{
  "id": "mean-value-theorem",
  "version": "5e7dd71f25d5",
  "status": "published",
  "name": "Mean Value Theorem Calculator",
  "question": "Which c fits the mean value theorem?",
  "summary": "Finds every c in (a, b) where f′(c) = (f(b) − f(a))/(b − a).",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/mean-value-theorem-calculator",
  "markdown": "https://www.acalculator.org/math/mean-value-theorem-calculator.md",
  "kind": "function",
  "method": "f′(c) = (f(b) − f(a))/(b − a) for some c in (a, b).",
  "assumptions": [
    "Angles in radians."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Function f(x)",
        "description": "The function of x.",
        "type": "string",
        "maxLength": 200
      },
      "a": {
        "title": "From a",
        "description": "Left end.",
        "type": "string",
        "maxLength": 40
      },
      "b": {
        "title": "To b",
        "description": "Right end.",
        "type": "string",
        "maxLength": 40
      }
    }
  },
  "outputs": {
    "c": {
      "label": "Values of c",
      "description": "Each c in (a, b) where f′(c) is the slope.",
      "format": "text"
    },
    "slope": {
      "label": "Slope of the secant",
      "description": "(f(b) − f(a))/(b − a).",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "sqrt(x)",
      "a": "0",
      "b": "9"
    },
    "outputs": {
      "c": "9/4",
      "slope": "1/3"
    },
    "text": "For sqrt(x) on [0, 9], c = 9/4."
  },
  "examples": [
    {
      "given": {
        "f": "sqrt(x)",
        "a": "0",
        "b": "9"
      },
      "expect": {
        "c": "9/4",
        "slope": "1/3"
      },
      "source": "OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.4 The Mean Value Theorem, Example 4.15"
    },
    {
      "given": {
        "f": "x^3 - 4x",
        "a": "-2",
        "b": "2"
      },
      "expect": {
        "c": "-2 sqrt(3)/3, 2 sqrt(3)/3",
        "slope": "0"
      },
      "source": "OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.4 The Mean Value Theorem, Example 4.14(b)"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 4.4 The Mean Value Theorem: https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem"
  ],
  "related": [
    "tangent-line",
    "instantaneous-rate-of-change",
    "critical-number",
    "slope"
  ],
  "changelog": []
}