Interpreting Results: Units, Sensitivity, and Reasonableness Checks

Why interpretation matters after the calculus is done

Derivative-based methods often produce a number quickly, but the real work is making that number meaningful and trustworthy. Interpreting results means: (1) attaching correct units, (2) explaining what the sign and size mean, (3) checking whether the answer fits the situation’s limits, (4) describing sensitivity (how outputs respond to small input changes), and (5) reporting a final statement with sensible rounding and a brief justification.

1) Unit analysis for derivatives

Core rule: “units of output per unit of input”

If a quantity y depends on x, then dy/dx has units (units of y)/(units of x). This is not optional; it is part of the meaning of the derivative.

• If C(q) is cost in dollars and q is items, then C'(q) is dollars per item.
• If s(t) is position in meters and t is seconds, then s'(t) is meters per second.
• If T(t) is temperature in °C and t is minutes, then T'(t) is °C per minute.

Step-by-step unit check

1. Write units next to each variable. Example: C in $, q in items.

2. Compute the derivative symbolically (as usual).

3. Attach units using the rule: output units divided by input units.

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4. Sanity-check the units against the story. “Dollars per item” sounds like a marginal cost; “items per dollar” would be a different concept.

Example: catching a unit mismatch

Suppose R(q) is revenue in dollars from selling q items. A student writes: “R'(50)=120 dollars.” The number might be correct, but the units are not. The corrected interpretation is: R'(50)=120 dollars per item, meaning revenue is increasing at about $120 for each additional item sold near 50 items.

2) Interpreting signs and magnitudes

Sign: direction of change

• Positive derivative: increasing output as input increases (locally).
• Negative derivative: decreasing output as input increases (locally).
• Zero derivative: locally flat; could indicate a peak, valley, or a flat spot depending on context.

Magnitude: “how fast” and “how sensitive”

The absolute value |dy/dx| tells how strongly y responds to changes in x. A derivative of 0.2 $/item is a small marginal change; 200 $/item is large. Always interpret magnitude relative to typical values in the context.

Example: interpreting a negative marginal value

Let P(q) be profit in dollars. If P'(300)=-15 $/item, then producing one more item near 300 items is expected to decrease profit by about $15. That does not mean profit is negative; it means profit is trending downward with increased production at that level.

3) Comparing to context limits (reasonableness checks)

Common constraints your answer must respect

• Nonnegativity: quantities like time, distance, mass, number of items, and many costs cannot be negative.
• Capacity limits: production cannot exceed machine capacity; occupancy cannot exceed seats; probability must stay between 0 and 1.
• Domain restrictions: the model may only apply on a stated interval (e.g., 0 ≤ t ≤ 10 minutes).
• Physical realism: speeds, rates, and sizes should be plausible for the situation.

Checklist for reasonableness

1. Check the domain: Is the input value you used allowed?

2. Check the output: Is the computed quantity allowed to be negative or exceed a maximum?

3. Check scale: Is the magnitude plausible (e.g., 10,000 m/s for a car is not)?

4. Check direction: Does the sign match what should happen in that scenario?

Example: capacity check

A model predicts optimal production q*=1200 items, but the factory can produce at most 900 items per day. The derivative-based optimum is not feasible. The correct action is to interpret the result as “the profit would keep increasing up to the capacity,” and then evaluate the objective at the feasible boundary (here, 900 items) rather than reporting 1200 as the final answer.

4) Brief sensitivity reasoning (marginal interpretation)

Using the linear approximation idea

For small changes, derivatives convert input changes into approximate output changes:

Δy ≈ (dy/dx) Δx

This is a communication tool: it turns a derivative into a “what if we change x a little?” statement.

