(D) 3 ∞ (E) Free Response Questions Show all of your work and clearly label any functions, graphs, tables, or other objects that you use. On the AP, your work will be scored on the correctness and completeness of your methods, as well as your actual answer. You will usually not be given credit for answers that don’t include supporting work. Let . (a) What is the average rate of change of on the interval ? f (b) What is ? (c) Is the line = 0 a vertical asymptote of Justify your answer using limits. x f? (d) Use the Squeeze Theorem to show the line = 0 is a horizontal asymptote of y f. Correct Answer (a) (b) (c) No, , so neither the left-hand limit nor the right hand limit is infinite, which is needed for the graph to have a vertical asymptote. (d) We know sin 1, so if > 0, then , and since , the Squeeze Theorem implies . This means the x x −1 ≤ ≤ line = 0 is a horizontal asymptote. y Let (a) Are the lines = 5 and = vertical asymptotes of Justify your answer. x x f? −5 (b) Identify all horizontal asymptotes of Justify your answer. f. (c) Is there a value of that makes continuous at = 5? A f x (d) Is there a value of that makes continuous at = B f x −5? Let (x) be a function defined for all with (x) 10. Also, does not exist but (0) = 3. f x, f f −5 ≤ ≤ (a) Let (x) = Show is continuous at = 0. g xf(x). g x (b) Does the graph of have a tangent line at (0, 0)? Explain. g
(D) 3 (E) ∞ Free Response Questions Show all of your work and clearly label any functions, graphs, tables, or other objects that you use. On the AP, your work will be scored on the correctness and completeness of your methods, as well as your actual answer. You will usually not be given credit for answers that don’t include supporting work. Let . (a) What is the average rate of change of f on the interval ? (b) What is ? (c) Is the line x = 0 a vertical asymptote of f? Justify your answer using limits. (d) Use the Squeeze Theorem to show the line y = 0 is a horizontal asymptote of f. Correct Answer (a) (b) (c) No, , so neither the left-hand limit nor the right hand limit is infinite, which is needed for the graph to have a vertical asymptote. (d) We know −1 ≤ sin x ≤ 1, so if x > 0, then , and since , the Squeeze Theorem implies . This means the line y = 0 is a horizontal asymptote. Let (a) Are the lines x = 5 and x = −5 vertical asymptotes of f? Justify your answer. (b) Identify all horizontal asymptotes of f. Justify your answer. (c) Is there a value of A that makes f continuous at x = 5? (d) Is there a value of B that makes f continuous at x = −5? Let f (x) be a function defined for all x, with −5 ≤ f (x) ≤ 10. Also, does not exist but f (0) = 3. (a) Let g (x) = xf(x). Show g is continuous at x = 0. (b) Does the graph of g have a tangent line at (0, 0)? Explain.