tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 51

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 51. 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 | Worked Examples Involving Differentiation 481 Exponent Laws Logarithm Laws i. bxby bx y i . logb MN logb M logb N Logarithm laws come from the exponent ii. by bx-y i i. logb n logb M logb N laws because log and exponential iii. b y by i ii. logb M logb M functions are inverse functions. Logarithms are useful when we re trying to solve for a variable in the exponent. If A B and A and B are positive then ln A ln B. Similarly if A B then eA eB. Caution 1 1 2soln 1 1 ln 2 but ln 1 ln 1 0 0 ln 2. 1 1 2 so 101 1 102 but 101 101 102. Changing the bases of logs y logb M To change to natural logs rewrite in exponential form and then take the natural logarithm of both sides. y logb M is equivalent to by M y ln b ln M ln M y ln b Derivatives A Summary 1. x g x f 2. k x k where k is any constant 3. - xn nx -1 where n is a constant We ve proven this for n any integer but in fact it s true for any constant n. 4 3E logb x 3E Kb ln x ih a X ln x 1 5. XbX ln b bx a ex ex b ekx kekx PROBLEMS FOR SECTION 1. If f x 3 what is f e For Problems 2 through 5 compute y . 2. y ln 3x2 Hint write ln 3x2 as ln 3 2 ln x. 3. y 4. y 5. y X ln 1 482 CHAPTER 14 Differentiating Logarithmic and Exponential Functions For Problems 6 through 13 differentiate the given function. 6. x x In 1 7. x 8. x e5x ln - 9. x 3x log x 10. x U 11. x x l 7 Conserve your energy. 12. x x x ln x 13. x x2 ln xy Q 14. Using what you know about the graph of In x sketch the graphs of the following. a y lnx b y ln x c y ln x d For parts a b and c locate all critical points and identify all local maxima and minima. 15. Graph x ln x 1 x for x 0 indicating all local maxima and minima and any points of inflection. While answering this problem do the following. a On a number line indicate the sign of . Above this number line draw arrows indicating whether is increasing or is decreasing. b On a number line indicate the sign of . Above this number line write is concave up and is concave down as appropriate. c Using all .