MATH 140A MIDTERM EXAMINATION I September 29, 2003Name ID # Section #There are 10 multiple choice questions, 9 True/False questions, and 3 free response questions.To receive full credit for free response questions (problems 13, 14 and 15) allwork must be shown.THE USE OF CALCULATORS IS NOT PERMITTED IN THISEXAMINATION.There are 15 problems on 12 pages, including this one.Check your booklet now.The space below is for the instructor’s use.MCT/F13.14.15.TotalMATH 140A MIDTERM EXAMINATION I PAGE 21. (5 pts.) The equation 2x2+ 2y2= 30 − 4x describesa) a circle: radius = 4, center = (−1, 0)b) a circle: radius = 4, center = (1, 1)c) a circle: radius = 2, center = (−2, 0)d) a parabola: vertex = (0,1)e) a parabola: vertex = (2,0)2. (5 pts.) Let f (x) =3x + 1x − 2. What is its inverse function, f−1(x)?a) f−1(x) =x − 23x + 1b) f−1(x) =2x + 1x − 3c) f−1(x) =x + 32x − 1d) f−1(x) =−3x − 1x − 2e) f−1(x) does not exist.MATH 140A MIDTERM EXAMINATION I PAGE 33. (5 pts.) limx→2x2− x − 2x2− 3x + 2=a) 0b)13c) 3d) ∞e) Does not exist.4. (5 pts.) limx→5x − 5(x − 5)3=a) 0b)15c)125d) ∞e) −∞MATH 140A MIDTERM EXAMINATION I PAGE 45. (5 pts.) limx→3√x2− 5 − 2x − 3=a)32b) 0c) 6d) ∞e) Does not exist.6. (5 pts.) limx→0x + 1x(x − 1)=a) 1b) −1c) −∞d) ∞e) Does not exist.MATH 140A MIDTERM EXAMINATION I PAGE 57. (5 pts.) limx→1−x2+ 1x − 1=a) −1b) −2c) 2d) −∞e) ∞8. (5 pts.) How many vertical asymptote(s) does the rational function f(x) =(x + 1)(x − 2)x(x + 1)(x − 2)2have?a) 0b) 1c) 2d) 3e) 4MATH 140A MIDTERM EXAMINATION I PAGE 69. (5 pts.) If the functionf(x) =x2+ cx − 2 x ≤ 1x + c3xx > 1is continuous at x = 1, then what is the value of c?a) c = 0b) c = 1c) c = 2d) c = 4e) No such value can be found for f (x) to be continous at x = 1.MATH 140A MIDTERM EXAMINATION I PAGE 710. (5 pts.) Let f be a function such that f(2) = 1 and f0(2) = 3. What is an equation of theline tangent to the graph y = f(x) at x = 2?a) y = x − 3b) y = 2x + 1c) y = 2x + 5d) y = 3x − 2e) y = 3x − 5MATH 140A MIDTERM EXAMINATION I PAGE 811. (8 pts. 2 pts. each) True or False: (Circle the appropriate letter.)Suppose f is a function defined on a subset of the real line.T F If f is continuous at x = c, then f (c) is defined.T F If limx→cf(x) exists, then f (c) is defined.T F If f is continuous at x = c, then it is continuous from the right and continuous fromthe left at x = c.T F Any rational function is continuous at each point where the function is defined.MATH 140A MIDTERM EXAMINATION I PAGE 912. (10 pts. 2 pts. each) True or False: (Circle the appropriate letter.)Suppose f is a polynomial function such that f (−1) = 1, f(0) = 5, f(2) = 2 and f (4) = −3.T F The average rate of change of f on the interval [0, 2] is−32.T F limx→0f(x) = 5.T F f is continuous at x = −10.T F The equation f (x) = 0 must have a solution in the interval (−1, 2)T F The equation f (x) = 0 must have a solution in the interval (2, 4)MATH 140A MIDTERM EXAMINATION I PAGE 1013. (10 pts.) Supposef(x) =x2+ xx2− 1, x < 3−x2(x − 2)(x − 4), x ≥ 3Find all discontinuities. For each discontinuity, determine its type. You must justify youranswer.MATH 140A MIDTERM EXAMINATION I PAGE 1114. (10 pts.) Consider the parabola described by y + x2− 2x + 9 = 0.(a) Find the coordinates of the vertex.(b) Find the y−intercept and the x−intercepts, if any.(c) Sketch the graph of the parabola, labeling the coordinates of the vertex and the intercepts.MATH 140A MIDTERM EXAMINATION I PAGE 1215. (12 pts.) Suppose f (x) =ax + bx2+ cx + 3such thatI. the domain of f consists of all real numbers except x = 1 and x = 3,II. the graph of f has a vertical asymptote at x = 1, but not at x = 3 (where it has aremovable discontinuity),III. the graph of f has a y-intercept of 2.Find the values of a, b and