Can you evaluate definite and indefinite integrals of

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Nature of Mathematics
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Chapter 18 / Exercise 17
Nature of Mathematics
Smith
Expert Verified
Can you evaluate definite and indefinite integrals of algebraic and transcendental functions using the algebraic rules of antidifferentiation, especially the substitution method? (5.4, 5.5) 68. Evaluate the following integrals. a. 4 3 0 2 dx x b. 5 ln e e dx x x c. 1 5 2 3 0 1 2 x x dx d. 2 5 x dx x e. 2 1 4 dx x f. 2 5 9 dx x g. 2 1 x x dx h. 2 sec 2 tan 2 x x dx 1
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Nature of Mathematics
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Chapter 18 / Exercise 17
Nature of Mathematics
Smith
Expert Verified
Revised 03-2016 14 Can you find the area bounded by two or more curves? (6.2) 69. Sketch the region enclosed by the given curves. Draw a typical rectangle and label its height and width. Then find the area of the region. a. 2 y x , 2 y x b. 1 y x , 2 2 6 Can you find the average function value over an interval? (5.4) 70. Find the average value of the function on the given interval. a.   2 t f x te , 0,5 b.       4 cos sin h x x x , y x 0,
Revised 03-2016 15 Answers 1. (a) (i) 4.42 m/s (ii) 5.35 m/s (iii) 6.094 m/s (iv) 6.2614 m/s (v) 6.27814 m/s (b) The instantaneous velocity appears to be approximately 6.28 m/s. 2. (i)  (ii) 4 (iii) 1.8 (iv) 1 5 1.8 9 (v) 64 (vi) This limit does not exist, as the limit from the left, 0, does not equal the limit from the right, 2. 3. 0.866 (The exact value is 3 2 .) 4. (a) 2 (b) 1 8 (c) 1 (d) 1 2 (e) 6 5 5. (a) (b)  (c)  (d) does not exist (e) 6. horizontal asymptotes: 3 2   y ; vertical asymptote: 7 2 x 7.
Revised 03-2016 16 8. 9. 2   c 10. The function f is only discontinuous at 0 x . At 0 x f is continuous from the left. 11. (a) (i) 1 m/s (ii) 0.5 m/s (iii) 1 m/s (iv) 0.5 m/s (b) The instantaneous velocity is 0 m/s. 12. The function   y s t is graphed at right. The average velocity over 3,4 is represented by the slope of the secant line 6   y x . The average velocity over 4,4.5 is represented by the slope of the secant line 0.5 y x . The instantaneous velocity at 4 t is represented by the slope of the tangent line 2 y . The two secant lines and the tangent line intersect at 4,2 . 13. 5   y x 14.   3 ' 2 2 7 f 15. 2 ' 6 3 f
Revised 03-2016 17 16. (a) The function   ' f x gives the rate of change of the production cost per calculus book at x books. The units are dollars/book. (b) The statement ' 800 0.17   f means that the cost of producing each calculus book is decreasing at a rate of $0.17/book (17 cents per book) as of the production of the 800 th book. 17.   3 ' 2 3 1 f x x 18. 2 2 4 1 ' 2 x x y x 19. 1 ' 2 h x 20.   2 ' 3 3 f t t 21. 1 ' 1 2 y x 22. 23. The graph of   y f x is shown as a dashed curve.
Revised 03-2016 18 24. Algebraically, we find that 2 ) 2 ( ) ( lim 2 ) 2 ( ) ( lim 2 2 x f x f x f x f x x . Graphically, we see that the graph of   2 f x x has a corner at 2 x . 25. Since f x is not defined to the left of 2 x , the two-sided limit of 2 ) 2 ( ) ( x f x f does not exist at 2 x ; hence, the derivative fails to exist there. 26. (a) 2 2 1 ' x y x (b) 1 ' 1 2sin cos p q q (c)  2 2 3 ' 3 3 1 2 1   y x x x x x (d) 2 1 2 ' 1   z w w w (e) sin 2 ' 2cos 2 f e (f)   1 2 cos ' tan sin 1 m nz A z n mz nz mz (g)         ln ' ln 3 3 1 ln x x f x x (h) 4 9 2 2 2 6 5 2 2 ' 5 1 1 1 y y y y

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