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On a hot summer afternoon, engineers study how heat distributes along a new metal alloy rod. Let T(x) be the temperature in °C at a point x cm from one end of the rod, where 0≤x≤2. The temperature gradient satisfies
and it is observed T(2)=10.
Find T(x).
Find the average temperature Tˉ along the rod from x=0 to x=2.
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EcoTech Ltd. tracks its annual profit. Let P(t) (in thousands of dollars) be its profit t years after founding. The rate of change of profit is modeled by
Determine whether the profit is increasing or decreasing when t=2.
Given that one year after founding the annual profit was $2,000, find an expression for P(t) for when t≥0.
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