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Calculus IApplications

The Mean Value Theorem, from average to instant.

See why a tangent must match the slope between two endpoints. Learn the conditions, then find c and use the theorem to prove roots and inequalities.

The conditions and the picture · Step 1 of 7

Continuous means no hole, no jump, no blow-up

−6−5−4−3−2−101234567891011−5−4−3−2−10123456x
A function is continuous at x = a if you can draw through that point without lifting your pen. Precisely: f(a) is defined, the limit at a exists, and the two are equal, so limx → a f(x) = f(a). This graph from class breaks in five places.
Try it: Pick each x value. Which of the three conditions fail, and what kind of break is it?
Look at x =
  • 1. f(-4) is defined
  • 2. the limit at -4 exists
  • 3. the limit equals f(-4)
Type
Removable
x → −4⁻
2
x → −4⁺
2
The limit exists (2), but there is no point at x = −4. Define f(−4) = 2 and the hole is filled.
Removable: a hole
The limit exists but is not f(a), or f(a) is missing. Redefine one point and it is fixed.
Jump: the sides disagree
Both one-sided limits exist but are different.
Infinite: a vertical asymptote
One or both one-sided limits are +∞ or −∞.

Need a hand with Rolle’s theorem or an MVT proof?

Bring your practice questions. We work through them together until every type feels routine.

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