id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-010601 | Foundations of Calculus: Area vs Antiderivative | 6 | Start by stating any domain restrictions: Compute the definite integral
$$\int_0^23 (7x+(38))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=7x+(38)$ is a line on $[0,23]$.",
"Step 2: The integral equals the signed area: rectangle from... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{5451}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Su... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010602 | Calculus: Derivatives — Difference Quotient | 6 | Carefully track domains: Let $f(x)=10x^2+(-10)x$.
(a) Compute $f'(6)$ using differentiation rules.
(b) Compute $f'(6)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justi... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=10(a+h)^2+(-10)(a+h)=10(a^2+2ah+h^2)+(-10)a+(-10)h$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{110}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=10,n=-10,a=6$ both give 110.",
"robustness_analysis": "Sensitivity analysis: The rule-b... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010603 | Calculus: Derivatives — Power Rule | 6 | Write the solution set clearly: Let $f(x)=-2x^2+(16)x$.
(a) Compute $f'(4)$ using differentiation rules.
(b) Compute $f'(4)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-2)x+(16)=-4x+(16)$.",
"Step 2: Substitute $x=4$ to get $f'(4)=-4(4)+(16)=0$.",
"Final step: Therefore \\boxed{0}."
],
"final_answer": "\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{0}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-2,n=16,a=4$ both give 0.",
"robustness_analysis": "Generality note: The rule-based metho... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010604 | Calculus: Derivatives — Difference Quotient | 6 | Do not skip justification steps: Let $f(x)=1x^2+(-12)x$.
(a) Compute $f'(14)$ using differentiation rules.
(b) Compute $f'(14)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explici... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=1(a+h)^2+(-12)(a+h)=1(a^2+2ah+h^2)+(-12)a+(-12)h$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{16}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=1,n=-12,a=14$ both give 16.",
"robustness_analysis": "If the problem we... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010605 | Calculus: Differentiation — Cross-Validation | 6 | Solve and then verify: Let $f(x)=-4x^2+(24)x$.
(a) Compute $f'(13)$ using differentiation rules.
(b) Compute $f'(13)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justif... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-4(a+h)^2+(24)(a+h)=-4(a^2+2ah+h^2)+(24)a+(24)h$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-80}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-4,n=24,a=13$ both give -80.",
"robustness_analysis": "If the problem ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-80}$.) |
math-010606 | Calculus: Integrals — Linear Functions | 6 | Start by stating any domain restrictions: Compute the definite integral
$$\int_0^17 (1x+(-39))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signe... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{1}{2}x^2+(-39)x$ since $F'(x)=1x+(-39)$.",
"Step 2: By FTC, $\\int_0^17 (1x+(-39))dx = F(17)-F(0)$.",
"Final step: $F(17)-F(0)=... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{-1037}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=1,q=-39,t=1... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010607 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Give a fully justified solution: Let $f(x)=7x^2+(5)x$.
(a) Compute $f'(-9)$ using differentiation rules.
(b) Compute $f'(-9)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitl... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(7)x+(5)=14x+(5)$.",
"Step 2: Substitute $x=-9$ to get $f'(-9)=14(-9)+(5)=-121$.",
"Final step: Therefore \\boxed{-121}."
],
"final_answer"... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-121}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=7,n=5,a=-9$ both give -121.",
"robustness_analysis": "If the problem were perturbed: T... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010608 | Calculus: Derivatives — Power Rule | 6 | Provide a rigorous solution: Let $f(x)=11x^2+(-10)x$.
(a) Compute $f'(12)$ using differentiation rules.
(b) Compute $f'(12)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(11)x+(-10)=22x+(-10)$.",
"Step 2: Substitute $x=12$ to get $f'(12)=22(12)+(-10)=254$.",
"Final step: Therefore \\boxed{254}."
],
"final_an... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{254}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=11,n=-10,a=12$ both give 254.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{254}$.) |
math-010609 | Calculus: Integrals — Signed Area Interpretation | 6 | Work this out carefully: Compute the definite integral
$$\int_0^6 (10x+(-14))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles t... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=10x+(-14)$ is a line on $[0,6]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{96}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=10,q... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{96}$.) |
math-010610 | Calculus: Derivatives — Difference Quotient | 6 | Indicate where a theorem is used: Let $f(x)=12x^2+(2)x$.
(a) Compute $f'(15)$ using differentiation rules.
(b) Compute $f'(15)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explici... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=12(a+h)^2+(2)(a+h)=12(a^2+2ah+h^2)+(2)a+(2)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{362}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=12,n=2,a=15$ both give 362.",
"robustness_analysis": "Sensitivity analysis: The rule-ba... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{362}$.) |
math-010611 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Try to avoid pattern-matching; explain why: Compute the definite integral
$$\int_0^24 (-15x+(1))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'sig... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-15}{2}x^2+(1)x$ since $F'(x)=-15x+(1)$.",
"Step 2: By FTC, $\\int_0^24 (-15x+(1))dx = F(24)-F(0)$.",
"Final step: $F(24)-F(0)=... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-4296}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-15,q=1,t=24$ yields -42... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-4296}$.) |
math-010612 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Problem: Let $f(x)=8x^2+(-10)x$.
(a) Compute $f'(-18)$ using differentiation rules.
(b) Compute $f'(-18)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(8)x+(-10)=16x+(-10)$.",
"Step 2: Substitute $x=-18$ to get $f'(-18)=16(-18)+(-10)=-298$.",
"Final step: Therefore \\boxed{-298}."
],
"fina... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-298}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=8,n=-10,a=-18$ both give -298.",
"robustness_analysis": "Generality note: The rule-based met... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-298}$.) |
math-010613 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Prompt: Compute the definite integral
$$\int_0^40 (-3x+(-6))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when the ... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-3}{2}x^2+(-6)x$ since $F'(x)=-3x+(-6)$.",
"Step 2: By FTC, $\\int_0^40 (-3x+(-6))dx = F(40)-F(0)$.",
"Final step: $F(40)-F(0)=... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-2640}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-3,q=-6,t=40$ yiel... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-2640}$.) |
math-010614 | Foundations of Calculus: Derivative Definition | 6 | Do not skip justification steps: Let $f(x)=4x^2+(-26)x$.
