WebAssign hw2.2 (Homework)

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1 WebAssign hw2.2 (Homework) Current Score : / 92 Due : Wednesday, May :25 AM PDT Michael Lee Math261(Calculus I), section 1049, Spring 2017 Instructor: Michael Lee 1. /2 pointsscalc Explain what is meant by the equation = 8. x 1 The values of f(x) can be made as close to 1 as we like by taking x sufficiently close to 8. If x 1 1 < x 2 1, then f(x 1 ) 8 < f(x 2 ) 8. If x 1 1 < x 2 1, then f(x 1 ) 8 f(x 2 ) 8. f(x) = 8 for all values of x. The values of f(x) can be made as close to 8 as we like by taking x sufficiently close to 1. Is it possible for this statement to be true and yet f(1) = 9? Explain. Yes, the graph could have a hole at (1, 8) and be defined such that f(1) = 9. Yes, the graph could have a vertical asymptote at x = 1 and be defined such that f(1) = 9. No, if f(1) = 9, f(x) then = 9. x 1 No, f(x) if = 8, then f(1) = 8. x 1 2. /2 pointsscalc Explain what it means to say that = 9 and = 6. x 8 x 8 + As x approaches 8 from the right, f(x) approaches 9. As x approaches 8 from the left, f(x) approaches 6. As x approaches 8, f(x) approaches 6, but f(8) = 9. As x approaches 8 from the left, f(x) approaches 9. As x approaches 8 from the right, f(x) approaches 6. As x approaches 8, f(x) approaches 9, but f(8) = 6. In this situation is it possible that exists? Explain. x 8 Yes, f(x) could have a hole at (8, 9) and be defined such that f(8) = 6. Yes, f(x) could have a hole at (8, 6) and be defined such that f(8) = 9. Yes, if f(x) has a vertical asymptote at x = 8, it can be defined such that = 9, = 6, f(x) and exists. x 8 x 8 + x 8 No, cannot exist f(x) if. x 8 x x Responses/last?dep= /18

2 3. /2 pointsscalc Explain the meaning of each of the following. (a) = x 4 The values of f(x) can be made arbitrarily close to 4 by taking x sufficiently large. The values of f(x) can be made arbitrarily large by taking x sufficiently close to (but not equal to) 4. The values of f(x) can be made arbitrarily close to 0 by taking x sufficiently close to (but not equal to) 4. f( 4) = (b) = x 2 + The values of f(x) can be made negative with arbitrarily large absolute values by taking x sufficiently close to, but greater than, 2. As x approaches 2, f(x) approaches. The values of f(x) can be made arbitrarily close to by taking x sufficiently close to 2. f(2) = 4. /6 pointsscalc Use the given graph of f to state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.) (a) x 2 (b) x 2 + (c) x 2 (d) f(2) (e) x 4 (f) f(4) Responses/last?dep= /18

3 5. /5 pointsscalc For the function f whose graph is given, state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.) (a) x 1 (b) x 3 (c) x 3 + (d) x 3 (e) f(3) Responses/last?dep= /18

4 6. /12 pointsscalc For the function h whose graph is given, state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.) (a) h(x) x 3 (b) h(x) x 3 + (c) h(x) x 3 (d) h( 3) (e) h(x) x 0 (f) h(x) x 0 + (g) h(x) x 0 (h) h(0) (i) h(x) x 2 (j) h(2) (k) h(x) x 5 + (l) h(x) x 5 Responses/last?dep= /18

5 7. /8 pointsscalc For the function g whose graph is given, state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.) (a) g(t) t 0 (b) g(t) t 0 + (c) g(t) t 0 (d) g(t) t 2 (e) g(t) t 2 + (f) g(t) t 2 (g) g(2) (h) g(t) t 4 Responses/last?dep= /18

6 8. /9 pointsscalc For the function f whose graph is shown, state the following. (If an answer does not exist, enter DNE.) (a) x 7 (b) x 3 (c) x 0 (d) x 6 (e) x 6 + (f) The equations of the vertical asymptotes. x = (smallest value) x = x = x = (largest value) Responses/last?dep= /18

7 9. /2 pointsscalc MI. A patient receives a 150 mg injection of a drug every 4 hours. The graph shows the amount f(t) of the drug in the bloodstream after t hours. Find f(t) t 12 f(t) = t 12 f(t) = t 12 + and f(t). t 12 + mg mg Responses/last?dep= /18

8 10. /2 pointsscalc Sketch the graph of the function. f(x) = 3 + x if x < 2 x 2 if 2 x < 2 6 x if x 2 Use the graph to determine the values of a for which a = x a does not exist. (Enter your answers as a comma separated list.) Responses/last?dep= /18

9 11. /3 pointsscalc Use the graph of the function f to state the value of each it, if it exists. (If an answer does not exist, enter DNE.) f(x) = (a) /x x 0 (b) x 0 + (c) x 0 Responses/last?dep= /18

