Video Transcript
The work done by an engine at time
𝑡 is given by the relation 𝑊 of 𝑡 is equal to two 𝑡 cubed six 𝑡 joules. Find the power of the engine as a
function of time.
We begin by recalling that power is
the derivative of work such that 𝑃 is equal to d𝑊 by d𝑡. In this question, we’re given an
expression for the work done 𝑊 in terms of time. It is equal to two 𝑡 cubed plus
six 𝑡. We can therefore differentiate this
function term by term to find a function for the power 𝑃. Recalling that if 𝑦 is equal to 𝑎
multiplied by 𝑥 to the power of 𝑛, then d𝑦 by d𝑥 is equal to 𝑛 multiplied by 𝑎
multiplied by 𝑥 to the power of 𝑛 minus one. We can differentiate two 𝑡 cubed
to give us six 𝑡 squared as three multiplied by two is six, and we decrease the
power by one. Differentiating six 𝑡 simply gives
us six as when we decrease the power, we get 𝑡 to the power of zero, which is equal
to one. The power of the engine as a
function of time is equal to six 𝑡 squared plus six with the standard unit of
watts.