(a) Compute $f'(-8)$ using differentiation rules.
(b) Compute $f'(-8)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explici... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(4)x+(-26)=8x+(-26)$.",
"Step 2: Substitute $x=-8$ to get $f'(-8)=8(-8)+(-26)=-90$.",
"Final step: Therefore \\boxed{-90}."
],
"final_answe... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-90}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=4,n=-26,a=-8$ both give -90.",
"robustness_analysis": "Robustness note... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-90}$.) |
math-010615 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Solve with verification: Let $f(x)=-1x^2+(12)x$.
(a) Compute $f'(-14)$ using differentiation rules.
(b) Compute $f'(-14)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly ju... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-1)x+(12)=-2x+(12)$.",
"Step 2: Substitute $x=-14$ to get $f'(-14)=-2(-14)+(12)=40$.",
"Final step: Therefore \\boxed{40}."
],
"final_answ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{40}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-1,n=12,a=-14$ both give 40.",
"robustness_analysis": "Robustn... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010616 | Foundations of Calculus: Area vs Antiderivative | 6 | Checkpoint: Compute the definite integral
$$\int_0^12 (-3x+(7))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when t... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-3}{2}x^2+(7)x$ since $F'(x)=-3x+(7)$.",
"Step 2: By FTC, $\\int_0^12 (-3x+(7))dx = F(12)-F(0)$.",
"Final step: $F(12)-F(0)=\\f... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-132}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-3... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-132}$.) |
math-010617 | Calculus: Derivatives — Power Rule | 6 | Try to avoid pattern-matching; explain why: Let $f(x)=-8x^2+(19)x$.
(a) Compute $f'(-1)$ using differentiation rules.
(b) Compute $f'(-1)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-8(a+h)^2+(19)(a+h)=-8(a^2+2ah+h^2)+(19)a+(19)h$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{35}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-8,n=19,a=-1$ both give 35.",
"robustness_analysis": "Generali... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{35}$.) |
math-010618 | Calculus: Derivatives — Difference Quotient | 6 | Prompt: Let $f(x)=-2x^2+(24)x$.
(a) Compute $f'(16)$ using differentiation rules.
(b) Compute $f'(16)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling $h... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-2(a+h)^2+(24)(a+h)=-2(a^2+2ah+h^2)+(24)a+(24)h$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-40}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-2,n=24,a=16$ both give -40.",
"robustness_analysis": "Genera... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010619 | Foundations of Calculus: Area vs Antiderivative | 6 | Indicate where a theorem is used: Compute the definite integral
$$\int_0^25 (7x+(15))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' h... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{7}{2}x^2+(15)x$ since $F'(x)=7x+(15)$.",
"Step 2: By FTC, $\\int_0^25 (7x+(15))dx = F(25)-F(0)$.",
"Final step: $F(25)-F(0)=\\f... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{5125}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Su... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{\frac{5125}$.) |
math-010620 | Foundations of Calculus: Derivative Definition | 6 | Compute the requested quantity: Let $f(x)=11x^2+(27)x$.
(a) Compute $f'(-20)$ using differentiation rules.
(b) Compute $f'(-20)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explic... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(11)x+(27)=22x+(27)$.",
"Step 2: Substitute $x=-20$ to get $f'(-20)=22(-20)+(27)=-413$.",
"Final step: Therefore \\boxed{-413}."
],
"final_... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-413}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=11,n=27,a=-20$ both give -413.",
"robustness_analysis": "If ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010621 | Calculus: Integrals — Exact Arithmetic | 6 | Give a fully justified solution: Compute the definite integral
$$\int_0^11 (14x+(5))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' ha... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{14}{2}x^2+(5)x$ since $F'(x)=14x+(5)$.",
"Step 2: By FTC, $\\int_0^11 (14x+(5))dx = F(11)-F(0)$.",
"Final step: $F(11)-F(0)=\\f... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{902}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=14,... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010622 | Calculus: Integrals — Exact Arithmetic | 6 | Compute the requested quantity: Compute the definite integral
$$\int_0^27 (8x+(4))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' hand... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{8}{2}x^2+(4)x$ since $F'(x)=8x+(4)$.",
"Step 2: By FTC, $\\int_0^27 (8x+(4))dx = F(27)-F(0)$.",
"Final step: $F(27)-F(0)=\\frac... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{3024}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=8,... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010623 | Calculus: Derivatives — Power Rule | 6 | Solve and then verify: Let $f(x)=5x^2+(-5)x$.
(a) Compute $f'(11)$ using differentiation rules.
(b) Compute $f'(11)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(5)x+(-5)=10x+(-5)$.",
"Step 2: Substitute $x=11$ to get $f'(11)=10(11)+(-5)=105$.",
"Final step: Therefore \\boxed{105}."
],
"final_answer... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{105}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=5,n=-5,a=11$ both give 105.",
"robustness_analysis": "Robustness note: The rule-based method ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{105}$.) |
math-010624 | Calculus: Integrals — Exact Arithmetic | 6 | Solve (and briefly cross-validate): Compute the definite integral
$$\int_0^14 (13x+(37))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=13x+(37)$ is a line on $[0,14]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{1792}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=13,q=37,t=14$ yield... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010625 | Calculus: Integrals — Linear Functions | 6 | Solve and include a self-check: Compute the definite integral
$$\int_0^17 (-4x+(-6))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' ha... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-4x+(-6)$ is a line on $[0,17]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-680}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-4,q=-6,t=17$ yields -680... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-680}$.) |
math-010626 | Calculus: Derivatives — Power Rule | 6 | Work carefully and justify each inference: Let $f(x)=-4x^2+(17)x$.
(a) Compute $f'(-11)$ using differentiation rules.
(b) Compute $f'(-11)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-4(a+h)^2+(17)(a+h)=-4(a^2+2ah+h^2)+(17)a+(17)h$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{105}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-4,n=17,a=-11$ both give 105.",
"robustness_analysis": "Generality note: The rule-based metho... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010627 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Carefully track domains: Let $f(x)=-4x^2+(22)x$.
(a) Compute $f'(-1)$ using differentiation rules.
