tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 98

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 98. 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 | Taylor Series 951 22. a Expand f x a x 4 by multiplying out or by using Pascal s triangle. b Rewrite f x as a 1 X 4 a4 1 a 4- Use the binomial series to expand 1 X 4 multiply by a4 and demonstrate that the result is the same as in part a . 23. Find the Maclaurin series for J- What is the radius of convergence 24. Use the binomial series to flnd the Maclaurin series for --What is the radius of i x2 convergence In Problems 25 through 34 use any method to find the Maclaurin series for f x . Strive for efficiency. Determine the radius of convergence. 25. f x xe x 26. f x sin 3x 27. f x cos 2 28. f x 3e2x 29. f x cos x2 30. f x 3x 31. f x x2 cos x 32. f x cos2 x Hint use a trigonometric identity 33. f x a x p where a and p are constants and p is not a positive integer. 34. f x 2TT3 i 35. Pathological Example Let x e - 2 for x 0 0 for x 0. a Graph x on the following domains -20 20 -2 2 and . A graphing instrument can be used. b It can be shown that is infinitely differentiable at x 0 and that 0 0 for all fc. Conclude that the Maclaurin series for x converges for all x but only converges to x at x 0. 36. Find the Maclaurin series for . What is its radius of convergence 37. For x e -1 1 ln 1 x x - 22 3 - Xr 1 2 . . a Find the Maclaurin series for ln 1 2x . What is its interval of convergence b Find the Maclaurin series for ln e ex . What is its interval of convergence c Find the Maclaurin series for log10 1 x . 952 CHAPTER 30 Series 38. Discover something wonderful. We know eu 1 u J2 3- j for all real u. Now define e raised to a complex number a bi where i 1 to be ea ebi where ebi 1 bi a Use the fact that i2 -1 i3 i and i4 1 to simplify the expression for etb. Gather together the real terms the ones without i s and the terms with a factor of i. Express elb as a sum of two familiar functions one of them multiplied by i . b Use your answer to part a to evaluate eX . 39. The hyperbolic functions hyperbolic cosine abbreviated cosh and hyperbolic sine abbreviated