MATHEMATICS 184 (Section 921) - TERM EXAM #3
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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.
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1. A spherical scoop of ice cream is melting in such a way that its radius is decreasing at a
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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.
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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)
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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.
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4. Differentiate the following functions: (Do not simplify your answers)
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(a) f (x) = tan(x) + ln(x) + arctan(x) + ex
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(b) y = x · sin2 (x)cos(x)
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(c) y = 5 · ln(7x3 ) + 3 · arcsin(6x − 2)
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2
x
2
(d) f (x) = ee
3
(e) g(t) =
4
(f) ey = ln(xy) + 2x2
(2t−5)4
(t 2 +1)3
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