Table Problems — Data Without Equations
The AP exam gives tables of values and asks you to apply calculus concepts.
Approximating derivatives from tables:
f'(a) ≈ [f(b) - f(a)] / (b - a) using nearby points.
For better accuracy, use the symmetric difference: [f(a+h) - f(a-h)] / 2h
Approximating integrals from tables:
Use Riemann sums (left, right, midpoint) or trapezoidal rule.
Intervals may be UNEQUAL — don't assume uniform Δx.
Justifying with MVT:
"Must there exist a time t where f'(t) = 3?" Calculate average rate between two points. If it equals 3, cite MVT (requires continuity + differentiability).
AP tip: Table problems appear on free response every year. Read carefully — they test whether you can apply calculus concepts without a formula.
Reference:
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