Mean Value Theorem Calculator

What is a Mean Value Theorem Calculator?

A Mean Value Theorem (MVT) Calculator analyzes a function on a closed interval and pinpoints the location(s) where the instantaneous rate of change equals the average rate of change. For a function continuous on \([a,b]\) and differentiable on \((a,b)\), the MVT guarantees a number \(c\in(a,b)\) such that the tangent slope at \(c\) equals the secant slope across the interval. The calculator verifies the hypotheses automatically, computes the secant slope, solves \(f'(x)=\frac{f(b)-f(a)}{b-a}\) for \(x\in(a,b)\), filters valid solutions, and shows each algebraic step. It can also illustrate Rolle’s Theorem as a special case and provide the generalized Cauchy MVT when two functions are supplied.

About the Mean Value Theorem Calculator

The tool accepts symbolic functions (polynomials, exponentials, logs, trig, rationals) or numeric black-box functions. It checks continuity on \([a,b]\) and differentiability on \((a,b)\) via domain tests and derivative existence; if assumptions fail, it flags the issue and explains why the theorem may not apply. When multiple points satisfy the condition, all solutions in \((a,b)\) are listed. For constant functions, any point in \((a,b)\) works; the calculator reports the entire interval or a representative set. For two functions \(f\) and \(g\) with \(g'\neq0\), it computes the Cauchy Mean Value Theorem ratio. Optional numeric sampling illustrates how the tangent at \(c\) parallels the secant from \((a,f(a))\) to \((b,f(b))\).

How to Use this Mean Value Theorem Calculator

1. Enter \(f(x)\) and the interval endpoints \(a<b\). (Optional: also enter \(g(x)\) to use the Cauchy form.)
2. Run the hypotheses check: continuity on \([a,b]\) and differentiability on \((a,b)\).
3. Compute the secant slope \(m_{\text{sec}}=\dfrac{f(b)-f(a)}{b-a}\).
4. Solve \(f'(x)=m_{\text{sec}}\) for \(x\in(a,b)\). Keep all solutions that lie strictly inside the open interval.
5. View step-by-step derivations and (optional) geometric notes connecting the tangent at \(c\) to the interval’s secant.

Core Formulas (LaTeX)

Average (secant) slope: \[ m_{\text{sec}}=\frac{f(b)-f(a)}{\,b-a\,}. \]

Mean Value Theorem: \[ \text{If } f\in C[a,b] \text{ and } f\in C^1(a,b),\ \exists\,c\in(a,b):\quad f'(c)=\frac{f(b)-f(a)}{b-a}. \]

Rolle’s Theorem (special case): \[ f(a)=f(b),\ f\in C[a,b],\ f\in C^1(a,b)\ \Rightarrow\ \exists\,c\in(a,b):\ f'(c)=0. \]

Cauchy Mean Value Theorem: \[ \exists\,c\in(a,b):\ \frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)},\ \text{with } g'(x)\ne0 \text{ on }(a,b). \]

Examples (Illustrative)

Example 1 — \(f(x)=x^2\) on \([1,3]\)

\(m_{\text{sec}}=\dfrac{9-1}{3-1}=4\). Solve \(f'(x)=2x=4\Rightarrow c=2\in(1,3)\). The tangent at \(x=2\) has slope \(4\), matching the secant.

Example 2 — \(f(x)=\sin x\) on \([0,\pi]\)

\(m_{\text{sec}}=\dfrac{0-0}{\pi-0}=0\). Solve \(f'(x)=\cos x=0\Rightarrow c=\tfrac{\pi}{2}\in(0,\pi)\).

Example 3 — Necessity of differentiability (\(f(x)=|x|\) on \([-1,1]\))

\(m_{\text{sec}}=\dfrac{1-1}{2}=0\). But \(f'(x)=-1\) for \(x<0\), \(f'(x)=1\) for \(x>0\); no \(c\) with \(f'(c)=0\). The MVT hypotheses fail because \(f\) is not differentiable at \(0\).

FAQs

What conditions must hold for the Mean Value Theorem?

Continuity on \([a,b]\) and differentiability on \((a,b)\). Without both, the guarantee may fail.

Can there be multiple points \(c\)?

Yes. The equation \(f'(x)=\dfrac{f(b)-f(a)}{b-a}\) can have several solutions inside \((a,b)\).

How is this different from Rolle’s Theorem?

Rolle’s is the special case \(f(a)=f(b)\), guaranteeing a point with zero derivative.

What does the calculator do if hypotheses fail?

It explains which condition breaks and shows why a point \(c\) may not exist (e.g., corners, cusps, discontinuities).

Can I apply the theorem when \(a>b\)?

Swap endpoints to enforce \(a<b\); slopes and conclusions are unchanged.

Does it support a two-function version?

Yes. With \(f\) and \(g\), it applies the Cauchy Mean Value Theorem provided \(g'(x)\neq0\) on \((a,b)\).