mat196f_2002_exam - Student Name Student Number University...

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Unformatted text preview: Student Name: Student Number: University of Toronto Faculty of Applied Science and Engineering MAT196F - Final Exam Friday, December 6, 2002 Examiners: S. Abou-Ward, S. Homayouni, P. Milman, A. Toms Duration: 2 1/2 hours * NO AIDS ALLOWED 1. a) Define continuity for a real valued function f whose domain is the real numbers. (4) 1+cx x <1 , . . . . Find a value of c such that f is continuous, if b) Let f(x)={ 2 c x3 + x — 46 x 21 possible. (5) Answer: c) Can you choose c above such that f is differentiable everywhere? (Show your work). (6) Answer: 2. Find the following indefinite integrals: 7x +ldl (5) Answer: b) [‘13 -(—cscx)cosxdx Sm x (5) Answer: C) Icos‘xdx (6) Answer: 3. An underground oil tank is spherical and is half full. Over time, settling and separation have caused the density of the oil to change. At the top of the oil, the density is .8 kg/L. As one descends, the density of the oil increases by an amount proportional to the square of the depth. The proportionality constant is k = .1 kg”. - ml. If the tank has a three metre radius. calculate the work done in pumping the contents of the tank to a point 4 metres above the top of the tank via the following steps. a) Write down a function describing the mass density of the oil as a function of depth (depth measured from the half-full line of the tank). (6) Answer: b) Convert your answer in a) to a weight density, and use it to write down an expression for the weight of the oil in a infinitesimally thin layer of oil sitting at depth x and with height air. (4) Answer: c) Find the work done in pumping out the tank by multiplying your answer in b) by the distance that such a layer must travel, and integrating over all such layers. (7) Answer: 4. Assume that f is a continuous function and that j tf(r)dr = sinx —xcosx 0 8) Determine 4512-]. (4) Answer: b) Find f‘(x). (3) Answer: c) Show that f(x) is one-to-one on the intervat 0 s x s g . (3) Answer: d) Find the derivative (Fm/Vii). (6) Answer: 5. a) Calculate the area A of the region bounded by the graph of f(x) = i, and the X x-axis for x e[1,b]. (4) Answer: b}‘ Catculate the volume of the solid of revolution generated by rotating the area in a) about the y-axis. (7) Answer: 0) Calculate the centroid of the region in a) above, (7) Answer: 6. 3) Find the minimum value of y = Ae’” + Be'”, 0,!) > 0. (5) Answer: b) A particle moves on a coordinate line with its position at time t given by the function x(t) = Ae" + Be‘". Show that the acceleration of the particle is proportional to its position. (4) Answer: c) Evaluate the following limit (if possible) 3 I . . e' -8 I 1 0 :23 H (4) Answer: (ii) lime "6 1—)1 lnx (4) Answer: ...
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