(b) Compute $f'(-1)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly just... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-4)x+(22)=-8x+(22)$.",
"Step 2: Substitute $x=-1$ to get $f'(-1)=-8(-1)+(22)=30$.",
"Final step: Therefore \\boxed{30}."
],
"final_answer"... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{30}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-4,n=22,a=-1$ both give 30.",
"robustness_analysis": "Sensitivity analy... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010628 | Foundations of Calculus: Area vs Antiderivative | 6 | Use two approaches if possible: Compute the definite integral
$$\int_0^31 (-9x+(38))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' ha... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-9x+(38)$ is a line on $[0,31]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{-6293}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituti... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{\frac{-6293}$.) |
math-010629 | Calculus: Derivatives — Difference Quotient | 6 | Try to avoid pattern-matching; explain why: Let $f(x)=2x^2+(-18)x$.
(a) Compute $f'(-6)$ using differentiation rules.
(b) Compute $f'(-6)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=2(a+h)^2+(-18)(a+h)=2(a^2+2ah+h^2)+(-18)a+(-18)h$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-42}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=2,n=-18,a=-6$ both give -42.",
"robustness_analysis": "Generality note: The rule-based method... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-42}$.) |
math-010630 | Calculus: Differentiation — Cross-Validation | 6 | Give reasoning, not just computation: Let $f(x)=-5x^2+(12)x$.
(a) Compute $f'(1)$ using differentiation rules.
(b) Compute $f'(1)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and expl... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-5(a+h)^2+(12)(a+h)=-5(a^2+2ah+h^2)+(12)a+(12)h$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{2}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-5,n=12,a=1$ both give 2.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{2}$.) |
math-010631 | Calculus: Differentiation — Cross-Validation | 6 | Use two approaches if possible: Let $f(x)=11x^2+(-3)x$.
(a) Compute $f'(10)$ using differentiation rules.
(b) Compute $f'(10)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicit... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=11(a+h)^2+(-3)(a+h)=11(a^2+2ah+h^2)+(-3)a+(-3)h$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{217}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=11,n=-3,a=10$ both give 217.",
"robustness_analysis": "Generality note... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{217}$.) |
math-010632 | Calculus: Differentiation — Cross-Validation | 6 | Make each step logically reversible (or explain if not): Let $f(x)=-7x^2+(-17)x$.
(a) Compute $f'(-16)$ using differentiation rules.
(b) Compute $f'(-16)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-7)x+(-17)=-14x+(-17)$.",
"Step 2: Substitute $x=-16$ to get $f'(-16)=-14(-16)+(-17)=207$.",
"Final step: Therefore \\boxed{207}."
],
"fin... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{207}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-7,n=-17,a=-16$ both give 207.",
"robustness_analysis": "If the proble... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{207}$.) |
math-010633 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Keep the final answer in boxed form: Compute the definite integral
$$\int_0^17 (-15x+(27))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed ar... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-15x+(27)$ is a line on $[0,17]$.",
"Step 2: The integral equals the signed area: rectangle fr... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{-3417}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-15,q=27,t=17$ yi... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010634 | Calculus: Differentiation — Cross-Validation | 6 | Prompt: Let $f(x)=10x^2+(-7)x$.
(a) Compute $f'(-15)$ using differentiation rules.
(b) Compute $f'(-15)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling ... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(10)x+(-7)=20x+(-7)$.",
"Step 2: Substitute $x=-15$ to get $f'(-15)=20(-15)+(-7)=-307$.",
"Final step: Therefore \\boxed{-307}."
],
"final_... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-307}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=10,n=-7,a=-15$ both give -307.",
"robustness_analysis": "If ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-307}$.) |
math-010635 | Foundations of Calculus: Derivative Definition | 6 | Warm-up: Let $f(x)=-6x^2+(-1)x$.
(a) Compute $f'(-4)$ using differentiation rules.
(b) Compute $f'(-4)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling $... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-6)x+(-1)=-12x+(-1)$.",
"Step 2: Substitute $x=-4$ to get $f'(-4)=-12(-4)+(-1)=47$.",
"Final step: Therefore \\boxed{47}."
],
"final_answe... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{47}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-6,n=-1,a=-4$ both give 47.",
"robustness_analysis": "If the problem were perturbed: The rule-... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{47}$.) |
math-010636 | Calculus: Integrals — Linear Functions | 6 | Work this out carefully: Compute the definite integral
$$\int_0^9 (-11x+(38))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles t... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-11x+(38)$ is a line on $[0,9]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{-207}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-11,q=38,t=9... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010637 | Calculus: Integrals — Linear Functions | 6 | Where appropriate, name the theorem you use: Compute the definite integral
$$\int_0^18 (-7x+(-27))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 's... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-7}{2}x^2+(-27)x$ since $F'(x)=-7x+(-27)$.",
"Step 2: By FTC, $\\int_0^18 (-7x+(-27))dx = F(18)-F(0)$.",
"Final step: $F(18)-F(... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-1620}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-1620}$.) |
math-010638 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Question: Compute the definite integral
$$\int_0^24 (-12x+(-9))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when t... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-12}{2}x^2+(-9)x$ since $F'(x)=-12x+(-9)$.",
"Step 2: By FTC, $\\int_0^24 (-12x+(-9))dx = F(24)-F(0)$.",
"Final step: $F(24)-F(... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-3672}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-12,q=-9,t=24$ yields -3... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-3672}$.) |
math-010639 | Calculus: Derivatives — Power Rule | 6 | Carefully track domains: Let $f(x)=6x^2+(-5)x$.
(a) Compute $f'(5)$ using differentiation rules.
(b) Compute $f'(5)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=6(a+h)^2+(-5)(a+h)=6(a^2+2ah+h^2)+(-5)a+(-5)h$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{55}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=6,n=-5,a=5$ both give 55.",
"robustness_analysis": "If the pro... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{55}$.) |
math-010640 | Calculus: Derivatives — Power Rule | 6 | Work carefully and justify each inference: Let $f(x)=-6x^2+(23)x$.
(a) Compute $f'(-16)$ using differentiation rules.
