MathTrigonometry
Graph of Sin: Find the Equation from the Original Curve
Read the original sine graph to determine its amplitude and period, then check the equation against its intercepts and turning points.
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MathAlgebra
Inequality for Absolute Value: Solve |4x + 1| − 3 > 2 Graphically
Graph the absolute-value function and horizontal comparison line, find both intersections, and mark the two open-ended solution intervals.
MathCalculus
Limit Notation: Estimate a Rational Limit with Eight Table Values
Complete the eight prescribed table entries on both sides of x = 2, interpret two-sided limit notation, and verify the limiting value algebraically.
MathCalculus
Calculus Theorems: Why the Mean Value Theorem Fails at a Corner
Prove that the derivative never matches the secant slope for a shifted absolute-value function, and identify the failed Mean Value Theorem hypothesis.
MathCalculus
Partial Fraction Decomposition of (x² + 1)/[x(x + 1)(x + 2)]
Decompose the rational function with three distinct linear factors, solve its coefficients, and check both the full decomposition and a two-expression grouping.
MathCalculus
Anti Derivative: Solve f′(x) = x⁻³ with f(1) = 1
Integrate x⁻³, determine the constant from f(1)=1, and verify the solution on the interval containing the initial point.
MathCalculus
Second Fundamental Theorem of Calculus: Approximate and Exact Area under y = x²
Compute L₁₀, R₁₀ and their average for y=x² on [0,4], then use the second fundamental theorem of calculus to obtain the exact area.
MathCalculus
Antiderivative of 1/2 x: Evaluate ∫dx/(2x)
Read the integrand as 1/(2x), apply the logarithmic rule, and check the antiderivative on each side of zero.
StatisticsProbability
Poisson Errors: Estimating Typos Missed by Two Proofreaders
Estimate the unseen typo distribution with Poisson splitting and two readers, then calculate the expected benefit of a third reader.