Finding c with the Mean Value Theorem

Calculus 1 · Axiom Academy

EXAMPLE Finding c with the Mean Value Theorem Applying the MVT to f(x) = x^3 on [0, 3] — locate the point where the tangent is parallel to the secant. The function f(x) = x^3 is continuous on [0, 3] and differentiable on (0, 3) , so the Mean Value Theorem applies. Find the value of c in (0, 3) for which f'(c) equals the average rate of change of f over [0, 3] . Nicely done. You applied the Mean Value Theorem end to end: average rate, then instantaneous rate, then solve. The MVT equation: the guaranteed c solves — the instantaneous rate equal to the average rate. Mind the interval: 3c^2 = 9 has two roots , but only lies in (0, 3) ; discard . Result: , the input where the tangent to y = x^3 is parallel to the secant across [0, 3] . Same three moves work for any MVT "find c" problem — compute the secant slope, differentiate, then solve f'(c) = that slope and keep the roots inside the interval.

This is the written version of the interactive lesson above. See the full Calculus 1 course.