(b) Compute $f'(-16)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-6(a+h)^2+(23)(a+h)=-6(a^2+2ah+h^2)+(23)a+(23)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{215}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-6,n=23,a=-16$ both give 215.",
"robustness_analysis": "Sensitivity analysis: The rule-... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{215}$.) |
math-010641 | Calculus: Derivatives — Power Rule | 6 | Where appropriate, name the theorem you use: Let $f(x)=-2x^2+(-14)x$.
(a) Compute $f'(10)$ using differentiation rules.
(b) Compute $f'(10)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefull... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-2(a+h)^2+(-14)(a+h)=-2(a^2+2ah+h^2)+(-14)a+(-14)h$.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-54}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-2,n=-14,a=10$ both give -54.",
"robustness_analysis": "Sensi... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-54}$.) |
math-010642 | Foundations of Calculus: Area vs Antiderivative | 6 | Write the solution set clearly: Compute the definite integral
$$\int_0^14 (7x+(3))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' hand... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{7}{2}x^2+(3)x$ since $F'(x)=7x+(3)$.",
"Step 2: By FTC, $\\int_0^14 (7x+(3))dx = F(14)-F(0)$.",
"Final step: $F(14)-F(0)=\\frac... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{728}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=7,q=3,t=14$ yields 7... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{728}$.) |
math-010643 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Track units/moduli carefully: Compute the definite integral
$$\int_0^18 (-8x+(27))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' hand... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-8x+(27)$ is a line on $[0,18]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-810}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitut... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-810}$.) |
math-010644 | Calculus: Integrals — Signed Area Interpretation | 6 | Derive the result step-by-step: Compute the definite integral
$$\int_0^20 (-11x+(9))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' ha... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-11}{2}x^2+(9)x$ since $F'(x)=-11x+(9)$.",
"Step 2: By FTC, $\\int_0^20 (-11x+(9))dx = F(20)-F(0)$.",
"Final step: $F(20)-F(0)=... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-2020}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitu... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-2020}$.) |
math-010645 | Calculus: Integrals — Signed Area Interpretation | 6 | Explain why your operations are valid: Compute the definite integral
$$\int_0^14 (-13x+(-11))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-13}{2}x^2+(-11)x$ since $F'(x)=-13x+(-11)$.",
"Step 2: By FTC, $\\int_0^14 (-13x+(-11))dx = F(14)-F(0)$.",
"Final step: $F(14)... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-1428}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-1428}$.) |
math-010646 | Calculus: Integrals — Exact Arithmetic | 6 | Give a fully justified solution: Compute the definite integral
$$\int_0^8 (1x+(2))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' hand... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=1x+(2)$ is a line on $[0,8]$.",
"Step 2: The integral equals the signed area: rectangle from t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{48}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=1,q=2,t=8$ yields 48.",
"... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{48}$.) |
math-010647 | Calculus: Differentiation — Cross-Validation | 6 | Indicate where a theorem is used: Let $f(x)=9x^2+(13)x$.
(a) Compute $f'(-12)$ using differentiation rules.
(b) Compute $f'(-12)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and expli... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(9)x+(13)=18x+(13)$.",
"Step 2: Substitute $x=-12$ to get $f'(-12)=18(-12)+(13)=-203$.",
"Final step: Therefore \\boxed{-203}."
],
"final_a... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-203}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=9,n=13,a=-12$ both give -203.",
"robustness_analysis": "Generality note: The rule-base... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-203}$.) |
math-010648 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Checkpoint: Compute the definite integral
$$\int_0^5 (7x+(-24))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when t... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{7}{2}x^2+(-24)x$ since $F'(x)=7x+(-24)$.",
"Step 2: By FTC, $\\int_0^5 (7x+(-24))dx = F(5)-F(0)$.",
"Final step: $F(5)-F(0)=\\f... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{-65}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=7,q=-24,t=5$ yields... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010649 | Calculus: Differentiation — Cross-Validation | 6 | Work carefully and justify each inference: Let $f(x)=-1x^2+(-30)x$.
(a) Compute $f'(-3)$ using differentiation rules.
(b) Compute $f'(-3)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-1(a+h)^2+(-30)(a+h)=-1(a^2+2ah+h^2)+(-30)a+(-30)h$.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-24}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-1,n=-30,a=-3$ both give -24.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-24}$.) |
math-010650 | Calculus: Integrals — Exact Arithmetic | 6 | Question: Compute the definite integral
$$\int_0^39 (9x+(35))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when the... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=9x+(35)$ is a line on $[0,39]$.",
"Step 2: The integral equals the signed area: rectangle from... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{16419}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=9,q=35,t=39... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{\frac{16419}$.) |
math-010651 | Calculus: Derivatives — Power Rule | 6 | Answer using clear logical steps: Let $f(x)=-9x^2+(-29)x$.
(a) Compute $f'(18)$ using differentiation rules.
(b) Compute $f'(18)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and expli... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-9)x+(-29)=-18x+(-29)$.",
"Step 2: Substitute $x=18$ to get $f'(18)=-18(18)+(-29)=-353$.",
"Final step: Therefore \\boxed{-353}."
],
"fina... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-353}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-9,n=-29,a=18$ both give -353.",
"robustness_analysis": "Generality note: The rule-based met... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-353}$.) |
math-010652 | Foundations of Calculus: Derivative Definition | 6 | Checkpoint: Let $f(x)=-6x^2+(28)x$.
(a) Compute $f'(-10)$ using differentiation rules.
(b) Compute $f'(-10)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancell... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-6(a+h)^2+(28)(a+h)=-6(a^2+2ah+h^2)+(28)a+(28)h$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{148}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-6,n=28,a=-10$ both give 148.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010653 | Calculus: Derivatives — Difference Quotient | 6 | Use two approaches if possible: Let $f(x)=-10x^2+(24)x$.
(a) Compute $f'(-16)$ using differentiation rules.
(b) Compute $f'(-16)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and expli... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-10)x+(24)=-20x+(24)$.",
"Step 2: Substitute $x=-16$ to get $f'(-16)=-20(-16)+(24)=344$.",
"Final step: Therefore \\boxed{344}."
