4.3 Rates of Change in Applied Contexts Other Than Motion

• The derivative $dy / dx$ gives the instantaneous rate of change of $y$ with respect to $x$, which is a measure of how quantity changes over time or with respect to another variable.
• What does the rate of change of profit with respect to production quantity represent in economics?

  Marginal profit
• In economics, if $Profit =$$P(x)$, then the marginal profit is represented by the derivative $dP / dx$.
• What does the rate of change of temperature with respect to time measure in physics?

  Cooling/Heating rate
• The derivative $dy / dx$ provides the instantaneous rate of change of $y$ with respect to $x$, indicating how a quantity changes with respect to another variable.
• What does the rate of change of population size with respect to time represent in biology?

  Population growth rate
• Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function.
• What is the derivative of $f(x) =$$x^{3}$ using the power rule?


  $3x^{2}$
• The chain rule is used when differentiating a composite function.
• What does the derivative $dy / dx$ measure in general?


  Instantaneous rate of change
• Match the applied context with its rate of change example:

  Physics ↔️ Rate of cooling/heating

  Biology ↔️ Population growth rate

  Economics ↔️ Marginal profit
• Steps to solve a rate of change problem in applied contexts:

  1️⃣ Identify the variables and their notations

  2️⃣ Set up the equation involving rates of change

  3️⃣ Differentiate the equation

  4️⃣ Substitute given values

  5️⃣ Solve for the unknown rate of change
• What is the marginal profit in economics defined as?

  Rate of change of profit
• What does the rate of change measure?

  Change of quantity
• The derivative $dy / dx$ gives the instantaneous rate of change of $y$ with respect to x</latex> over time
• In economics, the marginal profit is the rate of change of profit with respect to production quantity
• In physics, what does the rate of change of temperature represent?

  Cooling/heating rate
• In biology, the population growth rate is an example of a rate of change
• What is the process of finding the derivative of a function called?

  Differentiation
• Match the differentiation rule with its formula:

  Power Rule ↔️ $f(x) =$$x^{n} \implies f'(x) =$$nx^{n - 1}$

  Product Rule ↔️ $f(x) =$$u(x)v(x) \implies f'(x) =$$u'(x)v(x) +$$u(x)v'(x)$

  Quotient Rule ↔️ $f(x) =$$\frac{u(x)}{v(x)} \implies f'(x) =$$\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^{2}}$

  Chain Rule ↔️ $f(x) =$$g(h(x)) \implies f'(x) =$$g'(h(x)) \cdot h'(x)$
• In economics, what does $C(x)$ represent?


  Total cost
• In physics, $T(t)$ denotes the temperature as a function of time
• In biology, $N(t)$ represents the number of individuals in a population at time $t$.
• Match the context with its equation and key variables:

  Economics ↔️ $P =$$R - C$, Profit, Revenue, Cost

  Physics ↔️ $T(t) =$$T_{0}e^{ - kt}$, Temperature, Time, Initial temperature

  Biology ↔️ $N(t) =$$N_{0}e^{rt}$, Population, Time, Initial population
• Steps for solving rate of change problems:

  1️⃣ Identify variables and relationships

  2️⃣ Set up the equation

  3️⃣ Differentiate the equation

  4️⃣ Solve for the rate of change

  5️⃣ Interpret the result
• What is the equation relating the area of a circle to its radius?

  $A =$$\pi r^{2}$
• Differentiating the equation $A =$$\pi r^{2}$ with respect to $r$ gives $\frac{dA}{dr} =$$2\pi$r
• The area of a circle is increasing at a rate of $10\pi$ cm²/cm when the radius is 5 cm.