Đang chuẩn bị liên kết để tải về tài liệu:
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 42

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 42. 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 | 11.3 Polynomial Functions and Their Graphs 391 11.3 POLYNOMIAL FUNCTIONS AND THEIR GRAPHS EXAMPLE 11.6 SOLUTION Graphing Polynomials We ll begin with a couple of examples and then arrive at a general strategy for understanding the graphs of polynomials. Graph x x3 - 2x2 - 8x identifying the x-coordinates all zeros local extrema and inflection points. Zeros. This is the function from Example 11.2. In factored form x x x 4 x 2 it has zeros at x 2 x 0 and x 4. The graph of x does not cross the x-axis anywhere between these zeros. is continuous so its sign can t change between the zeros. Therefore between x 2 and x 0 for example x must be always positive or always negative. By determining the sign of x for just one test value of x on the interval 2 0 we can determine the sign of x on the entire interval 2 0 . We draw a number line showing the zeros of x and the sign of x between those zeros. Sign Computations suggested method Work with in factored form x x x 4 x 2 . We ll start with x very large and then look at the effect on the sign of the factors as x decreases and passes by each zero. 0 0 0 Sign of f - -2 0 - t For x very large all factors are positive As x drops below 4 the factor x 4 changes sign As x drops below 0 the factor x changes sign As x drops below 2 the factor x 2 changes sign A spot-check can be used to confirm this. y x Figure 11.10 Local Extrema. We need to look at the first derivative to determine critical points x 3x2 4x 8. We use the quadratic formula to determine where x 0 and obtain x 2 7 2.43 and x 2 2 7 1.10. By drawing a number line showing these points where x 0 and indicating the sign of x between them we can determine whether these stationary points are local minima local maxima or neither. 392 CHAPTER 11 A Portrait of Polynomials and Rational Functions graph of f x sign of f x 2 - 2 7 - 2 2 7 Computations If x is large enough in magnitude x2 dominates and is positive. For x 2 3 between the zeros is negative. So x has a local maximum at x