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M137.F08.assmt1 - Math 137 ASSIGNMENT 1 Fall 2008 Submit...

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Unformatted text preview: Math 137 ASSIGNMENT 1 Fall 2008 Submit all boxed problems and all extra problems by 8:20 am, September 19. All solutions must be clearly stated and fully justified. Use the format given on UWACE under Content, in the file Assignment Format for Math 135 and Math 137. NOTE : The TEXT problems without boxes are suggested for use in preparing for your As- signments, and/or for review before tests and examinations. TEXT PROBLEMS : Section 1.1 — 11, 12, 16, 17, 20, I, 29, I, 39, 50, 59, 64, 68 Section 1.2 — 5, 8, 17, 19, 20 Section 1.3 — 4a),d), I, 16, I, 24, I, I, 44, 48, 51, 53, 56, 61. (For problem 28, see page 16 of your Course Notes for guidance.) Appendix A — 23, 25, I 38, 46, 56, 57, 5.9, I, 61,-, 68 EXTRA PROBLEMS: 1 E11 Consider the function f (z) = m + E' a) State the domain Df. b) (i) On the same axes, sketch y = I and y = i for so > 0. (ii) Use your graphs to explain why, as it goes from 0 t0 1, f (a?) ranges from 00 to 2, While as 9; goes from 1 to 00, f (:r) ranges from 2 to 00 . c) Show that f is an odd function. d) Sketch the graph of y = f (at), and state the range Rf. E12 Sketch the given function using the definition of the absolute value function and/or graphical operations on basic functions. a) (4x—5)% b) 24122—1] E13 a) Find the domain of the function f(r) = M33: + 4 + i x [HINTz Rewrite f as ¥ and consider a: > 0, and :1: < 0 separately] 1‘4 — 4:132 b) Solve the inequality m > 0 by Sketching both y = m4 — 4x2 and y = $2 + 2:1: — 3 on the same axes, showing all intercepts. E1.4 A cylindrical tank has diameter 2 1n and height 6 m. a) Find a function V(d) 1113 which represents the total volume of fluid in the tank when it is filled to a depth of d In. State the domain and range of V(d), and sketch its graph. b) Find the depth d(V) in of the fluid if its volume is V m3. c) Suppose the tank is initially empty at time t = 0 and is filled at a rate of 10 litres per minute. Find the volume V(t) in3 after t minutes, and hence determine how long it takes to fill the tank. [HINT: Recall l m3 = 1000 litres]. E1.5* According to J. Maynard Smith (in Models in Ecology, Oxford : Cambridge University Press, 1974), in population models, the size f(a:) of the next generation in terms of the size :1: of the current generation can be modelled by A2: f($l=m a where A, a, and b are positive constants. a) Suppose a = b = 1, and /\ = 3 for a certain cell population, with :1: and f (CL) measured in millions of cells. Show that f (x) can be written as f(:c)=3(1- 1 ), 1+2: and hence find the limiting size of the population as 2: increases. 1 Sketch a qualitative graph of f(:1:). [ HINT: Start with y = — for w > 0 ; you will a: need a reflection, two shifts, and one scale, not necessarily in that order.] Suppose instead that a = 1, and b = A = 3. State the resulting model, sketch its graph for x > 0, and discuss how its predictions differ from those of the model in a), b). [HINT: See Example 7, pages 17—18 of your Course Notes for help with the qualitative graph] *Throughout your assignments in Math 137, problems marked with an * were adapted from problems in Calculus with Applications (81th. ed.) by M. Lial, R. Greenwell. and N. Ritchey. 2005. Pearson/Addison— Wesley. ...
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