Integrals with initial condition

At the end of Integrals, we took the intergral of the marginal cost function, x/5...

...resulting in the following equation for total cost.

The +C (constant of integration) here is a little bit of an issue, because it prevents us from knowing the exact total cost at a given output level.

However, what if we were given the following intel when we received the marginal cost function?

Intel: At 1 unit, the total cost equals $1.10. In mathematical terms, f(1) = 1.10.

This embodies an initial condition!

An initial condition gives you a point on the integrated function that exists, therefore enabling you to solve for the constant of integration (+C).

Solving with an initial condition

The process is quite simple:

When faced with an integral with an initial condition, first evaluate it as an indefinite integral. Then, plug in the initial condition (x, y) to solve for C, which can then be placed in the final integral expression.

So in the case of our marginal cost formula above, we've already evaluated the indefinite integral...

Which we can formalize with f(x) like so...

and from here, we can plug in our initial condition, f(1) = 1.10...

...which enables us to solve for C...

...and get a value of 1!

This means that we can update our total cost, f(x) equation like so and remove the "+ C"!

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