AP Calculus AB · Unit 5 Analytical Applications of Differentiation
Learning Objectives
Preconditions: To use MVT, $f(x)$ MUST be BOTH continuous on $[a,b]$ and differentiable on $(a,b)$. Corners and cusps will break the theorem!
The Mean Value Theorem (MVT): Guarantees there is AT LEAST ONE point $c$ where the instantaneous rate of change ($f'(c)$) perfectly equals the average rate of change between $a$ and $b$.
Rolle's Theorem: A special case wrapper of MVT. If $f(a) = f(b)$ (average slope is zero), then there must be at least one horizontal tangent line $f'(c) = 0$.
Tags
DerivativesMean Value TheoremRolle'sTangents
DRAG THE ENDPOINTS A AND B
Select Guarantee Scenario
Interval Evaluation [A, B]
$f'(c) = \frac{f(b)-f(a)}{b-a}$
AVERAGE SLOPE (SECANT)
$m_{sec}$0.00
Interval[-1.00, 1.00]
GUARANTEED TANGENT POINTS $c$
None
Mean Value Theorem: Drag points A and B. The engine will instantly calculate the secant slope (average rate of change) and automatically find all tangent points $c$ inside the interval that perfectly match it!
Rolle's Theorem Activated!
Because $f(a) = f(b)$, the SECANT slope is exactly 0. Therefore, a horizontal tangent is perfectly guaranteed!
MVT DOES NOT APPLY
Because the function has a sharp cusp, it is NOT differentiable on the open interval $(a, b)$. Therefore, NO tangential match is mathematically guaranteed (though one might accidentally exist!).