An equilibrium stock is Q=a/b, with positive input rate a and removal coefficient b. A one-unit change in a and a one-unit change in b are not comparable: the parameters have different units and scales. Begin with derivatives, then normalize or multiply by plausible changes before ranking importance.
At a=10,b=2, Q=5. The derivatives are Qₐ=1/2 and Qᵦ=−5/2. The second is numerically larger, but its interpretation depends on the unit in which b is expressed.
Predict
Predict the effect of a 10% increase in a and a 10% increase in b. Are the two finite percentage effects exactly opposite? A first-order elasticity is a local approximation, so check the exact ratios as well.
Experiment
The elasticities are (a/Q)Qₐ=1 and (b/Q)Qᵦ=−1.
Change
Exact new Q
Relative change
a from 10 to 11
11/2
1/10
b from 2 to 11/5
50/11
−1/11
The second exact change is about −9.09%, not −10%. Its first-order prediction approaches the exact result as the perturbation shrinks. Relative sensitivity is undefined when its normalizing output or parameter is zero; use an absolute or otherwise justified scale there.
Understand
The total differential is dQ≈da/b−a db/b². It describes nearby changes and requires a nonzero denominator. A small denominator can make both output and sensitivity large. Comparing only nominal outputs can hide that instability.
Local derivatives can miss interactions. For f(x,y)=xy at (0,0), both partial derivatives vanish and changing either coordinate alone leaves the output zero. Changing both to h gives h². This is not a contradiction: the first-order approximation has zero linear term, while the leading change is second order. Over a broad range, that interaction can dominate the decision.
A global sensitivity exploration specifies a joint admissible parameter set and examines outputs across it. A rectangular grid is useful when all combinations are allowed, but parameter dependence may exclude some corners. Monotonicity can certify extrema between sampled points; without such an argument or a numerical error bound, a finite grid is exploratory evidence only.
Sensitivity and uncertainty are distinct. A large derivative in a nearly known parameter can matter less than a moderate derivative in a poorly known one. Correlation can reinforce or cancel combined changes. The next lesson propagates a covariance matrix or an admissible interval set rather than treating every parameter direction independently.
Apply
Calculate two local derivatives for your model, attach units and multiply them by plausible input changes. Then choose a small two-parameter grid and compare the joint response with the sum of separate effects. Record whether boundaries or interactions change the conclusion. If you claim a global maximum, provide an argument covering points between the grid samples.
Continue
Uncertainty propagation combines sensitivity with information about inputs. A scenario range, confidence interval and predictive interval are different objects and must be labeled according to how they were constructed.
Apply · explain · revise
Practice and reasoning
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For Q=a/b at a=12,b=3, find ∂Q/∂a.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For positive a,b and Q=a/b, find the elasticity (b/Q)∂Q/∂b.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Derive first-order output change for Q=a/b.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show why one-at-a-time perturbations at (0,0) miss sensitivity of f(x,y)=xy.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Design a bounded global sensitivity check with two uncertain parameters.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does a large derivative not necessarily identify the most important uncertainty source?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
