Introduction

This topic extends the use of derivatives beyond motion problems to other real-world applications. The essential knowledge tells us that the derivative can be used to solve problems involving rates of change in applied contexts. This builds on our understanding from previous topics to analyze rates of change in economics, biology, chemistry, and other fields.

Economic Applications

Cost and Revenue Functions

For economic functions, derivatives represent marginal values:

Marginal Cost: $C^{\prime }(x)$ = additional cost to produce one more unit

Marginal Revenue: $R^{\prime }(x)$ = additional revenue from selling one more unit

Marginal Profit: $P^{\prime }(x)=R^{\prime }(x)-C^{\prime }(x)$ = additional profit from one more unit

Example 1:

A company's cost function is $C(x)=0.05x^2+20x+500$ dollars for producing $x$ units.

$C^{\prime }(x)=0.1x+20$ dollars per unit

When producing 100 units: $C^{\prime }(100)=0.1(100)+20=30$ dollars per unit

This means the 101st unit costs approximately $30$ to produce.

Demand and Supply Functions

For demand function $D(p)$ where $p$ is price:

$D^{\prime }(p)$ represents the rate of change of demand with respect to price

Example 2:

If demand is $D(p)=1000-5p^2$ units when price is $p$ dollars:

$D^{\prime }(p)=-10p$ units per dollar

At $p=8$ dollars: $D^{\prime }(8)=-80$ units per dollar

This means demand decreases by 80 units for each dollar increase in price.

Population and Growth Models

Population Dynamics

For population function $P(t)$:

$P^{\prime }(t)$ represents the rate of population change (growth or decline)

Example 3:

A town's population follows $P(t)=5000+200t-2t^2$ people after $t$ years.

$P^{\prime }(t)=200-4t$ people per year

At $t=10$ years: $P^{\prime }(10)=200-40=160$ people per year

The population is growing at 160 people per year after 10 years.

At $t=60$ years: $P^{\prime }(60)=200-240=-40$ people per year

After 60 years, the population is declining at 40 people per year.

Temperature and Heat Transfer

Temperature Distribution

For temperature function $T(x)$ or $T(t)$:

$T^{\prime }(x)$ represents temperature gradient (spatial rate of change)

$T^{\prime }(t)$ represents heating or cooling rate (temporal rate of change)

Example 4:

The temperature of a cooling object follows $T(t)=80e^{-0.1t}+20$ degrees Fahrenheit after $t$ minutes.

$T^{\prime }(t)=80\cdot (-0.1)\cdot e^{-0.1t}=-8e^{-0.1t}$ degrees per minute

At $t=5$ minutes: $T^{\prime }(5)=-8e^{-0.5}\approx -4.85$ degrees per minute

The object is cooling at approximately 4.85 degrees per minute after 5 minutes.

Non-Standard Independent Variables

When Time is the Dependent Variable

Sometimes the independent and dependent variables are reversed from typical expectations. Instead of time being the independent variable, it might be the dependent variable.

Example 5:

A chemical process has time as a function of pressure: $t(p)=0.2p^2-3p+15$ minutes when pressure is $p$ atmospheres.

$t^{\prime }(p)=0.4p-3$ minutes per atmosphere

At $p=10$ atmospheres: $t^{\prime }(10)=0.4(10)-3=1$ minute per atmosphere

This means that for each additional atmosphere of pressure, the process takes 1 additional minute.

Key Point: Always pay attention to which variable is independent. The units of the derivative are always: $\text{Units of derivative}=\frac{\text{units of dependent variable}}{\text{units of independent variable}}$

Problem-Solving Strategy

General Approach

Key Interpretation Points

Positive derivative: The quantity is increasing

Negative derivative: The quantity is decreasing

Zero derivative: The quantity is momentarily constant

Large absolute value: Rapid rate of change

Small absolute value: Slow rate of change

Practice Section