Download 6 Questions on Calculus I - Exam 1 | MATH 131 and more Exams Calculus in PDF only on Docsity! Name (Last, First) ID # Signature Lecturer Section # university of massachusetts amherst department of mathematics and statistics Math 131 Exam 1 October 6, 2005 6:00-7:30 p.m. Instructions • Turn off all cell phones and watch alarms! Put away cell phones, iPods, etc. • There are eight (8) questions. • Do all work in this exam booklet. You may continue work to backs of pages and the blank page at the end, but if you do so indicate where. • Do not use any other paper except this exam booklet and the one-page “cheat sheet” that you prepared. • Organize your work in an unambiguous order. Show all necessary steps. • Answers given without supporting work may receive 0 credit! • Do not write anything in the table below. • Be prepared to show your UMass ID card when you hand in your exam booklet. QUESTION PER CENT SCORE 1 16 2 16 3 20 4 16 5 16 6 16 Extra credit 6 TOTAL 106 1. (4 × 4% = 16%) Let f(x) = √ 1 − 2 x x . (a) Determine the domain of f . (b) Derive a formula for the inverse function f−1 of f . (c) What is f ( f−1 ( 2 π )) ? (An exact value, not a numeric approximation!) (d) If g(y) = y2 for all y, then what is (g ◦ f)(3 x)? 1 4. (2×8% = 16%) Use the definition of the derivative in terms of limits to find: (a) f ′ (3) where f(x) = x2 − 4 x + 5. (b) g ′ (x) where g(x) = 1 3 − x . 4 5. (16%) Through decay, Chemical Q loses one-third of its mass every 7 years. Twenty-five years ago, a container of Chemical Q was buried. The EPA has just dug it up and found 1,000 grams remaining. How much Chemical Q was originally buried? (Carefully identify the variables you use!) 5 6. (2 × 8% = 16%) A magnetic field is induced by electrical current moving through a wire. The strength B of the magnetic field at a distance r from the center of a wire is given by: B(r) = { B0 r r0 if r ≤ r0, B0 r0 r if r > r0 for certain non-zero constants B0 and r0. (a) Is the function B(r) continuous at r0? Why or why not? (b) Is the function B(r) differentiable at r0? Why or why not? 6
