00:01
All right, so in this problem, we are asked to find the average daily consumption for the home in kilowatt hours.
00:09
And our function here, as such, c of t equals 3 .9 minus 2 .4, sine pi t over 12.
00:26
And if we think about this function, our y -axis is measured in the units of kilowatts and our x -axis t is measured in the units of hours.
00:38
Why don't we make a quick little sketch of what that graph might look like, kilowatt hours in the y, hours on the x.
00:49
And our graph is going to look a bit something like this.
00:56
And now we are finding the average daily consumption.
01:00
So we're going to go from zero hours up to 24, 24 hours in one day.
01:06
And since our answer is asking us to find a number measured in kilowatt hours, we are looking for the integral of this function.
01:16
If you think about it, the area under the curve is going to be kilowatts times hours, giving us kilowatt hours.
01:23
So why don't we take the integral of both sides right here? so the integral of c of t is going to come out to be, well, what is the integral of 3 .9? 3 .9t, just basic using the power rule to find that.
01:42
And then over here, we have a negative sign.
01:46
So if we remember our basic trigonometric derivatives, we know that a cosine function gives you a negative sign derivative.
01:55
So we're going to have a positive cosine and then why don't i give myself a little bit more space here.
02:04
So we have space for the coefficient of cosine and then our argument is going to stay the same for all derivatives and integrals, pi t over 12...