tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 55

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 55. 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 | The Derivative of r where n is any Real Number 521 22. x x x 3 2 23. x x - 2 x 1 2 24. x 3 - x 2 x - 1 25. x ex x - 3 3 26. Prove that if x x - a 3 x - b where a b 0 and a b then has a point of inflection at x a. 27. a Which of the following are equal to e-x2 Identify all correct answers. i. e- x x ii. e-x x iii. -1 Y iv. 4 gx gx v. e-x 2 vi. e-2 x vii. e-2x viii. e -x 2 b Differentiate e-x2. 28. Graph x e-x2 and answer the following questions. a What is the domain of The range b Is an even function an odd function or neither c For what values of x is increasing Decreasing d Find all relative maximum and minimum points. e Does have an absolute maximum value An absolute minimum value A greatest lower bound If any of these exist identify them. f Find the x-coordinates of the points of inflection. Note This function is a good one to keep in mind. It is extremely useful in both probability and statistics because with some minor adjustments it gives us a normal distribution curve. THE DERIVATIVE OF x WHERE n IS ANY REAL NUMBER At this point we have proven that x mx -1 for any integer m. In fact this formula holds if m is any real number. We can use the Chain Rule to prove this fact. Suppose that r is any real number and we want to flnd Xxr. The key is to rewrite xr so we can use derivative formulas that we already know. xr eln x Z In x ex and ln x are inverses so e z z. Use log rule iii . Now we take the derivative. 522 CHAPTER 16 Taking the Derivative of Composite Functions ln x er lnx . r in x Use the Chain Rule emess emess mess . Jx Jx er ln x Keep in mind that r is a constant. Rewrite er ln x as xr. -1 We now can take the derivative of xn where n is any real number. Jx nx 1 EXERCISE Find y if y x - fx 2 1. Answer y Ä 2 X 1 x x xX ln x xx 224x1-1 PROBLEMS FOR SECTION For Problems 1 through 3 find y . 1. y x2 2x2 2. y 2x2 1 3 3. y 3x 1 U 4. Differentiate the following simplifying the expression flrst if useful. a y W d y l b y ln ez 1 e y ln .