memorization quiz sometime second week of may 2025

Mean Value Theorem (MVT) — Conditions

Function must be continuous on [a, b] and differentiable on (a, b)

Mean Value Theorem (MVT) — Conclusion

There exists c in (a, b) such that f′(c) = (f(b) - f(a)) / (b - a)

Intermediate Value Theorem (IVT) — Conditions

Function must be continuous on [a, b]; N is between f(a) and f(b)

Intermediate Value Theorem (IVT) — Conclusion

There exists c in (a, b) such that f(c) = N

L'Hôpital's Rule — Conditions

lim f(x) and lim g(x) as x → a give 0/0 or ∞/∞; f and g differentiable near a; g′(x) ≠ 0 near a; lim f′(x)/g′(x) exists

L'Hôpital's Rule — Conclusion

lim f(x)/g(x) = lim f′(x)/g′(x) as x → a

Definition of Differentiability at a Point — Condition

The limit limₕ→0 [f(a + h) - f(a)] / h exists

Definition of Differentiability at a Point — Conclusion

f is differentiable at a and f′(a) = limₕ→0 [f(a + h) - f(a)] / h

Squeeze Theorem — Conditions

g(x) ≤ f(x) ≤ h(x) near a (except possibly at a); lim g(x) = lim h(x) = L as x → a

Squeeze Theorem — Conclusion

lim f(x) = L as x → a

Definition of Continuity at a Point — Conditions

lim f(x) as x → a exists; f(a) is defined; lim f(x) = f(a)

Definition of Continuity at a Point — Conclusion

f is continuous at a