],
"final... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{344}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-10,n=24,a=-16$ both give 344.",
"robustness_analysis": "Robustness note: The rule-based meth... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{344}$.) |
math-010654 | Foundations of Calculus: Derivative Definition | 6 | Complete the analysis: Let $f(x)=-1x^2+(13)x$.
(a) Compute $f'(3)$ using differentiation rules.
(b) Compute $f'(3)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-1(a+h)^2+(13)(a+h)=-1(a^2+2ah+h^2)+(13)a+(13)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{7}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-1,n=13,a=3$ both give 7.",
"robustness_analysis": "If the problem were perturbed: The ru... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{7}$.) |
math-010655 | Calculus: Derivatives — Difference Quotient | 6 | Carefully track domains: Let $f(x)=7x^2+(18)x$.
(a) Compute $f'(-16)$ using differentiation rules.
(b) Compute $f'(-16)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly jus... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(7)x+(18)=14x+(18)$.",
"Step 2: Substitute $x=-16$ to get $f'(-16)=14(-16)+(18)=-206$.",
"Final step: Therefore \\boxed{-206}."
],
"final_a... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-206}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=7,n=18,a=-16$ both give -206.",
"robustness_analysis": "Sensitivity analysis: The rule... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-206}$.) |
math-010656 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Solve and sanity-check: Let $f(x)=-5x^2+(4)x$.
(a) Compute $f'(-10)$ using differentiation rules.
(b) Compute $f'(-10)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly just... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-5(a+h)^2+(4)(a+h)=-5(a^2+2ah+h^2)+(4)a+(4)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{104}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-5,n=4,a=-10$ both give 104.",
"robustness_analysis": "Robustness note: The rule-based ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{104}$.) |
math-010657 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Answer with a short justification: Let $f(x)=-1x^2+(-7)x$.
(a) Compute $f'(-20)$ using differentiation rules.
(b) Compute $f'(-20)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and exp... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-1(a+h)^2+(-7)(a+h)=-1(a^2+2ah+h^2)+(-7)a+(-7)h$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{33}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-1,n=-7,a=-20$ both give 33.",
"robustness_analysis": "Sensiti... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{33}$.) |
math-010658 | Calculus: Differentiation — Cross-Validation | 6 | Derive the result step-by-step: Let $f(x)=5x^2+(25)x$.
(a) Compute $f'(7)$ using differentiation rules.
(b) Compute $f'(7)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=5(a+h)^2+(25)(a+h)=5(a^2+2ah+h^2)+(25)a+(25)h$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{95}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=5,n=25,a=7$ both give 95.",
"robustness_analysis": "Sensitivit... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{95}$.) |
math-010659 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Prompt: Let $f(x)=-3x^2+(-9)x$.
(a) Compute $f'(-10)$ using differentiation rules.
(b) Compute $f'(-10)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-3(a+h)^2+(-9)(a+h)=-3(a^2+2ah+h^2)+(-9)a+(-9)h$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{51}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-3,n=-9,a=-10$ both give 51.",
"robustness_analysis": "Sensitivity anal... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{51}$.) |
math-010660 | Calculus: Differentiation — Cross-Validation | 6 | Answer using clear logical steps: Let $f(x)=9x^2+(4)x$.
(a) Compute $f'(7)$ using differentiation rules.
(b) Compute $f'(7)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=9(a+h)^2+(4)(a+h)=9(a^2+2ah+h^2)+(4)a+(4)h$.",
"... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{130}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=9,n=4,a=7$ both give 130.",
"robustness_analysis": "Generality note: The rule-based method is... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{130}$.) |
math-010661 | Foundations of Calculus: Area vs Antiderivative | 6 | Use two approaches if possible: Compute the definite integral
$$\int_0^7 (3x+(29))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' hand... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{3}{2}x^2+(29)x$ since $F'(x)=3x+(29)$.",
"Step 2: By FTC, $\\int_0^7 (3x+(29))dx = F(7)-F(0)$.",
"Final step: $F(7)-F(0)=\\frac... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\frac{553}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=3,q=29,t=7$ y... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{\frac{553}$.) |
math-010662 | Calculus: Integrals — Exact Arithmetic | 6 | Exercise: Compute the definite integral
$$\int_0^10 (-10x+(3))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when th... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-10x+(3)$ is a line on $[0,10]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-470}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-1... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-470}$.) |
math-010663 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Work carefully and justify each inference: Let $f(x)=-4x^2+(-16)x$.
(a) Compute $f'(-19)$ using differentiation rules.
(b) Compute $f'(-19)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefull... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-4)x+(-16)=-8x+(-16)$.",
"Step 2: Substitute $x=-19$ to get $f'(-19)=-8(-19)+(-16)=136$.",
"Final step: Therefore \\boxed{136}."
],
"final... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{136}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-4,n=-16,a=-19$ both give 136.",
"robustness_analysis": "Sensitivity analysis: The rule... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010664 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Proceed methodically: Compute the definite integral
$$\int_0^34 (9x+(-7))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the c... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=9x+(-7)$ is a line on $[0,34]$.",
"Step 2: The integral equals the signed area: rectangle from... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{4964}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=9,q=-7,t=34$ yields 4964.... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010665 | Foundations of Calculus: Area vs Antiderivative | 6 | Find the exact value: Compute the definite integral
$$\int_0^2 (-9x+(-24))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the ... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-9}{2}x^2+(-24)x$ since $F'(x)=-9x+(-24)$.",
"Step 2: By FTC, $\\int_0^2 (-9x+(-24))dx = F(2)-F(0)$.",
"Final step: $F(2)-F(0)=... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-66}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-9,q=-24,t=2$ yields -66."... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-66}$.) |
math-010666 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Question: Compute the definite integral
$$\int_0^23 (3x+(28))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when the... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=3x+(28)$ is a line on $[0,23]$.",
"Step 2: The integral equals the signed area: rectangle from... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\frac{2875}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=3,q=28,t=23$ yield... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010667 | Calculus: Differentiation — Cross-Validation | 6 | Complete the analysis: Let $f(x)=-10x^2+(-13)x$.
(a) Compute $f'(0)$ using differentiation rules.
