export const mockCalculusUnit = { unitId: "calc-u1", unitTitle: "Limits & Derivatives", nodes: [ { id: "calc-n1", title: "Concept of Limits", description: "Classify scenarios where limits exist or fail.", type: "concept", status: "available", stages: [ { stageId: "calc-s1", topic: "Limit Existence", module: "Instruction", component: "TaxonomyMatrix", skin: "Code", config: { data: { buckets: ["Limit Exists", "Limit Does Not Exist"], items: [ { id: "limit1", content: "Left limit = 5, Right = 5" }, { id: "limit2", content: "Left limit = 3, Right = -3 (Jump)" }, { id: "limit3", content: "Approaches infinity (Asymptote)" }, { id: "limit4", content: "Continuous polynomial" } ] }, initialState: { assignments: {} } }, validation: { type: "exact", condition: {} }, feedback: { success: "Excellent classification! Limits require agreement from both sides.", error: "Check the definitions of jumps and asymptotes.", hint: "If left and right don't match, it doesn't exist." } } ] }, { id: "calc-n2", title: "Derivative Rules", description: "Match base functions to their derivatives.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s2", topic: "Basic Differentiation", module: "Practice", component: "PatternMatcher", skin: "Scientific", config: { data: { pairs: [ { id: "diff1", left: "f(x) = x³", right: "f'(x) = 3x²" }, { id: "diff2", left: "f(x) = sin(x)", right: "f'(x) = cos(x)" }, { id: "diff3", left: "f(x) = eˣ", right: "f'(x) = eˣ" }, { id: "diff4", left: "f(x) = ln(x)", right: "f'(x) = 1/x" } ] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "All matched! You master the basic power and transcendental rules.", error: "Review your trig and exponential derivatives.", hint: "The derivative of eˣ is special!" } } ] }, { id: "calc-n3", title: "Squeeze Theorem", description: "Apply the squeeze theorem to find limits.", type: "concept", status: "locked", stages: [ { stageId: "calc-s3", topic: "Squeeze Theorem Application", module: "Instruction", component: "LogicChain", skin: "Code", config: { data: { nodes: ["Find lower bound g(x)", "Find upper bound h(x)", "Show g(x) ≤ f(x) ≤ h(x)", "Evaluate lim g(x) = lim h(x)", "Conclude lim f(x)"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Perfect squeeze theorem steps!", error: "Review the theorem conditions.", hint: "Both bounds must converge to the same value." } } ] }, { id: "calc-n4", title: "Continuity", description: "Determine where functions are continuous.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s4", topic: "Continuity Conditions", module: "Practice", component: "TaxonomyMatrix", skin: "Code", config: { data: { buckets: ["Continuous", "Discontinuous"], items: [ { id: "cont1", content: "f(a) is defined, lim = f(a)" }, { id: "cont2", content: "Jump at x = 2" }, { id: "cont3", content: "Removable hole at x = 0" }, { id: "cont4", content: "Polynomial on all reals" } ] }, initialState: { assignments: {} } }, validation: { type: "exact", condition: {} }, feedback: { success: "Great continuity analysis!", error: "Check the three conditions.", hint: "All three conditions must hold." } } ] }, { id: "calc-n5", title: "Power Rule", description: "Master the power rule for differentiation.", type: "concept", status: "locked", stages: [ { stageId: "calc-s5", topic: "Power Rule Practice", module: "Instruction", component: "LogicChain", skin: "Code", config: { data: { nodes: ["f(x) = xⁿ", "Bring down exponent", "Multiply by coefficient", "Reduce exponent by 1", "f'(x) = nxⁿ⁻¹"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Power rule mastered!", error: "Review the power rule steps.", hint: "d/dx[xⁿ] = nxⁿ⁻¹" } } ] }, { id: "calc-n6", title: "Product Rule", description: "Apply the product rule to differentiate.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s6", topic: "Product Rule", module: "Practice", component: "LogicChain", skin: "Scientific", config: { data: { nodes: ["Identify u and v", "Find u'", "Find v'", "Apply u'v + uv'", "Simplify"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Product rule applied correctly!", error: "Remember: (uv)' = u'v + uv'", hint: "Don't forget both terms." } } ] }, { id: "calc-n7", title: "Quotient Rule", description: "Differentiate rational functions.", type: "concept", status: "locked", stages: [ { stageId: "calc-s7", topic: "Quotient Rule", module: "Instruction", component: "LogicChain", skin: "Code", config: { data: { nodes: ["Identify numerator u", "Identify denominator v", "Compute u'v - uv'", "Divide by v²", "Simplify"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Quotient rule nailed!", error: "Review: (u/v)' = (u'v - uv')/v²", hint: "Lo dHi minus Hi dLo over Lo Lo." } } ] }, { id: "calc-n8", title: "Chain Rule", description: "Differentiate composite functions.