Original study example
For a hypothetical cost model C(x) = 100 + 5x + x², calculate C(3) and the derivative C′(3). Explain why these numbers mean different things.
Work through the example
C(3) = 100 + 15 + 9 = 124. The derivative is C′(x) = 5 + 2x, so C′(3) = 11. One is a total cost; the other is a local rate of change per unit. Compare C(4) - C(3) = 12 to see why a finite increment need not equal the derivative exactly.