4.4 Introduction to Related Rates

• What are related rates used for in calculus?

  Finding rates of change
• The equation $\frac{dV}{dt} =$$4 \pi r^{2} \frac{dr}{dt}$ relates the rate of change of the volume of a sphere to the rate of change of its radius.
• When tackling related rates problems, identifying the given rates and desired rates is crucial
• In the ladder problem, what is the given rate?

  \frac{dx}{dt} = 2 ft / s</latex>
• Related rates problems involve understanding how mathematical relationships between variables are used to determine their rates of change.
• If the radius of a sphere is increasing at a rate of $2 cm / s$, the rate of change of the volume can be found using $\frac{dV}{dt} =$$4 \pi r^{2} \frac{dr}{dt}$ at a given radius
• What is the given rate in the cylindrical tank problem?

  $\frac{dV}{dt} =$$5 cm^{3} / s$
• Match the quantity with its corresponding rate and units:

  Volume of water ($V$) ↔️ $\frac{dV}{dt} =$$5 cm^{3} / s$

  Height of water ($h$) ↔️ $\frac{dh}{dt}$
• In related rates problems, relating variables with an equation is crucial for finding the rate of change of one quantity based on the rates of other related quantities.
• What is the equation for the volume of a sphere?

  V = \frac{4}{3} \pi r^{3}</latex>
• In related rates problems, differentiating an equation with respect to time is used to connect the rates of change of different quantities.
• Steps for differentiating an equation with respect to time in related rates problems:

  1️⃣ Identify the equation

  2️⃣ Differentiate each term with respect to $t$

  3️⃣ Simplify the resulting equation
• When differentiating with respect to time, the chain rule must be used for terms involving functions of $t$.
• In related rates problems, we differentiate equations with respect to time to connect the rates of change of different quantities.
• Steps to solve related rates problems

  1️⃣ Identify the equation

  2️⃣ Differentiate each term with respect to $t$

  3️⃣ Simplify the resulting equation
• What is $\frac{dV}{dt}$ in terms of $\frac{dr}{dt}$ for the equation $V =$$\frac{4}{3} \pi r^{3}$?


  $\frac{dV}{dt} =$$4 \pi r^{2} \frac{dr}{dt}$
• Related rates problems always involve the chain rule and implicit differentiation.
• Match the variable with its rate of change in related rates problems:

  Volume ↔️ $\frac{dV}{dt}$

  Radius ↔️ $\frac{dr}{dt}$

  Height ↔️ $\frac{dh}{dt}$
• The equation \frac{dV}{dt} = 4\pi r^{2} \frac{dr}{dt}</latex> links the rates of change of volume and radius
• Steps to identify given and desired rates in related rates problems

  1️⃣ Read the problem carefully

  2️⃣ List known rates with their units

  3️⃣ Identify the desired rate
• Given rates in related rates problems are the rates of change you need to find.

  False
• In related rates problems, it is essential to find an equation that connects the variables
• Match the quantity with its rate of change and units for a cylindrical tank problem:

  Volume of water ↔️ $\frac{dV}{dt} =$$5 \, cm^{3} / s$

  Height of water ↔️ $\frac{dh}{dt} \, cm / s$

  Radius of the tank ↔️ $r =$$3 \, cm$
• Steps to differentiate an equation with respect to time in related rates problems

  1️⃣ Identify the equation

  2️⃣ Differentiate each term using the chain rule

  3️⃣ Simplify the resulting equation
• The chain rule is used in related rates problems when differentiating composite functions with respect to time.
• Steps to solve related rates problems

  1️⃣ Identify the equation

  2️⃣ Differentiate each term

  3️⃣ Simplify