Calculus I · Unit 2B · lesson
Marginal Cost, Revenue, Profit, and Elasticity
Learning objectives
Interpret marginal functions, optimize profit, and distinguish absolute from relative sensitivity.
Derivatives in Business Models
Before the formulas
In Marginal Cost, Revenue, Profit, and Elasticity, the derivative converts a formula into a decision-relevant local statement. The useful question is often not merely "what is the rate?" but "how much does a small change matter here, in these units, under these assumptions?"
Build the model in stages and retain intermediate quantities. This makes unit checks possible and reveals which parameter drives the result. If measured data are involved, report uncertainty and avoid claiming precision the inputs do not support.
Read this graph as text
A marginal value is the slope of a total curve at one production level. The tangent to total cost at q has slope C'(q) . Over one additional unit, that slope approximates the actual cost increase C(q+1)-C(q) when the scale is appropriate. At q=5 , the tangent slope is 8 dollars per unit. That predicts that the next unit will add about 8 dollars to total cost. The actual finite increase is close but not necessarily equal, because marginal cost is an instantaneous rate while producing one full unit is a nonzero change.
Every relationship in a marginal value is the slope of a total curve at one production level is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: This graph makes the connection between a derivative and "one more unit" precise while preserving the approximation language. It should discourage the false identity C'(q)=C(q+1)-C(q) .
The tangent to total cost at q has slope C'(q). Over one additional unit, that slope approximates the actual cost increase C(q+1)-C(q) when the scale is appropriate.
Marginal means the predicted effect of one additional unit near the current level
If is cost, then estimates the additional cost of producing one more unit near output . It is not generally the same as average cost , which spreads total cost across all units produced.
Marginal analysis is local. The derivative predicts a small change near the current production level; it should not be multiplied across a huge change without checking whether the rate remains stable.
Suppose demand is
and cost is
Revenue is price times quantity:
Profit is
Differentiate:
The critical production level is
Since , the profit function is concave down and gives the unique maximum on the feasible domain where price remains nonnegative.
Marginal means local change
is marginal cost, marginal revenue, and marginal profit. Near integer production levels, these derivatives approximate the change caused by producing one additional unit.
Relative sensitivity is measured by elasticity:
It estimates the percentage output change caused by a one-percent input change.
After the explanation
Use the section idea
Treat each derivative model as a conditional claim whose variables, units, assumptions, calibration range, and limitations remain visible.
A useful model connects a measurable input to a measurable output, while its derivative describes local sensitivity inside a stated domain.
Define the relationship and objective, differentiate, evaluate candidates or rates, then test sign, scale, units, and assumption sensitivity.
Extending a fitted model outside its data range or presenting medication, stopping-distance, or business outputs without the assumptions that shape them.
What observation would falsify the model, and how would the conclusion change if its strongest assumption failed?
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