MATH 140A MIDTERM EXAMINATION I September 27, 2004Name ID # Section #There are 12 multiple-choice questions, 9 True/False questions, and 2 partial credit / shortanswer questions. For the partial credit problems you must present your work clearly andunderstandably; no credit will be given for unsupported answers. For each multiple-choiceproblem five possible answers are given, only one of which is correct. Circle the correctanswer in your exam booklet and blacken the corresponding space on the scantron form.Mark only one choice; darken the circle completely (you should not be able to see the letterafter you have darkened the circle). For True/False problems, please circle the correct answerin each question.THE USE OF CALCULATORS IS NOT PERMITTED IN THISEXAMINATION.There are 15 problems on 10 pages, including this one.Check your booklet now.The area below is for the instructor’s use.MC .................... (60)T/F ................... (18)14 ...................... (10)15 ...................... (12)Total ................. (100)MATH 140A MIDTERM EXAMINATION I PAGE 21. (5 pts.) A polynomial function always has all the properties listed below, EXCEPTa) Its domain is (−∞, ∞).b) It is continuous everywhere.c) Its graph has exactly one y-intercept.d) Its graph does not have any vertical asymptote.e) Its range is (−∞, ∞).2. (5 pts.) Suppose f (x) = x3+ 1 and h 6= 0, find the difference quotientf(x + h) − f (x)h.a) 3x2+ 3hx + h2b) 3x2c)x3+ 3hx2+ 3h2x + h3hd) h2e)1hMATH 140A MIDTERM EXAMINATION I PAGE 33. (5 pts.) limx→3x2+ x − 12x2− 5x + 6=a) 7b) 0c) 1d) +∞e) −∞4. (5 pts.) limx→1−x + 3x2(x − 1)=a) 0b) 3c) 4d) −∞e) +∞MATH 140A MIDTERM EXAMINATION I PAGE 45. (5 pts.) limx→4√x2+ 9 − 5x − 4=a) 0b)−15c)45d) +∞e) −∞6. (5 pts.) limx→−2x + 5(x + 2)2=a) −∞b) −2c) 3d) 0e) +∞MATH 140A MIDTERM EXAMINATION I PAGE 57. (5 pts.) limx→3+9 − x2|3 − x|=a) −6b) 0c) 6d) −∞e) +∞8. (5 pts.) Iff(x) =(−x2+ c , x < −1cx + 3 , x ≥ −1is a continuous function defined on (−∞, ∞), then what is the value of c?a) −1b) 2c) 3d) 4e) Not enough information is given to determine the value of c.MATH 140A MIDTERM EXAMINATION I PAGE 69. (5 pts.) According to the Intermediate Value Theorem, which of the polynomials below musthave a real root on the interval (−1, 0)?a) x3− 3x2+ 3x − 4b) x3− 3x2+ 3x − 1c) x3+ 2x2+ 4x − 1d) x3+ 2x2+ 4x + 2e) x3+ 2x2+ 4x + 510. (5 pts.) Which of the equations below represents the line tangent to the circle given byx2+ y2= 5, at the point (1, −2)?a) y = −12x −32b) y = −2x − 4c) y = −2xd) y =12x +32e) y =12x −52MATH 140A MIDTERM EXAMINATION I PAGE 711. (5 pts.) Which of the following is an infinite discontinuity (where it has a vertical asymptote)on the graph of the functionf(x) =x(x + 1)(x + 3)2x2(x2− 1)(x + 3)?I. x = −3, II. x = −1, III. x = 0, IV. x = 1.a) IV onlyb) I and III onlyc) III and IV onlyd) I, II, and III onlye) I, II, III, and IV12. (5 pts.) The average rate of change of the function f(x) = x3−5x+4, on the interval [−2, 2],isa) −4b) −1c) 0d) 1e) 4MATH 140A MIDTERM EXAMINATION I PAGE 813. (18 pts., 2 pts. each) True or False:a) T F For all real numbers a and b, a2− b2= (a + b)(a − b).b) T F For all real numbers a and b, |a + b| = |a|+ |b|.c) T F For all a, b ≥ 0,√a2+ b2= a + b.d) T F For all a > 0, ax+y= axay.e) T F If f (x) is not continuous at x = c, then f(c) does not exist.f) T F If limx→7f(x) = L and limx→7g(x) = M , then limx→7(3f(x) − 2g(x)) = 3L − 2M.g) T F If limx→c−f(x) 6= limx→c+f(x), then f(x) cannot be continuous at x = c.h) T F The graph of a rational function might have one or more discontinuities.i) T F If f(x) is a continuous function such that f (1) = 7 and f (5) = −3. Thenthere is always at least one point, x = c, on the interval (1, 5) such that f(c) = 6.MATH 140A MIDTERM EXAMINATION I PAGE 914. (10 pts.) Supposef(x) =x2+ 4 , x ≤ 02x − 2(x − 1)(x − 3), 0 < x ≤ 4−4x − 6, x > 4For each of the points given below, determine whether f (x) is continuous there. For eachdiscontinuity found, determine its type.At x = 0 Answer:At x = 1 Answer:At x = 3 Answer:At x = 4 Answer:At x = 6 Answer:MATH 140A MIDTERM EXAMINATION I PAGE 1015. (12 pts.) Considerf(x) =x2+ ax + bcx + 8such that:I. the domain of f consists of all real numbers except x = −4 (it has a removable discontinuitythere),II. the graph of f has an x-intercept of 2, andIII. the graph of f has a y-intercept of −1.Find the values of a, b and c.Answer: a = , b = , c =