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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 77

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 77. 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 | 22.4 Properties of the Definite Integral 741 f f t dt C f t dt. a b 6. fla f t dt 0 if f is odd a f t dt 2 f0 f t dt if f is even. PROBLEMS FOR SECTION 22.4 For Problems 1 and 2 evaluate the following. 1. a 2 x dx b 4 1 x 1 dx 2. a f V1 x2 dx b 1 1 V1 x2 dx Hint Think about circles. 3. Choose the correct answer and explain your reasoning. a f-a 1 dx i 0 ii 00 777 dx iii 2f0 1772 dx iv - r 7772 dx 742 CHAPTER 22 Net Change in Amount and Area Introducing the Definite Integral b f-a 1 2 dx i 0 ii fo .V dx iii 2 fo 1 2 dx iv - fo 1 dx 4. a Explain why f035 1 T dx f045 1 nf dx Hint Look at the sign of the integrand. b Put in ascending order f1 1 In x f2 1 In x e 1 In x f4 1 In x . dx .- ttt dx .- ttt dx .- ttt dx. 5. a Assume that a b. Insert a or sign between the expressions below as appropriate. Explain your reasoning clearly. f t dt and i f t dt J a J a b Under what circumstances will the two expressions be equal 6. For each of the following sketch a graph of the indicated region and write a definite integral or if you prefer the sum and or differences of definite integrals that gives the area of the region. a The area between the horizontal line y 4 and the parabola y x2 b The area between the line y x 1 and the parabola y x2 1 7. Put the following four integrals in ascending order from smallest to largest . Explain using graphs how you can be sure that the order you gave is correct. I sin t dt I 2 sin t dt I sin 2t dt I sin t dt Jo Jo Jo J- 8. Suppose f05 f t dt 10. Evaluate four out of the five expressions that follow. One of them you do not have enough information to evaluate. a f05 7f t dt b f05 f t 7 dt c f05 f t dt 7 d f0 7f t 7 dt e f 7f t 7 dt 9. If f05 f t dt 10 and f05 g t dt 3 does it follow that a f t g t for all t between 0 and 5 Explain. Mathematicians might write this statement in mathematical symbols as follows f t g t V t in 0 5 . The symbol V is read for all. b f t g t for some t between 0 and 5 Explain. Mathematicians might write this statement in