974 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 12. Repeat Problem 11 for ( ) = − + f x x2 8. In Problems 13–22, a function f is nonnegative and continuous on an interval [ ] a b , . (a) Graph f, indicating the area A under f from a to b. (b) Approximate the area A by partitioning [ ] a b , into four subintervals of equal width and choosing u as the left endpoint of each subinterval. (c) Approximate the area A by partitioning [ ] a b , into eight subintervals of equal width and choosing u as the left endpoint of each subinterval. (d) Express the area A as an integral. (e) Use a graphing utility to approximate the integral. 13. ( ) [ ] = + f x x 2, 0, 4 2 14. ( ) [ ] = − f x x 4, 2, 6 2 15. ( ) [ ] = f x x , 0, 4 3 16. ( ) [ ] = f x x , 1, 5 3 17. ( ) [ ] = f x x 1 , 1, 5 18. ( ) [ ] = f x x, 0, 4 19. ( ) [ ] = − f x e , 1, 3 x 20. ( ) [ ] = f x x ln , 3, 7 21. π ( ) [ ] = f x x sin , 0, 22. π ( ) = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ f x x cos , 0, 2 In Problems 23–30, an integral is given. (a) What area does the integral represent? (b) Graph the function, and shade the region represented by the integral. (c) Use a graphing utility to approximate this area. 23. ∫ ( ) +x dx 3 1 0 4 24. ∫ ( ) − +x dx 2 7 1 3 25. ∫ ( ) − x dx 1 2 5 2 26. ∫ ( ) −x dx 16 0 4 2 27. ∫ π x dx sin 0 2 28. ∫ π π − x dx cos 4 4 29. ∫ e dx x 0 2 30. ∫ x dx ln e e2 31. Confirm the entries in Table 6. [Hint: Review the formula for the sum of an arithmetic sequence.] 32. Consider the function ( ) = − f x x 1 2 whose domain is the interval [ ] −1, 1 . (a) Graph f. (b) Approximate the area under the graph of f from −1 to 1 by dividing [ ] −1, 1 into five subintervals, each of equal width. (c) Approximate the area under the graph of f from −1 to 1 by dividing [ ] −1, 1 into ten subintervals, each of equal width. (d) Express the area as an integral. (e) Evaluate the integral using a graphing utility. (f) What is the actual area? Retain Your Knowledge Problems 33–36 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for a final exam, or subsequent courses such as calculus. 33. Graph the function ( ) = f x x log . 2 34. If = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A 1 2 3 4 and = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ B 5 6 0 7 8 1 , find AB. 35. If ( ) = + + f x x x 2 3 1, 2 find ( ) ( ) + − f x h f x h and simplify. 36. Find the partial fraction decomposition of ( )( ) − + x x 16 2 2 . 2 ‘Are You Prepared?’ Answers 1. = A lw 2. 24 Chapter Review Things to Know Limit (p. 941) ( ) = → f x N lim x c As x gets closer to c, ≠ x c, the value of f gets closer to N. Basic limits (p. 946) = → A A lim x c The limit of a constant is the constant. = → x c lim x c The limit of x as x approaches c is c.
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