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s8", topic: "Chain Rule", module: "Practice", component: "LogicChain", skin: "Scientific", config: { data: { nodes: ["Identify outer function f", "Identify inner function g", "Differentiate f'(g(x))", "Multiply by g'(x)", "Final answer"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Chain rule mastered!", error: "Don't forget the inner derivative!", hint: "d/dx[f(g(x))] = f'(g(x))·g'(x)" } } ] }, { id: "calc-n9", title: "Implicit Differentiation", description: "Differentiate implicitly defined functions.", type: "concept", status: "locked", stages: [ { stageId: "calc-s9", topic: "Implicit Differentiation Steps", module: "Instruction", component: "LogicChain", skin: "Code", config: { data: { nodes: ["Differentiate both sides", "Apply chain rule to y terms", "Collect dy/dx terms", "Factor out dy/dx", "Solve for dy/dx"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Implicit differentiation complete!", error: "Remember to treat y as a function of x.", hint: "Every y term needs dy/dx." } } ] }, { id: "calc-n10", title: "Related Rates", description: "Solve related rates problems.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s10", topic: "Related Rates Strategy", module: "Practice", component: "LogicChain", skin: "Scientific", config: { data: { nodes: ["Draw diagram", "Write equation relating variables", "Differentiate with respect to t", "Substitute known values", "Solve for unknown rate"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Related rates solved!", error: "Make sure to differentiate with respect to time.", hint: "Everything changes with time." } } ] }, { id: "calc-n11", title: "Trig Derivatives", description: "Differentiate trigonometric functions.", type: "concept", status: "locked", stages: [ { stageId: "calc-s11", topic: "Trig Function Derivatives", module: "Instruction", component: "TaxonomyMatrix", skin: "Code", config: { data: { buckets: ["Positive Cosine Family", "Negative Sine Family"], items: [ { id: "trig1", content: "d/dx[sin x] = cos x" }, { id: "trig2", content: "d/dx[cos x] = -sin x" }, { id: "trig3", content: "d/dx[tan x] = sec²x" }, { id: "trig4", content: "d/dx[cot x] = -csc²x" } ] }, initialState: { assignments: {} } }, validation: { type: "exact", condition: {} }, feedback: { success: "Trig derivatives classified!", error: "Review co-function patterns.", hint: "Co-functions have negative derivatives." } } ] }, { id: "calc-n12", title: "Higher-Order Derivatives", description: "Compute second and third derivatives.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s12", topic: "Higher-Order Derivatives", module: "Practice", component: "LogicChain", skin: "Code", config: { data: { nodes: ["f(x)", "f'(x) — First derivative", "f''(x) — Second derivative", "f'''(x) — Third derivative"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "Higher-order derivatives done!", error: "Just keep differentiating!", hint: "Differentiate the previous result." } } ] }, { id: "calc-n13", title: "L'Hôpital's Rule", description: "Evaluate indeterminate limits.", type: "concept", status: "locked", stages: [ { stageId: "calc-s13", topic: "L'Hôpital's Rule", module: "Instruction", component: "LogicChain", skin: "Scientific", config: { data: { nodes: ["Check 0/0 or ∞/∞ form", "Differentiate numerator", "Differentiate denominator", "Re-evaluate limit", "Repeat if needed"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "L'Hôpital's applied correctly!", error: "Only works for indeterminate forms.", hint: "Must be 0/0 or ∞/∞ first." } } ] }, { id: "calc-n14", title: "Mean Value Theorem", description: "Understand and apply MVT.", type: "exercise", status: "locked", stages: [ { stageId: "calc-s14", topic: "MVT Application", module: "Practice", component: "LogicChain", skin: "Code", config: { data: { nodes: ["Verify continuity on [a,b]", "Verify differentiability on (a,b)", "Compute [f(b)-f(a)]/(b-a)", "Set f'(c) equal to slope", "Solve for c"] }, initialState: {} }, validation: { type: "exact", condition: {} }, feedback: { success: "MVT applied perfectly!", error: "Check the theorem conditions.", hint: "There exists a c where the tangent equals the secant." } } ] }, { id: "calc-n15", title: "Final Challenge", description: "Comprehensive limits & derivatives assessment.", type: "challenge", status: "locked", stages: [ { stageId: "calc-s15", topic: "Comprehensive Review", module: "Assessment", component: "TaxonomyMatrix", skin: "Code", config: { data: { buckets: ["Differentiation Technique", "Limit Technique"], items: [ { id: "final1", content: "Chain Rule" }, { id: "final2", content: "L'Hôpital's Rule" }, { id: "final3", content: "Product Rule" }, { id: "final4", content: "Squeeze Theorem" } ] }, initialState: { assignments: {} } }, validation: { type: "exact", condition: {} }, feedback: { success: "Congratulations! Calculus mastered!", error: "Review the course material.", hint: "Classify each technique by its primary use." } } ] } ] };