Mathematics 31B - Practice Midterm 1Exam Date: October 22, 2012Instructor: D.E.WeisbartNAME (please print legibly):Your University ID Number:Your Discussion Section and TA:Signature:• There are SEVEN questions on this examination.• Calculators, notes and books may not be used in this examination.• You may not receive full credit for a correct answer if insufficient work isshown.QUESTION VALUE SCORE1 122 123 124 125 126 127 12TOTAL 8411. (12 points)a. Simplify the following expression in terms of ln(2) and ln(5). Computeln(209) =—————————Answerb. Your answer to the following question should be expressed as a whole number. Computee(ln(10)+ln(23))=—————————Answerc. Computelimn→∞ 1 +43n2n!=—————————Answer22. (12 points)a. Suppose that f (x) is differentiable, that f (3) = 8 and that f0(3) = 4. Thenf−10(8) =—————————Answerb. Suppose that f (x) is differentiable, that f(2) = 4 and that f0(2) =12. Find the equationfor the line tangent to the graph of f−1(x) at the point (4, 2).—————————Answer33. (12 points)a. Compute the derivative of the functionf(x) = ln(x2+ 2) + e2x2+x.—————————Answerb. Compute the derivative of the functionf(x) = log4xe3x2.—————————Answerc. Compute the derivative of the functionf(x) = (x2+ 5x + 1)2x+1.—————————Answer44. (12 points)a. Suppose thaty0(x) = 2y − 1 and y(0) = 10.Compute y(4).—————————Answerb. A radioactive isotope has half-life 100 years. If you initially have 100 grams of thematerial, how much will you have after 750 years.—————————Answerc. Your friend gives you and unknown amount of a radioactive isotope. After 2 years youhave 100 grams and after 4 years you have 80 grams. How much will you have after 10 years?—————————Answer55. (12 points)a. Can you use l’Hopital’s rule to calculate the following limit.limx→1x2− 1x + 1=—————————Answerb.limx→1ex2− ex − 1=—————————Answerc.limx→∞ln xx=—————————Answer66. (12 points)a.Z2x2e3xdx =—————————Answerb.Zln(3x) dx =—————————Answer77. (12 points)a.Ze2xsin(3x) dx =—————————Answerb.Z1012 + 9x2dx =—————————Answerc.Z1−11√2 − x2dx