The marginal cost function is .

The fixed start-up cost is .

Find the cost of producing function for first using net change theorem.

Net change theorem:

Apply integral on each side.

Substitute .

Substitute .

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Substitute in .

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Net Change Theorem:

The definite integral of the rate of change of quantity gives the total change, or net change, in that quantity on the interval is .

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The cost of producing first juice is .

The cost of producing first juice is .