Mean value theorem & L'Hôpital's rule

Japanese school year: University year 1

What you learn

This topic covers the Mean Value Theorem, ensuring a point where the tangent slope equals the average rate of change, alongside L'Hôpital's Rule for indeterminate limits. These theorems are essential for proving monotonicity and simplifying complex limits. Prior proficiency in basic derivatives and limits is required.

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Key points

The Mean Value Theorem states that between endpoints $a$ and $b$ , there is at least one point $c$ where the instantaneous slope $f'(c)$ exactly equals the average slope between $a$ and $b$ .

\frac{f(b) - f(a)}{b - a} = f'(c) \quad (a < c < b)

Rolle's Theorem states that if the function values at endpoints $a$ and $b$ are equal ( $f(a) = f(b)$ ), there must be a point $c$ between them where the tangent is completely flat ( $f'(c) = 0$ ).

f(a) = f(b) \implies f'(c) = 0 \quad (a < c < b)

L'Hopital's rule helps evaluate indeterminate limits that result in $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . You can differentiate the numerator $f(x)$ and denominator $g(x)$ separately into $\frac{f'(x)}{g'(x)}$ before taking the limit.

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

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Practice 1Coming soon