tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 44

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 44. 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 | Rational Functions and Their Graphs 411 The graph will look like one of the following. Figure 2. Find x-intercepts. Where is the numerator of the simplified expression zero For a fraction to be zero its numerator must be zero. 3. Positive negative. Figure out the sign of x . The sign of any function can change only across a zero or a point of discontinuity. Draw a sign of x number line. The sign can change only on either side of an x-intercept or vertical asymptote. Observe the following. If the factor x - a occurs in either numerator of with odd multiplicity then changes sign around x a. If it occurs with even multiplicity then does not change sign around x a. This is consistent with the behavior we observed in polynomials. 4. Find the y-intercept. Set x 0. This merely gives a point of reference. 5. Look for horizontal asymptotes. A horizontal asymptote includes the behavior of x as x i x . A horizontal asymptote gives the behavior of x only for x of very large magnitude therefore it can be crossed. Find limx OT x . For rational functions if limx OT x is finite it will be equal to limx -OT x . degree top degree bottom no horizontal asymptote x rc degree top degree bottom horizontal asymptote at y 0 the x-axis degree top degree bottom horizontal asymptote at y fraction formed by leading coefficients 6. Consider symmetry. If the asymptotes and intercepts don t have the required symmetry then the function is neither even nor odd. If they do then consider whether the function has even symmetry odd symmetry or neither. You can look at the numerator and denominator separately first. even odd dd even odd even odd odd even 7. If you are interested in local extrema then compute the derivative. The graphs of some rational functions can be built up from the sum or difference of x2-l-1 1 other familiar rational functions. For example A x 1 x 1. Refer to Figure r2 1 to see how the graph of - can be built by summing the y-values of the graphs of x and 1 x. 412