NameMath 221 First Midterm Exam Thursday October 5 2006Circle your section:301 7:45 MW B305 VAN VLECK Murcko302 8:50 MW B333 VAN VLECK Ganguly303 8:50 MW B329 VAN VLECK Andrejko304 9:55 MW B219 VAN VLECK Andrejko305 9:55 MW B105 VAN VLECK Hu306 11:00 MW B105 VAN VLECK Hu308 12:05p MW B329 VAN VLECK Ganguly310 2:25p MW B203 VAN VLECK Murcko312 9:55-11:50 MWF 277 BASCOM Rouse313 12:15-2:10 MWF 3349 ENGR HALL OwenI 40 PointsII 40 PointsIII 35 PointsIV 40 PointsV 25 PointsVI 20 PointsTotal 200 PointsShow your reasoning.I. (40 points.) In each of the following, find dy/dx.(a) y = sin x.(b) y = (sin x)−1.(c) y = sin(x−1).(d) y = sin−1(x).II. (40 points.) Find the limit. Distinguish between an infinite limit and one which doesn’texist. (Give reasons!)(a) limx→0sin(6x)x(b) limx→0+sin 6x(c) limx→∞sin(6x)x(d) limh→0sin(6 + h) − sin 6hIII. (35 points.) Find an equation for the tangent line to the curvex2+ xy − y2= 1at the point (x, y) = (2, 3).IV. (40 points.) State and prove the formula for the derivative of the product of two functions.In your pro of you may use (without pr oof) the limit laws, the theorem that a differentiablefunction is continuous, and high school algebra.V. (25 points.) (a) Find f0(x) and g0(x) iff(x) =x + 1x − 1, g(x) =x + 2x − 2.(b) Let f(x) and g(x) be as in part (a) and p = f · g, i.e. p(x) = f(x) · g(x). Find the derivativep0(x) of p(x).(c) Let f(x) and g(x) be as in parts (a) and (b ) and let w = f ◦ g, i.e. w(x) = f(g(x)). Findthe value g(5) of g(u) when u = 5 and the value w0(5) of the derivative w0(x) when x = 5.VI. (20 points.) (a) Draw the graph y = f(x) where f(x) is the function defined byf(x) =2x for x < 0;3x for 0 ≤ x < 1;4x − 1 for 1 ≤ x.(b) Give a formula (like the above formula for f(x)) for the inverse function x =
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