(b) Compute $f'(0)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justif... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-10(a+h)^2+(-13)(a+h)=-10(a^2+2ah+h^2)+(-13)a+(-13)h$.... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-13}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-10,n=-13,a=0$ both give -13.",
"robustness_analysis": "Generality note: The rule-based... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010668 | Foundations of Calculus: Derivative Definition | 6 | Solve and sanity-check: Let $f(x)=11x^2+(-9)x$.
(a) Compute $f'(-18)$ using differentiation rules.
(b) Compute $f'(-18)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly jus... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=11(a+h)^2+(-9)(a+h)=11(a^2+2ah+h^2)+(-9)a+(-9)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-405}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=11,n=-9,a=-18$ both give -405.",
"robustness_analysis": "Robustness note: The rule-bas... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-405}$.) |
math-010669 | Foundations of Calculus: Area vs Antiderivative | 6 | Solve and justify each step: Compute the definite integral
$$\int_0^3 (-9x+(-33))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handl... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-9}{2}x^2+(-33)x$ since $F'(x)=-9x+(-33)$.",
"Step 2: By FTC, $\\int_0^3 (-9x+(-33))dx = F(3)-F(0)$.",
"Final step: $F(3)-F(0)=... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{-279}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitutin... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{\frac{-279}$.) |
math-010670 | Foundations of Calculus: Area vs Antiderivative | 6 | Checkpoint: Compute the definite integral
$$\int_0^21 (-5x+(21))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when ... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-5}{2}x^2+(21)x$ since $F'(x)=-5x+(21)$.",
"Step 2: By FTC, $\\int_0^21 (-5x+(21))dx = F(21)-F(0)$.",
"Final step: $F(21)-F(0)=... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\frac{-1323}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). S... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010671 | Calculus: Differentiation — Cross-Validation | 6 | Make each step logically reversible (or explain if not): Let $f(x)=10x^2+(26)x$.
(a) Compute $f'(-11)$ using differentiation rules.
(b) Compute $f'(-11)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=10(a+h)^2+(26)(a+h)=10(a^2+2ah+h^2)+(26)a+(26)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-194}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=10,n=26,a=-11$ both give -194.",
"robustness_analysis": "Generality note: The rule-bas... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-194}$.) |
math-010672 | Calculus: Integrals — Signed Area Interpretation | 6 | Prompt: Compute the definite integral
$$\int_0^5 (10x+(22))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when the l... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{10}{2}x^2+(22)x$ since $F'(x)=10x+(22)$.",
"Step 2: By FTC, $\\int_0^5 (10x+(22))dx = F(5)-F(0)$.",
"Final step: $F(5)-F(0)=\\f... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{235}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=10,... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010673 | Foundations of Calculus: Derivative Definition | 6 | Compute the requested quantity: Let $f(x)=-10x^2+(19)x$.
(a) Compute $f'(-5)$ using differentiation rules.
(b) Compute $f'(-5)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explici... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-10)x+(19)=-20x+(19)$.",
"Step 2: Substitute $x=-5$ to get $f'(-5)=-20(-5)+(19)=119$.",
"Final step: Therefore \\boxed{119}."
],
"final_an... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{119}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-10,n=19,a=-5$ both give 119.",
"robustness_analysis": "If the problem were perturbed: The ru... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010674 | Calculus: Differentiation — Cross-Validation | 6 | Complete the analysis: Let $f(x)=-2x^2+(24)x$.
(a) Compute $f'(-14)$ using differentiation rules.
(b) Compute $f'(-14)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly just... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-2)x+(24)=-4x+(24)$.",
"Step 2: Substitute $x=-14$ to get $f'(-14)=-4(-14)+(24)=80$.",
"Final step: Therefore \\boxed{80}."
],
"final_answ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{80}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-2,n=24,a=-14$ both give 80.",
"robustness_analysis": "Generality note: The rule-based method ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010675 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Challenge: Compute the definite integral
$$\int_0^23 (12x+(-12))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when ... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{12}{2}x^2+(-12)x$ since $F'(x)=12x+(-12)$.",
"Step 2: By FTC, $\\int_0^23 (12x+(-12))dx = F(23)-F(0)$.",
"Final step: $F(23)-F(... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{2898}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=12,q=-12,t=23$ yields 289... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{2898}$.) |
math-010676 | Calculus: Integrals — Linear Functions | 6 | Question: Compute the definite integral
$$\int_0^32 (7x+(-24))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when th... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{7}{2}x^2+(-24)x$ since $F'(x)=7x+(-24)$.",
"Step 2: By FTC, $\\int_0^32 (7x+(-24))dx = F(32)-F(0)$.",
"Final step: $F(32)-F(0)=... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{2816}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=7,q=-24,t=32$ yields 2816... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010677 | Calculus: Integrals — Linear Functions | 6 | Give a theorem-based solution: Compute the definite integral
$$\int_0^14 (-15x+(-29))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' h... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-15x+(-29)$ is a line on $[0,14]$.",
"Step 2: The integral equals the signed area: rectangle f... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-1876}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitu... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-1876}$.) |
math-010678 | Calculus: Derivatives — Power Rule | 6 | Use two approaches if possible: Let $f(x)=11x^2+(9)x$.
(a) Compute $f'(20)$ using differentiation rules.
(b) Compute $f'(20)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitl... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=11(a+h)^2+(9)(a+h)=11(a^2+2ah+h^2)+(9)a+(9)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{449}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=11,n=9,a=20$ both give 449.",
"robustness_analysis": "Generality note: The rule-based m... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010679 | Foundations of Calculus: Area vs Antiderivative | 6 | Explain what is being counted/optimized: Compute the definite integral
$$\int_0^12 (-15x+(15))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signe... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-15}{2}x^2+(15)x$ since $F'(x)=-15x+(15)$.",
"Step 2: By FTC, $\\int_0^12 (-15x+(15))dx = F(12)-F(0)$.",
"Final step: $F(12)-F(... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-900}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitut... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-900}$.) |
math-010680 | Foundations of Calculus: Derivative Definition | 6 | Question: Let $f(x)=4x^2+(-14)x$.
(a) Compute $f'(12)$ using differentiation rules.
