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math102_2000 - The University of British Columbia December...

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Unformatted text preview: The University of British Columbia December 14. 2000 Mathematics 102 Section 103, 104 Instructor: Dr. Leah Keshet Duration: 2 hours 30 minutes I Be sure that you have .10 pages in addition to this One. 0 Put your FULL name below and on the back of the other pages. 0 In all questions, you must show work — i.e. display intermediate results — to get full credit. 0 You may use a calculator and a ruler. 0 Be careful! Read the questions carefully. Follow instructions exactly. Check all work. 0 Be neat! I will not attempt to decipher messy calculations or interpret messy pic- tures. o All work you wish to be graded must be placed on these sheets. Work books should be used for scrap work only, and must not be handed in. I will simply throw away any other papers I receive. H D H H 'H I... l I. o CD ,_. H c N First Name Last Name Signature Student Number H C C Mathematics 102 “—m Section 103. 104 __._____~__~_ page 1 out of 10 [10] 1. Find the derivatives of the following functions. Do not simplify your answers. (a) f(a:) i 111(2: + v:::2 + 1) 0)) W) = t2?” (c) f (2:) = arctan(1 — 32) (d) f($)= (COS($))/(1~Sin($)) EBE) EEQE REMOVE M, m. _ “Mn" .1» .M Im “W: “ Mathematics 102 ______ Section 103. 104 _________ page 2 out of 10 [20] 2. Each of the following questions can be done with little or no computation. Enter your answers in the boxes provided. (a) Shown below is the graph of a function y = Rsin(w(t — b)) + c. From the graph determine the values of the constants R, w, b, c. (b) Shown below is the graph of the velocity of a particle moving in one dimension. Indicate DIRECTLY on the graph any time(s) at which the particle’s acceleration is zero. (c) A function f (1) satisfies f (1) = —1 and f’ (1) = 2. What is the equation of the tangent line of f (x) at :c = 1? Mathematics 102 —m Section 103. 104 —________ page 3 out of 10 (d) Shown below is the graph of y = Ce"t for some constants C, k, and a tangent line. Use data from the graph to determine 0 and k. (0, 4) “ (e) Use the data in the graph below to make the best approximation you can to f (2.01). . Mathematics 102 ' Section 103, 104 M.__ page 4 out of 10 [10] 3. Consider the curve 1:4 + y4 = 2. Your answers to the parts below MUST be supported using calculus. (a) What isk'the equation of the tangent line to the curve 3:4 + y“ = 2 at (1, 1)? (b) Determine if the curve 3:4 + y4 = 2 is concave up or down near the point (1, 1). Mathematics 102 Section 103. 104 page 5 out of 10 [10] 4. (a) Shown below is the graph of a function y = f(:c). Sketch the graph of f’($) on the grid to the right. (b) Sketch on the grid below the graph of a function F(:c) such that its derivative corresponds to the original function, f (2:), given in part (a). Mathematics 102 Section 103, 104 ____.___,__ page 6 out of 10 [8] 5. The half life of strontium 90 is 29.1 years. Suppose a pasture field has been contaminated with strontium and has a level of radioactivity that is 10 times the ”safe level”. How long will it take until the level of radioactivity in the field will return to the ”safe level” ? Mathematics 102 Wm Section 103, 104 __~_______ page 7 out of 10 [12] 6. Consider the differential equation dy/dt = 4y — y2. (a) Sketch the slope field corresponding to this equation in the range —1 S y S 5, and sketch on the same figure the graphs of the solutions that also satisfy y(0) = 1 and y(0) = 4. (b) Find all steady state (i.e constant or equilibrium) solutions of this differential equation. (c) Let y(t) be the solution of this differential equation with y(0) = 1. Use Euler’s method with step size 0.1 to estimate y(0.2). ' Mathematics 102 Section 103, 104 [10] 7. Use Newton’s method to find a value of :1: that satisfies em — 122/2 = 0 . You MUST follow these instructions: 0 Use the starting value .120 = 0. a Display your answer to 4 significant places. a Show all intermediate steps. page 8 out of 10 Mathematics 102 _________.______ Section 103. 104 ____m page 9 out of 10 [10] 8. A cannon—ball fired by a cannon at ground level at angle 0 to the horizon (0 S 0 S 7r/2) will travel a horizontal distance (called the range, R) given by the formula below: R = (1/16)vgsin0c030. Here 110 > 0, the initial velocity of the cannon—ball, is a fixed constant and air resistance is neglected. (See diagram below.) What is the maximum possible range? [10] Mathematics 102 Section 103. 104 page 10 out of 10 9. A wheel of radius 1 meter rolls on a flat surface without slipping. The wheel moves from left to right, rotating clockwise at a constant rate of 2 revolutions per second. Stuck to the rim of the wheel isa piece of gum; as the wheel rolls along, the gum follows a path shown by the wide arc (called a ”cycloid curve”) in the diagram. The (2:, y) coordinates of the gum are related to the wheel’s angle of rotation 0 by the formulae 22:9—sin0, yzl—cosfi, where 0 S 0 S 27r. How fast is the gum moving horizontally at the instant that it reaches its highest point? How fast is it moving vertically at that same instant? ...
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