10 12. /1 pointsscalc Sketch the graph of an example of a function f that satisfies all of the given conditions. x 0 f(x) = 2, x 0 + f(x) = 1, f(0) = 1 Responses/last?dep= /18

11 13. /1 pointsscalc Sketch the graph of an example of a function f that satisfies all of the given conditions. = 6, x 5 + = 4, x 5 = 4, x 2 f(5) = 5, f( 2) = /1 pointsscalc Use a table of values to estimate the value of the it. If you have a graphing device, use it to confirm your result graphically. (Round your answer to two decimal places.) θ 0 sin(5θ) tan(4θ) 15. /1 pointsscalc Determine the infinite it. x + 6 x 7 + x 7 Responses/last?dep= /18

12 16. /1 pointsscalc Determine the infinite it. x 8 7 x (x 8) /1 pointsscalc Determine the infinite it. 2 sec(x) x (π/2) + x 18. /1 pointsscalc Determine the infinite it. cot(x) x π /1 pointsscalc Determine the infinite it. x csc(x) x 2π 20. /1 pointsscalc Determine the infinite it. x 2 2x x 2 + x 2 4x Responses/last?dep= /18

13 21. /3 pointsscalc Find the vertical asymptotes of the function. y = x x 2x 2 x = (smaller value) x = (larger value) Confirm your answer by graphing the function. (A graphing calculator is recommended.) Responses/last?dep= /18

14 22. /2 pointsscalc Evaluate the function for values of x that approach 1 from the left and from the right. f(x) = = x 1 1 x 3 1 = x /7 pointsscalc (a) Evaluate h(1) = h(0.5) = h(0.1) = h(0.05) = h(0.01) = h(0.005) = h(x) = tan(x) x x 3 for x = 1, 0.5, 0.1, 0.05, 0.01, (Round your answers to six decimal places.) (b) Guess the value of x 0 tan(x) x x 3. (If an answer does not exist, enter DNE.) Responses/last?dep= /18

15 24. /1 pointsscalc A graphing calculator is recommended. Graph the function f(x) = sin(π/x) of the example in the viewing rectangle [ 1, 1] by [ 1, 1]. Then zoom in toward the origin several times. Comment on the behavior of this function. No matter how many times we zoom in toward the origin, the graphs of f(x) = sin(π/x) appear to consist of almost vertical lines. This indicates that f(x) has a vertical asymptote at x = 0. No matter how many times we zoom in toward the origin, the graphs of f(x) = sin(π/x) appear to consist of almost horizontal lines. This indicates that f(x) 0 as x 0. No matter how many times we zoom in toward the origin, the graphs of f(x) = sin(π/x) appear to consist of almost horizontal lines. This indicates that f(x) has a horizontal asymptote at y = 0. No matter how many times we zoom in toward the origin, the graphs of f(x) = sin(π/x) appear to consist of almost vertical lines. This indicates more and more frequent oscillations as x 0. No matter how many times we zoom in toward the origin, the graphs of f(x) = sin(π/x) appear to consist of almost vertical lines. This indicates that f(x) 0 as x 0. Responses/last?dep= /18

16 25. /9 pointsscalc (a) Use numerical and graphical evidence to guess the value of the it. x 1 x 3 1 x 1 (i) Let x 3 1 y =. x 1 five decimal places.) Fill out the table to find numerical evidence to help identify the value of the it. (Round your answers to x y (ii) Plot y = x3 1 x 1 to find graphical evidence to help identify the value of the it. (iii) Using the evidence found in (i) and (ii), estimate the it. (Give your answer to the nearest whole number.) x 1 x 3 1 = x 1 (b) How close to 1 does x have to be to ensure that the function in part (a) is within a distance 0.5 of its it? (Round your answer to four decimal places.) To ensure that y is within 0.5 of its it, it is required that x be within of 1. Responses/last?dep= /18

17 26. /5 pointsscalc XP.MI. Use the given graph of f to state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.) (a) x 5 (b) x 5 + (c) x 5 (d) x 9 (e) f(9) 27. /1 pointsscalc XP. Use a table of values to estimate the value of the it. If you have a graphing device, use it to confirm your result graphically. (Round your answer to two decimal places.) 8 x 5 x x 0 x 28. /1 pointsscalc XP. Determine the infinite it. x + 7 x 6 + x Responses/last?dep= /18

18 29. /1 pointsscalc XP.MI. Determine the infinite it. x + 5 x 6 x Responses/last?dep= /18

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1 of 10 2/7/ :49 AM Limit of a Function (Section 2.2) (2344034) Question 1 2 3 4 5 6 7 8 9 10 11 12 1. Question Details SCalcET7 2.2.001. [1733502] Explain what is meant by the equation f(x) = 5. x 9 If x 1 9 < x 2 9, then