(b) Compute $f'(12)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling ... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(4)x+(-14)=8x+(-14)$.",
"Step 2: Substitute $x=12$ to get $f'(12)=8(12)+(-14)=82$.",
"Final step: Therefore \\boxed{82}."
],
"final_answer"... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{82}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=4,n=-14,a=12$ both give 82.",
"robustness_analysis": "Generality note: The rule-based method i... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010681 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Work this out carefully: Let $f(x)=5x^2+(26)x$.
(a) Compute $f'(-19)$ using differentiation rules.
(b) Compute $f'(-19)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly jus... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(5)x+(26)=10x+(26)$.",
"Step 2: Substitute $x=-19$ to get $f'(-19)=10(-19)+(26)=-164$.",
"Final step: Therefore \\boxed{-164}."
],
"final_a... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-164}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=5,n=26,a=-19$ both give -164.",
"robustness_analysis": "Sensitivity analysis: The rule... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-164}$.) |
math-010682 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Compute the requested quantity: Let $f(x)=-12x^2+(-14)x$.
(a) Compute $f'(12)$ using differentiation rules.
(b) Compute $f'(12)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explic... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-12(a+h)^2+(-14)(a+h)=-12(a^2+2ah+h^2)+(-14)a+(-14)h$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-302}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-12,n=-14,a=12$ both give -302.",
"robustness_analysis": "Ro... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010683 | Calculus: Integrals — Exact Arithmetic | 6 | Give reasoning, not just computation: Compute the definite integral
$$\int_0^39 (-12x+(26))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed a... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{-12}{2}x^2+(26)x$ since $F'(x)=-12x+(26)$.",
"Step 2: By FTC, $\\int_0^39 (-12x+(26))dx = F(39)-F(0)$.",
"Final step: $F(39)-F(... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-8112}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-12,q=26,t=39$ yields -8... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Remember: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010684 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Write the solution set clearly: Compute the definite integral
$$\int_0^36 (9x+(-6))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' han... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{9}{2}x^2+(-6)x$ since $F'(x)=9x+(-6)$.",
"Step 2: By FTC, $\\int_0^36 (9x+(-6))dx = F(36)-F(0)$.",
"Final step: $F(36)-F(0)=\\f... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{5616}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=9,... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{5616}$.) |
math-010685 | Calculus: Integrals — Exact Arithmetic | 6 | Challenge: Compute the definite integral
$$\int_0^33 (2x+(-31))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles the case when t... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{2}{2}x^2+(-31)x$ since $F'(x)=2x+(-31)$.",
"Step 2: By FTC, $\\int_0^33 (2x+(-31))dx = F(33)-F(0)$.",
"Final step: $F(33)-F(0)=... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{66}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=2,q=-31,t=33$ yields ... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010686 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Make each step logically reversible (or explain if not): Let $f(x)=1x^2+(-1)x$.
(a) Compute $f'(-12)$ using differentiation rules.
(b) Compute $f'(-12)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=1(a+h)^2+(-1)(a+h)=1(a^2+2ah+h^2)+(-1)a+(-1)h$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-25}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=1,n=-1,a=-12$ both give -25.",
"robustness_analysis": "If the problem were perturbed: The rul... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-25}$.) |
math-010687 | Calculus: Derivatives — Difference Quotient | 6 | Exercise: Let $f(x)=-1x^2+(17)x$.
(a) Compute $f'(9)$ using differentiation rules.
(b) Compute $f'(9)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling $h... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-1)x+(17)=-2x+(17)$.",
"Step 2: Substitute $x=9$ to get $f'(9)=-2(9)+(17)=-1$.",
"Final step: Therefore \\boxed{-1}."
],
"final_answer": "... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-1}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-1,n=17,a=9$ both give -1.",
"robustness_analysis": "Generalit... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-1}$.) |
math-010688 | Calculus: Integrals — Signed Area Interpretation | 6 | Solve with verification: Compute the definite integral
$$\int_0^16 (12x+(25))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles t... | [
{
"method_name": "Antiderivative (FTC)",
"approach": "Find an antiderivative and evaluate $F(t)-F(0)$.",
"steps": [
"Step 1: An antiderivative is $F(x)=\\frac{12}{2}x^2+(25)x$ since $F'(x)=12x+(25)$.",
"Step 2: By FTC, $\\int_0^16 (12x+(25))dx = F(16)-F(0)$.",
"Final step: $F(16)-F(0)=... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{1936}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitut... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010689 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Solve with verification: Let $f(x)=4x^2+(-27)x$.
(a) Compute $f'(-5)$ using differentiation rules.
(b) Compute $f'(-5)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly just... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(4)x+(-27)=8x+(-27)$.",
"Step 2: Substitute $x=-5$ to get $f'(-5)=8(-5)+(-27)=-67$.",
"Final step: Therefore \\boxed{-67}."
],
"final_answe... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-67}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=4,n=-27,a=-5$ both give -67.",
"robustness_analysis": "Sensitivity ana... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Core principle: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-67}$.) |
math-010690 | Calculus: Derivatives — Difference Quotient | 6 | Warm-up: Let $f(x)=-2x^2+(4)x$.
(a) Compute $f'(8)$ using differentiation rules.
(b) Compute $f'(8)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling $h$ ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=-2(a+h)^2+(4)(a+h)=-2(a^2+2ah+h^2)+(4)a+(4)h$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-28}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-2,n=4,a=8$ both give -28.",
"robustness_analysis": "If the problem were perturbed: The rule-... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010691 | Calculus: Integrals — Linear Functions | 6 | Solve (and briefly cross-validate): Compute the definite integral
$$\int_0^31 (-8x+(-24))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed are... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-8x+(-24)$ is a line on $[0,31]$.",
"Step 2: The integral equals the signed area: rectangle fr... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-4588}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitu... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. |
math-010692 | Calculus: Derivatives — Power Rule | 6 | Task: Let $f(x)=-3x^2+(-13)x$.
(a) Compute $f'(20)$ using differentiation rules.
(b) Compute $f'(20)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling $h$... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(-3)x+(-13)=-6x+(-13)$.",
"Step 2: Substitute $x=20$ to get $f'(20)=-6(20)+(-13)=-133$.",
"Final step: Therefore \\boxed{-133}."
