workshop1

# workshop1 - Workshop 1 1 1 Sketch the region R deﬁned by...

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Unformatted text preview: Workshop 1 1 1. Sketch the region R deﬁned by 1 ≤ x ≤ 2 and 0 ≤ y ≤ x3 . a) Find (exactly) the number a such that the line x = a divides R into two parts of equal area. b) Find (to 3 places) the number b such that the line y = b divides R into two parts of equal area. 2. Suppose that the outdoor temperature (in approximated by the function ◦ F) on a particular day was πt T (t) = 50 + 14 sin( 12 ), where t is time (in hours) after 9AM. a)Find the maximum temperature Tmax , minimum temperature Tmin , and average temperature Taver Taver = 1 12 12 0 T (t)dt on that day during the period 9AM to 9PM. b)Show that Taver = Tmax +Tmin .(This is the deﬁnition that the weather bureau 2 uses for “average temperature”.) c)Show that if t is not given by the above formula, but rather T (t) is a linear function of t, then Taver = Tmax +Tmin .(Use either geometric reasoning or an 2 integral.) 3.Compare the masses of the following two objects. Explain intuitively why they should be equal. a) A triangular plate looks like the region bounded by the x-axis and the lines x = 4 and y = x2 . If the plate is 1 mm thick and has density 1 g /mm3 , compute the mass of the plate (the x- and y -axes have units in millimeters as well.) b) A wire is 4 mm and has density given by f (x) = x2 g /mm3 , where x is given in millimeters from one end. If the wire has constant cross-sectional area of 1 mm2 , compute the mass of the wire. ...
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