has a local minimum at x n but this local minimum is not an absolute minimum. ii. P has a local minimum at x n and this local minimum may be an absolute minimum. iii. P has a local maximum at x n but this local maximum is not an absolute maximum. iv. P has a local maximum at x n and this local maximum may be an .