],
"final_... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-133}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=-3,n=-13,a=20$ both give -133.",
"robustness_analysis": "Generality note: The rule-bas... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Key idea: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-133}$.) |
math-010693 | Calculus: Derivatives — Algebraic Expansion Pitfalls | 6 | Work carefully and justify each inference: Let $f(x)=10x^2+(-29)x$.
(a) Compute $f'(-4)$ using differentiation rules.
(b) Compute $f'(-4)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully ... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=10(a+h)^2+(-29)(a+h)=10(a^2+2ah+h^2)+(-29)a+(-29)h$.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-109}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=10,n=-29,a=-4$ both give -109.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-109}$.) |
math-010694 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Give a theorem-based solution: Compute the definite integral
$$\int_0^24 (14x+(-12))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' ha... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=14x+(-12)$ is a line on $[0,24]$.",
"Step 2: The integral equals the signed area: rectangle fr... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{3744}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=14... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{3744}$.) |
math-010695 | Calculus: Integrals — Fundamental Theorem of Calculus | 6 | Answer using clear logical steps: Compute the definite integral
$$\int_0^11 (-2x+(-32))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area'... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-2x+(-32)$ is a line on $[0,11]$.",
"Step 2: The integral equals the signed area: rectangle fr... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-473}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituting $p=-2... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Takeaway: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{-473}$.) |
math-010696 | Calculus: Differentiation — Cross-Validation | 6 | Derive the result step-by-step: Let $f(x)=8x^2+(-7)x$.
(a) Compute $f'(-14)$ using differentiation rules.
(b) Compute $f'(-14)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explici... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=8(a+h)^2+(-7)(a+h)=8(a^2+2ah+h^2)+(-7)a+(-7)h$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-231}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=8,n=-7,a=-14$ both give -231.",
"robustness_analysis": "Robustness note: The rule-base... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{-231}$.) |
math-010697 | Calculus: Derivatives — Difference Quotient | 6 | Question: Let $f(x)=11x^2+(7)x$.
(a) Compute $f'(5)$ using differentiation rules.
(b) Compute $f'(5)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and explicitly justify cancelling $h$... | [
{
"method_name": "Difference Quotient",
"approach": "Use the definition of derivative and simplify the algebra until the limit is immediate.",
"steps": [
"Step 1: Compute $\\frac{f(a+h)-f(a)}{h}$ for $h\\ne 0$.",
"Step 2: Expand: $f(a+h)=11(a+h)^2+(7)(a+h)=11(a^2+2ah+h^2)+(7)a+(7)h$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{117}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=11,n=7,a=5$ both give 117.",
"robustness_analysis": "Robustness note: The rule-based method i... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Remember: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. (Here the result is $\boxed{117}$.) |
math-010698 | Foundations of Calculus: Area vs Antiderivative | 6 | Give a theorem-based solution: Compute the definite integral
$$\int_0^40 (13x+(-3))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' han... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=13x+(-3)$ is a line on $[0,40]$.",
"Step 2: The integral equals the signed area: rectangle fro... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{10280}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substitu... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Key idea: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{10280}$.) |
math-010699 | Calculus: Derivatives — Power Rule | 6 | Explain what is being counted/optimized: Let $f(x)=2x^2+(7)x$.
(a) Compute $f'(1)$ using differentiation rules.
(b) Compute $f'(1)$ directly from the limit definition $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
(c) Explain why both computations must give the same number.
In part (b), expand $(a+h)^2$ carefully and exp... | [
{
"method_name": "Power Rule",
"approach": "Differentiate term-by-term and evaluate at $x=a$.",
"steps": [
"Step 1: Differentiate: $f'(x)=2(2)x+(7)=4x+(7)$.",
"Step 2: Substitute $x=1$ to get $f'(1)=4(1)+(7)=11$.",
"Final step: Therefore \\boxed{11}."
],
"final_answer": "\\boxe... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{11}$.\nThe difference quotient simplifies to $2ma+n$, which matches evaluating the rule-based derivative $2mx+n$ at $x=a$. With $m=2,n=7,a=1$ both give 11.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Expanded $(a+h)^2$ as $a^2+h^2$ (missing $2ah$).",
"why_plausible": "A common incorrect 'distribution' mistake.",
"why_wrong": "The missing $2ah$ term changes the linear part and therefore the derivative.",
"which_method_catches_it": "Power-rule method avoids this expansion e... | Takeaway: Derivatives computed by rules should match the limit definition because the rules are proved from that definition; using both is an excellent self-check against algebra mistakes. |
math-010700 | Calculus: Integrals — Linear Functions | 6 | Solve and sanity-check: Compute the definite integral
$$\int_0^31 (-5x+(-13))\,dx.$$
(a) Compute using an antiderivative and the Fundamental Theorem of Calculus.
(b) Compute by interpreting the integral as signed area under a line (decompose into a triangle + rectangle).
(c) Explain briefly how 'signed area' handles t... | [
{
"method_name": "Geometry (Triangle + Rectangle)",
"approach": "Interpret the integral as signed area under a line segment and compute using basic area formulas.",
"steps": [
"Step 1: The graph $y=-5x+(-13)$ is a line on $[0,31]$.",
"Step 2: The integral equals the signed area: rectangle fr... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\frac{-5611}$.\nBoth methods compute the same quantity: the FTC gives $\\frac{p}2t^2+qt$, and the geometry method computes the same expression as triangle area plus rectangle area (with sign). Substituti... | [
{
"error_description": "Integrated $px$ as $p\\ln x$ (confusing with $1/x$).",
"why_plausible": "The log antiderivative is heavily memorized.",
"why_wrong": "$\\int x\\,dx=\\frac{x^2}{2}$; $\\ln x$ corresponds to $1/x$, not $x$.",
"which_method_catches_it": "Geometric method makes it clear the resul... | Core principle: Definite integrals can be computed analytically (FTC) or geometrically when the graph is simple; treating the integral as signed area keeps formulas valid even when the function dips below the axis. (Here the result is $\boxed{\frac{-5611}$.) |
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