The total cost in dollars to produce Q units of a good is given by the
function C of Q equals twenty point five Q plus eighty-four thousand.
Part A: What is the total cost to produce eight hundred units? So for part A,
they're telling us that Q equals eight hundred. We want to find C of Q,
which is the total cost. So, C of eight hundred would be equal to twenty point five
times eight hundred plus eighty-four thousand, which is equal to ten thousand four hundred, which
means the total cost to produce eight hundred units is eleven thousand four hundred dollars.
For part B: What is the cost of the eight hundred first item? So,
to find the cost of just the eight hundred first item, we'll find the total cost of
eight hundred one items and then subtract the total cost of eight hundred items.
This is often called the marginal cost at Q equals eight hundred, which is the cost of
producing the next item when Q is equal to eight hundred. So again, we'll take
the total cost of eight hundred one items and subtract the total cost of eight hundred items.
So, when Q is equal to eight hundred one, we'd have twenty point
five times eight hundred one plus eighty-four thousand. And then when Q is eight hundred, which
we found above, we'd have twenty point five times eight hundred plus eighty-four thousand. We
found C of eight hundred above, that's one hundred thousand four hundred, and
now we'll find C of eight hundred one on the calculator, which is one
hundred thousand twenty point five, giving us a difference of twenty point five, which
means the cost of the eight hundred first item is twenty dollars and fifty cents.
There's a couple of things I do want to mention about this. Notice how the cost function is
linear, where twenty point five would be the slope of the linear equation. So, as Q increases by one,
notice how the cost increases by twenty point five or twenty dollars and fifty cents. And what
we'll find is that we often approximate the marginal cost, or this value here,
by evaluating the derivative of the total cost function at Q equals eight hundred.
If we look at our cost function for a moment, notice the derivative of the cost function,
or C prime of Q, would be equal to the derivative of twenty point five Q, which is just twenty point
five, plus the derivative of eighty-four thousand, which would be zero. So notice, for any value of
Q, the marginal cost of the cost of the next item would just be twenty point five or twenty dollars
and fifty cents. So, the marginal cost using the derivative function and the marginal cost
using the change of the total cost function will not always be the same, but often C prime of Q,
the derivative of the total cost function, is used to approximate the marginal cost.
I hope you found this helpful.