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MATHEMATICS 184 (Section 921) - TERM EXAM #3 NAME: STUDENT ID NUMBER: SIGNATURE: INSTRUCTIONS: Notes, books and calculators are not permitted. Answers may receive no credit if accompanying work is not provided. When indicated, simplify answers as much as possible, and be sure to clearly indicate the final answer for each question. Please use the back of the pages if you need extra space. 8 1. A spherical scoop of ice cream is melting in such a way that its radius is decreasing at a 3 constant rate of 0.1 cm/minute. When the volume of the scoop is 32π 3 cm , at what rate is its volume decreasing? Hint: Volume of a sphere = 34 πr3 , where r = radius in cm, note that here radius is a function of time. Page 2 8 2. A lighthouse is on a small island 4 km. from the nearest point P of a straight shoreline. The light of this lighthouse makes 2 revolutions per minute, moving a beam of light along the shoreline as it revolves. How fast is the beam of light moving along the shoreline when it is 2 km. from the point P? Hint: One revolution of the light is 2π radians (do not use degrees) Page 3 11 3. Find the equations of the tangent lines to the curve x2 − xy + y2 = 4 at the points where the curve crosses the x-axis. Page 4 4. Differentiate the following functions: (Do not simplify your answers) 4 (a) f (x) = tan(x) + ln(x) + arctan(x) + ex 3 (b) y = x · sin2 (x)cos(x) 2 (c) y = 5 · ln(7x3 ) + 3 · arcsin(6x − 2) Page 5 2 x 2 (d) f (x) = ee 3 (e) g(t) = 4 (f) ey = ln(xy) + 2x2 (2t−5)4 (t 2 +1)3 Page 6