mat196f_2004_exam - Last Name: First Name: Student Number:...

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Unformatted text preview: Last Name: First Name: Student Number: V University of Toronto Faculty of Applied Science and Engineering MAT196F — Final Exam Thursday, December 16, 2004 Examiners: S. Abou-Ward, X. Kang Duration: 2 1/2 hours NO AIDS ALLOWED i 1. (a) Find values of constants a, b and c for whitzh the graphs of the two polynomials f(x) = x2 + ax + b and g(x) = x3 — c will intersect at the point (1,2) and have the same tangent line at that point.“ (5) for all real x. (b) Let f(x) = (HXZ) (i) find all points x such that f '(x) = 0. (3) (ii) examine the sign f‘ and determine these intervals in which f is monotonic. (4) Answer: Monotonically increasing: Interval: Monotonically decreasing: Interval: (4) (3) (iii) examine the sign of f " and determine the intervals in which f is concave up or concave down. Answer: Concave up: Concave down: (iv) identify ail point(s) x where f has a iocai maximum or iocai minimum, or inflection points. Answer: Local max at x= Local min at x= Inflection point at x= t 2. (a) Let f(x)=%(a"+a"‘) ifa>0- |ff(x+3\x)+f(x—y)=kf(x)f(y) constant, find that constant k. (5) , where k is a (b) Find f '(x) in (a) above. (4) 1 (0) Evaluate I 0 ax _a—X a" +a"‘ dx. (4) 3, Afunction f is defined for all real x by the formula 1+sint 2+t2 dz. X f(x)=3+j 0 Without attempting to evaluate this integrai, find a quadratic polynomial p(x) = a + bx + cx2 such that p(0) = f(0), p'(0) = f'(0), and p"(0) = f"(0). (15) 5 4. A particle is to be moved along the x-axis by a qUadratic propelling force f (x) = ax2 +bx dynes. Calculate a and b so that 900 ergs of work are required to move the particle 10 centimeters (cm) from the origin, if the force is 65 dynes when x z: 5 cm. (10) 5. (a)‘ The graphs of f (x) = x2 and g(x) = cx3, where c > 0, intersect at the points (0,0) and (1/0 ,1/c2) . Find c so that the region which lies between these graphs and over the interval [0,1/c] has area 2/3. (6) (b) Find the centroid of the region described in (a) above assuming that c = 1/ 2'. (3) © Use Pappus theorem (or any other method) to find the volume of the solid of revolution obtained by revolving the region described in (a) above about the y ~axis. (Again let c = 1/2). (5) 6. (i) Find the following indefinite integrals: i (a) IV1+3coszx sin2x dx (4) Answer: sinJ; b dx ( > i J; (4) Answer: (c) [x3 x2 +2 dx (6) Answer: (ii) Evaluate the integrals: (6) Answer: 1 —+cosx 2 dx 71' (b) j 0 (6) Answer: ...
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mat196f_2004_exam - Last Name: First Name: Student Number:...

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