Name: ID#:Midterm IMath 1aIntroduction to CalculusMarch 7, 2005Show all of your work. Full credit may not be given for an answer alone.You may use the backs of the pages or the extra pages for scratch work. Do notunstaple or remove pages.This is a non-calculator exam.Students who, for whatever reason, submit work not their own willordinarily be required to withdraw from the College.—Handbook for StudentsProblem Possible PointsNumber Points Earned1 202 83 104 105 10Total 581 11. (20 Points) Letf(x) =−1xif x < 0;x2− x if 0 ≤ x ≤ 1;x − 1 if 1 < x < 2;0 if x ≥ 2..(a) Find the left- and right-hand limits at 0, 1, and 2. Justify your answers.11 1(b) At what points is f not continuous? Justify.(c) Does the graph of f have any asymptotes? Justify./ 2022 22. (8 Points) Here is a table the percentage of low -weight (< 2500 grams)births among African-American mothers in various years.Year % low birthweight1990 13.251991 13.551992 13.311993 13.341994 13.241995 13.13199613.011997 13.011998 13.051999 13.112000 12.992001 12.95200213.29(i) What is the average rate of change in the low-birthweight percentage rateover the years 1992 to 1998?(ii) Estimate the (instantenous) rate of change in the low-birthweight percent-age rate in 1994./ 833 33. (10 Points)(i) Let f b e a function. State the definition of the derivative of f at a.(ii) Let f (x) =1x − 2. Use the definition of the derivative to find f0(a).Note. No credit will be given for answers which use the product rule,quotient rule, power rule, et cetera./ 1044 44. (10 Points) Find the following derivatives. Use any method you like.(i)ddx4x3+ 3x2+ 2x + 1(ii)ddxex(x − 2).(iii) f0(x), where f(x) =2x + 34x − 2. Simplify your answer./ 1055 55. (10 Points)(i) State the Intermediate Value Theorem.(ii) Use it to prove that there exists a number c between 0 and 1 which solvesthe equationarctan x = 2x −π3./ 106(This page intentionally left
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