{
  "title": "Calculus I — Midterm Practice",
  "subtitle": "Limits, derivatives, and applications",
  "author": "Dept. of Mathematics",
  "timeLimitMinutes": 45,
  "feedbackMode": "end",
  "passingScore": 70,
  "categories": ["Limits", "Derivatives", "Applications"],
  "questions": [
    {
      "id": "q1",
      "type": "multiple-choice",
      "category": "Limits",
      "points": 2,
      "prompt": "Evaluate $\\displaystyle\\lim_{x \\to 0} \\frac{\\sin(3x)}{x}$.",
      "choices": [
        { "id": "a", "text": "$0$" },
        { "id": "b", "text": "$1$" },
        { "id": "c", "text": "$3$" },
        { "id": "d", "text": "$\\infty$" }
      ],
      "answer": "c",
      "explanation": "Using the standard limit $\\lim_{u\\to 0}\\frac{\\sin u}{u}=1$ with $u=3x$ gives $3\\cdot 1 = 3$."
    },
    {
      "id": "q2",
      "type": "multiple-select",
      "category": "Derivatives",
      "points": 3,
      "prompt": "Which of the following functions are differentiable at $x=0$? Select all that apply.",
      "choices": [
        { "id": "a", "text": "$f(x) = |x|$" },
        { "id": "b", "text": "$f(x) = x^2 \\sin(1/x)$ with $f(0)=0$" },
        { "id": "c", "text": "$f(x) = \\sqrt{x}$" },
        { "id": "d", "text": "$f(x) = e^x$" }
      ],
      "answer": ["b", "d"],
      "explanation": "$|x|$ has a corner, $\\sqrt{x}$ is not defined for $x<0$. Both $x^2\\sin(1/x)$ (extended) and $e^x$ are differentiable at 0."
    },
    {
      "id": "q3",
      "type": "math-input",
      "category": "Derivatives",
      "points": 3,
      "prompt": "Compute $\\dfrac{d}{dx}\\left[x^2 \\ln x\\right]$. Enter your answer in LaTeX.",
      "answer": "2x\\ln x + x",
      "acceptedForms": ["2x\\ln(x) + x", "x(2\\ln x + 1)", "x + 2x\\ln x"],
      "explanation": "By the product rule: $2x\\ln x + x^2 \\cdot \\frac{1}{x} = 2x\\ln x + x$."
    },
    {
      "id": "q4",
      "type": "short-answer",
      "category": "Limits",
      "points": 2,
      "prompt": "What is the value of $\\displaystyle\\lim_{x \\to \\infty}\\left(1 + \\frac{1}{x}\\right)^x$? Give the standard symbol.",
      "answer": "e",
      "acceptedForms": ["E", "e", " e "],
      "caseSensitive": false,
      "explanation": "This is the definition of Euler's number, $e \\approx 2.71828$."
    },
    {
      "id": "q5",
      "type": "order",
      "category": "Applications",
      "points": 3,
      "prompt": "Order these steps to find the absolute maximum of $f$ on $[a,b]$ (first → last):",
      "items": [
        { "id": "s1", "text": "Confirm $f$ is continuous on $[a,b]$" },
        { "id": "s2", "text": "Find critical points in $(a,b)$" },
        { "id": "s3", "text": "Evaluate $f$ at critical points and endpoints" },
        { "id": "s4", "text": "Compare values; the largest is the absolute max" }
      ],
      "answer": ["s1", "s2", "s3", "s4"],
      "explanation": "The Extreme Value Theorem requires continuity on a closed interval; then candidates are critical points and endpoints."
    },
    {
      "id": "q6",
      "type": "match",
      "category": "Derivatives",
      "points": 4,
      "prompt": "Match each function to its derivative.",
      "left": [
        { "id": "l1", "text": "$\\sin x$" },
        { "id": "l2", "text": "$\\tan x$" },
        { "id": "l3", "text": "$\\ln x$" },
        { "id": "l4", "text": "$e^{2x}$" }
      ],
      "right": [
        { "id": "r1", "text": "$\\dfrac{1}{x}$" },
        { "id": "r2", "text": "$\\cos x$" },
        { "id": "r3", "text": "$2e^{2x}$" },
        { "id": "r4", "text": "$\\sec^2 x$" }
      ],
      "answer": { "l1": "r2", "l2": "r4", "l3": "r1", "l4": "r3" }
    },
    {
      "id": "q7",
      "type": "essay",
      "category": "Applications",
      "points": 5,
      "prompt": "Explain, in your own words, the Mean Value Theorem and give a geometric interpretation.",
      "rubric": [
        "States hypotheses (continuous on $[a,b]$, differentiable on $(a,b)$)",
        "States conclusion: $\\exists c \\in (a,b)$ with $f'(c) = \\frac{f(b)-f(a)}{b-a}$",
        "Explains geometric meaning (tangent parallel to secant)",
        "Gives or references an example"
      ]
    }
  ]
}