Math 100 Dec 93 - Math 100 Exam/Dec 1993 Math 100 Exam Dec 1993 PART A DO ALL 8 QUESTIONS IN PART A J 1[12 marks Find the derivatives of the

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Unformatted text preview: Math 100 Exam/Dec. 1993 Math 100 Exam Dec. 1993 PART A: DO ALL 8 QUESTIONS IN PART A. J 1. [12 marks] Find the derivatives of the following functions. You do not have to simplify your answers. (a) y = J21 +1 tan21: (b) y = tan“‘(a:"’) Note: tan‘1 5 arctan (c) u = (2+ 1)“ (d) “191%;351) 2. [10 marks] Find the derivative of 1(3) = \lz’ + .1: directly from the J definition. No credit will be given if you use the rules of differentiation. 1 3. [10 marks] The equation 2:”; + 21113 = 8 defines y as a function of 1:, \] y=f(:r),nearz=2.y=l. (a) Find the slope of the curve :21; + 2x113 = 8 when a: = 2, y = l. (b) Use the linear approximation (tangent line approximation) to find an approximate value for f (1.92). (c) Given that y” = % for a: = 2 and y = 1, does the tangent line approxi- mation yield a bigger or smaller answer than the actual value in some small neighborhood of a: = 2? You must give reasons for your answer. 4. [8 marks] A half mile race track consists of two opposite sides of a rect- Page 1 angle with semi-circular ends as shown in the following diagram. Find the dimensions a: and y which will maximize the area of the rectangle. 5. [8 marks] A 50 foot ladder is placed against a wall of a large building. The base of the ladder, resting on slippery ground, slips away from the wall at a rate of 3 feet per second. Find the rate of change of the height of the ladder top when the base of the ladder is 30 feet away from the wall. 6. [12 marks] Evaluate the following limits. t_ i J? (‘l 332.. ——1‘- s/i + —- fl sin" -l- — 1 - (b) hm ———23’—§- Note: sin"l .=.. arcsin 2—01 I — l . . 1/, (c) [1330 + sin 2:) . a' id) 21320 1 “1,, the value of a. where a > 0 is a constant. Hint: The answer depends on 1 7. [8 marks] Let f(:r) = ze' — 2. (a) Prove that f (1:) = 0 has exactly one solution r between 1: = 0 and z = l. Math 100 Exam/Dec. 1993 (b) Give the “Newton’s method" iteration formula for finding 1'. You do not actually have to find the root. (c) Show that for any initial value so, 0 5 1:0 3 1, the value of 11 from Newton's method satisfies 2:1 2 r. 8. [12 marks] Let y = 22"” + 22/3. (a) Find all intervals on which the function is increasing, decreasing, concave up or concave down, and find all inflection points (if any). (b) Determine the coordinates of all local or absolute maxima and minima (if any). (c) Determine the asymptotes of the graph (if any). ((1) Plot the graph. Notes: You must show all your work to get full marks. No marks will be given for a graph that does not match the calculations in parts a, b and c. PART B: CONSISTS OF 4 QUESTIONS. YOU ARE TO DO 2. YOU MUST SPECIFY WHICH 2 YOU WANT MARKED. l 2 a: sin— forty“) 9. [10 marks] Let f(z) be defined by f(a:) = {0 x f o 01' 3 = . (a) Use the definition of the derivative to find f’(0). (b) Explain why f'(:r) is not continuous at z = 0. 10. [10 marks] Prove that: (a) sinzzzsfor031sg. Page 2 2 (b) coszgl—i-forOst 1. 1r 2 11. [10 marks] Let f (2:) be a function defined on the real line such that f’(::) and f"(::) exist for all 2. Suppose f(1) = 0, f(2) = 3, f(3) = 2 and f(4) = 4. (a) Prove that f'(a) = 0 for some a. (b Prove that f”(b) = 0 for some b. 12. [10 marks] A lake of constant volume V gallons contains Q(t) pounds of pollutant at time t, evenly distributed throughout the lake. Water containing a concentration of 1: pounds per gallon of pollutant enters the lake at a rate of r gallons per minute, and the well mixed solution leaves at the same rate. (a) If 00 is the amount of pollutant in the lake at time t = 0 find Q(t) for any time I. (b) Determine the limiting amount of pollutant in the lake (i.e. find guano 0(a)). (c) If k = 0 find the time T for the amount of pollutant to be reduced to 50% of its original value. ...
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This note was uploaded on 01/30/2011 for the course MATH 100 taught by Professor Lamb during the Fall '08 term at The University of British Columbia.

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Math 100 Dec 93 - Math 100 Exam/Dec 1993 Math 100 Exam Dec 1993 PART A DO ALL 8 QUESTIONS IN PART A J 1[12 marks Find the derivatives of the

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