Using Derivatives to Model Change and Make Decisions

1) Quick review: function notation, inputs/outputs, and units

A function connects an input variable to an output quantity. We often write this as y = f(x), where x is the independent variable (what you choose or measure) and y is the dependent variable (what results).

Why units matter

Always attach units to both variables. Units tell you what “change” means and prevent misinterpretation.

• If x is time in seconds (s) and f(x) is position in meters (m), then f'(x) has units m/s (velocity).
• If x is number of items produced (items) and f(x) is cost in dollars ($), then f'(x) has units $/item (marginal cost).
• If x is time in years and f(x) is population in people, then f'(x) has units people/year (instantaneous growth rate).

In practical terms, the derivative f'(x) tells you how fast the output is changing right now as the input changes, and it matches the slope of the tangent line to the graph of f at that input.

2) Interpreting derivative statements like f'(3)=2 (with units)

A statement like f'(3)=2 is incomplete until you know what x and f(x) represent. The meaning is:

At input value x=3, the output f(x) is increasing at a rate of 2 output-units per 1 input-unit.

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Example A: motion (velocity)

Let s(t) be position (meters) at time t (seconds). If s'(3)=2, then:

• Interpretation: at t=3 s, the object’s instantaneous velocity is 2 m/s.
• Decision meaning: near 3 seconds, each additional second increases position by about 2 meters (assuming the rate doesn’t change much over that tiny interval).

Example B: economics (marginal cost)

Let C(q) be total cost (dollars) to produce q items. If C'(3)=2, then:

• Units: $/item.
• Interpretation: when producing q=3 items, the cost is increasing at about $2 per additional item (for very small changes around 3).
• Decision meaning: near 3 items, producing one more item is expected to add roughly $2 to total cost (a local approximation).

Example C: biology (instantaneous growth rate)

Let P(t) be a population (people) at time t (years). If P'(3)=2, then:

• Units: people/year.
• Interpretation: at year 3, the population is increasing at about 2 people per year at that instant.

Sign and magnitude: what the number tells you

• If f'(3)>0, f is increasing at x=3.
• If f'(3)<0, f is decreasing at x=3.
• A larger absolute value |f'(3)| means faster change (steeper slope) at that point.
• If f'(3)=0, the output is momentarily not changing with respect to the input (flat tangent), which often signals a possible “best/worst” point in decision problems.

3) Distinguishing average vs instantaneous rates

Average rate of change (over an interval)

The average rate of change from x=a to x=b is the slope of the secant line:

Average rate = (f(b) - f(a)) / (b - a)

Units: (output units)/(input units), same as the derivative, but it describes the whole interval, not a single moment.

Instantaneous rate of change (at a point)

The instantaneous rate at x=a is the derivative f'(a). Conceptually, it is what the average rate approaches as the interval shrinks around a:

f'(a) = lim_{h→0} (f(a+h) - f(a)) / h

Practical comparison with a motion example

Suppose s(t) is position in meters.

• Average velocity from t=3 to t=5: (s(5)-s(3))/(5-3) in m/s. This tells you the overall pace during that 2-second window.
• Instantaneous velocity at t=3: s'(3) in m/s. This tells you the speed at the exact moment t=3.

Decision tip: If you need a “right now” rate (e.g., current speed, current marginal cost, current growth rate), you want the derivative. If you need performance over a period (e.g., average speed over a trip segment), you want the average rate.

4) Estimating rates from a table or graph

In many real situations you don’t have a formula for f(x). You may have measured data (a table) or a plotted curve. You can still estimate rates.

A) Estimating from a table (difference quotients)

To estimate f'(a), use nearby values of f and compute slopes. The closer the points are to a, the better the estimate (assuming the data is reliable).

Step-by-step: estimate f'(3) from data

1. Locate the input value of interest (x=3).
2. Choose nearby points around 3 (ideally one on each side if available).
3. Compute a slope using a secant line: (f(x2)-f(x1))/(x2-x1).
4. Use a symmetric estimate when possible: pick x1=3-h and x2=3+h to balance error.
5. Attach units to the final rate: (output units)/(input units).

Example table: cost vs quantity

Estimate C'(3) using a symmetric difference around 3:

C'(3) ≈ (C(4) - C(2)) / (4 - 2) = (48 - 41)/2 = 3.5

Interpretation with units: at about 3 items, marginal cost is approximately $3.50 per item.

Decision meaning: near 3 items, producing one more item is expected to add about $3.50 to total cost (locally).

B) Estimating from a graph (tangent slope)

If you have a graph of y=f(x), the derivative at a point is the slope of the tangent line there. To estimate it:

1. Mark the point on the curve at the input value of interest.
2. Sketch the tangent line (a line that just “kisses” the curve at that point and matches its direction).
3. Pick two readable points on the tangent line (not necessarily on the curve) and compute slope: rise/run.
4. Attach units: (vertical units)/(horizontal units).

Common pitfall: using two points on the curve near the point gives an average rate (secant slope). That can be a good approximation, but it is not exactly the tangent slope unless the interval is extremely small.

5) Practice set: interpret meaning (not computation)

For each item: (i) state what the derivative means in words, (ii) include units, and (iii) say whether the quantity is increasing or decreasing at that point.

1. Position and velocity

Given: s(t) is position in meters, t in seconds, and s'(8) = -4.

• What is happening at t=8?
• What does the negative sign indicate physically?

2. Temperature change

Given: T(h) is temperature in °C, h is time in hours, and T'(2)=1.5.

• Interpret T'(2)=1.5 with units.
• About how much would you expect the temperature to change over the next 0.2 hours, if the rate stays roughly constant near h=2?

3. Marginal cost

Given: C(q) is total cost in dollars to produce q units, and C'(120)=0.80.

• Interpret the statement in a production setting.
• Is cost rising or falling at q=120?

4. Revenue sensitivity

Given: R(p) is revenue in dollars when price is p dollars per item, and R'(10)=-200.

• Interpret the meaning of the derivative at p=10.
• What does it suggest about raising the price slightly above $10 (locally)?

5. Estimating a derivative from a table

Given: P(t) is population (thousands of people) at time t (years):

• Estimate P'(5) using a symmetric difference.
• Interpret your estimate with units.

6. Average vs instantaneous (concept check)

Scenario: A car’s position is s(t). Over the interval [10, 12] seconds, the average velocity is 20 m/s, but s'(11)=5 m/s.

• Explain how both statements can be true.
• Which value describes the car’s speed exactly at 11 seconds?

Checklist: interpreting derivatives in context

• Identify variables: What is the input? What is the output? Write y=f(x) with meaning.
• Attach units: Input units, output units, and therefore derivative units = (output)/(input).
• Interpret the point: f'(a)=k means “at x=a, f is changing at k output-units per input-unit.”
• Use the sign: Positive = increasing, negative = decreasing, zero = momentarily flat.
• Rate type: Average rate uses two points (secant slope); instantaneous rate uses the derivative (tangent slope).
• Context meaning: Translate “rate” into the real quantity (velocity, marginal cost, growth rate, temperature change, etc.).