tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 33

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 33. 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 | Derivatives of Sums Products Quotients and Power Functions 301 13. f x xI - x 14. f x x2 x 0 x x 0 15. The function f x x x is continuous at x 0 but not differentiable at x 0. Explain using the definitions of continuity at a point and differentiability at a point. 16. Let f x x5L_. a Sketch the graph of f. Do this by graphing x2 1 and looking at the reciprocal. Check your answer with a computer or calculator. b Make a rough sketch of f based on your graph of f. c Find f x analytically using the Quotient Rule. Graph f . 17. Let f x x3 x - 1 I kx x 1 a What values of k makes f x a continuous function b If k is chosen so that f is continuous at x 1 is f differentiable there 18. Find the equation of the tangent line to f x x x2 2 at x 1. 19. For what value s of x is the slope of the tangent line to f x 1 x3 equal to 1 For Problems 20 through 23 find the following d x 1 20. 3 dx yx3 3x 1J d 21. ---------- dx y x d x2 5x 23. dx 2x10 PART III Exponential Polynomial and Rational Functions with Applications H R Exponential Functions EXPONENTIAL GROWTH GROWTH AT A RATE PROPORTIONAL TO AMOUNT Look around you at quantities in the world that change. While some functions are linear the rate of change of output with respect to input is constant you don t need to look far to see some very different patterns of growth and decay. Leave a piece of bread on a shelf for too long and mold will grow not only will the amount of mold increase with time but the rate at which the mold grows will also increase with time. Or observe the early stages of the spread of infectious disease in a large susceptible population the rate at which the disease spreads increases as the number of infected people increases. Think about the rate of growth of money in an interest-earning bank account the more money in the account the faster additional money will accumulate. For instance 3 of 1000 is greater than 3 of 100. The growth of bread mold the spread of disease and the accumulation of money in a