Show method selection, derivation, conditions, units, verification, and interpretation.
U03 · L07
Name
Date
1
State the central method of Back-Substitution, Domain Analysis & Verification and list the conditions that must be checked.
Faculty solution: Use reconstructing x-forms and checking interval conditions. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
For x=3tanθ, express secθ in x.
Faculty solution: secθ=√(x²+9)/3 on the principal interval.
3
Why must domain restrictions be stated after trigonometric substitution?
Faculty solution: Inverse-trigonometric and square-root identities can change sign outside the selected interval; restrictions preserve the equivalence.
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 Back-Substitution, Domain Analysis & Verification 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 Back-Substitution, Domain Analysis & Verification.
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.