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Math 1210-001 Lab 2 Spring 2016 Due Date: 01/28/2016 Name: uID: Instructions: This lab relates to sections 1.5 (limits with infinity), 1.6 (continuity), with some material from section 1.4 (trigonometric limits). Please show all of your work as partial credit will be given where appropriate, and there may be no credit for problems where there is no work shown. All answers should be completely simplified, unless otherwise stated. No calculators or electronics of any kind are allowed. This assignment is due at the start of your lab meeting on Thursday January 28. 1. (16 points) Continuity State whether the indicated function is continuous at c = 1. Justify your answer. (a) f (x) = (x − 1)(x + 3) (b) g(x) = |x − 1| (c) h(x) = 6 x−1 (d) h(t) = t2 −1 t−1 2 t 6= 1 . t=1 2. Continuity (Graph-Based) (a) (8 points) We say that a function is continuous on an open interval if it is continuous at each point in that interval. If the interval includes one or more of its endpoints, then continuity on the interval means that the function is also continuous from the right at any left-hand endpoint, and continuous from the left at any right-hand endpoint. From the graph of h given below, indicate the intervals on which h is continuous. Indicate left or right continuity when relevant. Page 2 (b) (8 points) Sketch the graph of a function f that satisfies the following conditions: • • • • Its domain is [-2,2]. It satisfies f (−2) = f (−1) = f (1) = f (2) = 1. It is discontinuous only at at −1 and 1. It is right continuous at -1 and left continuous at 1. Page 3 3. (10 points) In this problem, each given function is not defined at a certain point. State the point at which it is not defined. How should the function be defined in order to make it continuous at that point? (a) g(x) = (b) h(t) = x2 +6x+8 x+2 √ t−1 t−1 4. (10 points) Identifying Discontinuities State the points, if any, at which the following functions have discontinuities. Are they removable or nonremovable? (a) g(u) = u2 +|u−1| √ 3 u+1 2 x<0 x −x 0 ≤ x ≤ 1 . (b) h(x) = x x>1 Page 4 5. (10 points) Find the values of a and b such that the function x+1 x<1 ax + b 1 ≤ x < 2 f (x) = 3x x≥2 is continuous everywhere. Page 5 6. (14 points) A cab company charges $2.50 for the first 14 mile and $0.40 for each additional 1 mile. Sketch a graph of the cost of a cab ride as a function of the number of miles 8 driven (up to 2 miles). Discuss the continuity of this function. Page 6 7. (24 points) x3 x→0 cos x + sin x (a) lim sin x x→0 x + 2 (b) lim r (c) lim x→∞ 3 1 + 64x2 8x2 + 5 Page 7 (d) lim− x→5 cos (x − 5) x−5 sin(t) t→2 (t − 2)2 (e) lim sin2 (3t) t→0 4t (f) lim Page 8