Name of Student: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(Your Instructor’s Name: . . . . . . . . . . . . . . . . . . . . . . . . . . )First Mid-Term of Math 1aOctober 17, 2000 (Tuesday)7 p.m. - 9 p.m., Science Center Hall C & EInstructors: Yum-Tong Siu (course head), Peter Clark, Kim FroyshovDeepee Khosla, Yang Liu, Russell MannDavid Savitt, Kiril Selverov, Yuhan ZhaQuestion Points Score1 132 133 134 125 136 127 128 12Total 100• You have TWO hours to complete this examination.• No calculators are allowed.• No partial credit can be given for unsubstantiated answers.• Use the back of the page if more space is needed for your answer(with an indication that your answer is continued on the back of the page).1. Let f(x) = |x + 2|− 2|x| + |x − 3|.(a) Evaluate f (−5), f(12), and f(3).(b) Sketch the graph of f (x).(c) Let g(x) =1xand h(x) =√x − 3. Find the natural domains ofg ◦ f and h ◦ f . (Hint: refer to your graph!)12. Sketch the graph off(x) =x2− 1|x| − 1.Where is f continuous? Are there any removable discontinuities? (Re-call: a point is a removable discontinuity if the function f becomescontinuous after we change its value at that point.)23. Let f(x) =3−xx2−2x−8. Evaluate the following limits.(a) limx→∞f(x). (b) limx→−∞f(x). (c) limx→1+f(x).(d) limx→1−f(x). (e) limx→4+f(x). (f) limx→4−f(x).34. Compute the derivative of f(x) =x3−1x2+x.45. Compute f0(x) and f00(x) when f(x) = sin2(x4+ 1).56. Find all lines tangent to the graph of y = x2which pass through thepoint (1, −3).67. Suppose g(1) = 4, g0(1) = 3, and g00(1) = −2. Suppose also thatf(4) = 6, f0(4) = −1, and f00(4) = 5. What are the values of the firstand second derivatives of (f ◦ g)(x) at x = 1?78. Letf(x) =xnsin1x2for x 6= 00 for x = 0.(a) Find the smallest integer value for n such that f(x) is continuousat x = 0.(b) Find the smallest integer value for n such that f(x) is differentiableat x = 0.(c) Find the smallest integer value for n such that f00(x) exists atx =
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