Complete solutions, method expectations, validity conditions, and verification guidance.
Faculty keySix matching responsesPrint safe
Khaemenes Higher Learning
Faculty Key · Substitution for √(a²−x²)
Show method selection, derivation, conditions, units, verification, and interpretation.
U03 · L04
Name
Date
1
State the central method of Substitution for √(a²−x²) and list the conditions that must be checked.
Faculty solution: Use using x=a sin θ and a reference triangle. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Evaluate ∫dx/√(9−x²).
Faculty solution: arcsin(x/3)+C.
3
Evaluate ∫√(16−x²)dx.
Faculty solution: (x/2)√(16−x²)+8arcsin(x/4)+C.
4
A learner makes an error by using a trigonometric identity or substitution without matching the algebraic form. Explain the first invalid step and repair the reasoning.
Faculty solution: The first invalid step is using a trigonometric identity or substitution without matching the algebraic form. Return to the governing definition or theorem, restore the missing condition, and verify the corrected result.
5
Create a short application of Substitution for √(a²−x²) in periodic energy, circular geometry, radical models, and exact measurement. Define variables and identify the quantity an integral, differential equation, sequence, or series would measure.
Faculty solution: A defensible response defines variables and units, states assumptions, builds the correct mathematical object, and interprets its value in periodic energy, circular geometry, radical models, and exact measurement.
6
Describe two independent ways to verify a result from Substitution for √(a²−x²).
Faculty solution: Use an exact structural check such as differentiation, substitution, partial sums, or theorem hypotheses; then compare with numerical, graphical, dimensional, or limiting evidence.