Step-by-step sensitivity statement

1. Identify the derivative value at the operating point. Example: C'(200)=3.5 $/item.

2. Choose a small change in input that matches the question (e.g., Δq=+10 items).

3. Compute the approximate change: ΔC ≈ 3.5 × 10 = 35 dollars.

4. State the approximation clearly and note it is local (valid for small changes near the point).

Example: marginal cost sensitivity

If C'(200)=3.5 $/item, then increasing production from 200 to 210 items changes cost by approximately ΔC ≈ (3.5 $/item)(10 items)=35 $. This does not claim the cost per item is exactly $3.50 for all quantities; it describes the local behavior near 200 items.

Example: motion sensitivity

If v(4)=s'(4)=-2 m/s, then over a short time interval of Δt=0.5 s near t=4, the position changes by approximately Δs ≈ (-2 m/s)(0.5 s)=-1 m. The negative sign indicates motion in the negative direction.

5) Presenting results: rounding, units, and justification

Rounding guidelines

• Match precision to data: If inputs are measured to the nearest unit, reporting 8 decimal places is misleading.
• Use context-appropriate rounding: items are often whole numbers; money is often to cents; time might be to tenths of a second depending on measurement.
• Round at the end to reduce accumulated rounding error.

What a complete final statement should include

• The value (number).
• The units (correct derivative units if it’s a rate).
• The meaning in a full sentence.
• A brief reasonableness note (fits constraints, sign makes sense, magnitude plausible).

Template sentences

• Rate statement: “At x=a, dy/dx = k (units of y)/(units of x), meaning that near x=a, increasing x by 1 (unit of x) changes y by about k (units of y).”

• Feasibility statement: “This value is feasible because it lies within the allowed range […] and respects the constraint that …”

• Rounding justification: “Rounded to the nearest … because the inputs were given to the nearest …”

Exercises: critique, correct, and rewrite

Exercise 1: Unit correction (marginal cost)

A student computes C'(80)=4.2 and writes: “The cost at 80 items is $4.20.”

• Identify the unit error.
• Rewrite the interpretation in a complete sentence with correct units.
• Use ΔC ≈ C'(80)Δq to estimate the cost change if production increases from 80 to 85 items.

Exercise 2: Sign and meaning (profit)

A report states: “Since P'(500)=-30, the profit is -$30 at 500 units.”

• Explain why this is incorrect.
• Provide the correct interpretation of P'(500)=-30 with units.
• What does the sign suggest about producing slightly more than 500 units?

Exercise 3: Context limits (capacity)

An optimization result gives a recommended production level of 1,400 units/day. The plant can produce at most 1,000 units/day.

• Explain what is wrong with reporting 1,400 as the final answer.
• State what should be checked/evaluated instead (describe the action, not the full optimization process).
• Write a final recommendation sentence that respects the capacity limit.

Exercise 4: Sensitivity with units (temperature change)

A cooling model gives T'(12)=-0.8 °C/min.

• Interpret the derivative in words.
• Estimate the temperature change over the next 3 minutes starting at t=12 minutes.
• Explain briefly why this is an approximation and when it might be unreliable.

Exercise 5: Critique a flawed conclusion (rounding and units)

A student writes: “The velocity is v(2)=3.14159265, so the object moves 3.14159265 meters.”

• List two distinct issues in this statement (units/meaning and rounding/interpretation).
• Rewrite the statement correctly if v(2)=3.14159265 m/s and the context measurements are to the nearest 0.1 s and 0.1 m.

Exercise 6: Rewrite for clarity (complete sentences)

You are given the computed results below. Rewrite each as a clear final statement with units and a brief reasonableness check.

• R'(200)=12 (revenue in dollars, quantity in items)
• s'(5)=-1.6 (position in meters, time in seconds)
• V'(10)=0.05 (volume in liters, time in minutes)

Exercise 7: Spot the hidden unit inconsistency

A model uses t measured in minutes, but a student substitutes t=30 when the situation describes 30 seconds.

• Explain how this affects the numerical result.
• State the correct substitution.
• Describe one habit that prevents this mistake (write it as a checklist item).