Name the function and its units
For a total-cost function C(q), q is output and C′(q) is the local rate of change of cost with output. Average cost C(q)/q answers a different question. Both expressions need units and an applicable domain.
MIT single-variable calculus materials provide the mathematical reference for derivatives as rates of change. Economic interpretation adds what changes, what is held fixed and where the function applies.
Differentiate an original function
Use the fictional exercise C(q)=40+8q+0.5q² for q≥0. Differentiating term by term gives C′(q)=8+q and C″(q)=1. These are teaching units, not service prices.
At q=4, total cost is 80, average cost is 20 and marginal cost is 12. The fixed term affects total and average cost but drops out of this derivative. Explain that distinction before introducing further rules.
Compare a derivative with a finite change
Increasing q from 4 to 5 raises cost from 80 to 92.5, an actual increase of 12.5. The derivative at 4 is 12; it is a local rate, not the exact cost of every finite increase.
For an increase from 4 to 4.1, the actual change is 1.205 and the linear approximation C′(4)×0.1 is 1.2. The difference is 0.005. In this quadratic example, smaller changes improve the local approximation.
Explain the answer without symbols
A marginal cost of 12 means cost changes locally at about 12 cost units per output unit. It does not mean total cost is 12. State the point at which the derivative is evaluated.
Self-check at q=6: marginal cost is 14, total cost is 106 and average cost is approximately 17.67. Reproduce these results independently on permitted practice material and bring your attempted reasoning to a tutorial.
References
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