4.4 Introduction to Related Rates

• Related rates describe the relationship between two or more changing quantities
• Differentiating an equation with respect to time is a key step in solving related rates problems.


  True
• Related quantities depend on the rates of change of each other.


  True
• Writing an equation relating the quantities is a necessary step in solving related rates problems.

  True
• What is the volume formula for a sphere?

  $V =$$\frac{4}{3} \pi r^{3}$
• Steps to solve a related rates problem

  1️⃣ Identify given information and unknown quantity

  2️⃣ Draw a diagram to visualize the problem

  3️⃣ Write an equation relating the quantities

  4️⃣ Differentiate the equation with respect to time

  5️⃣ Substitute known rates to solve for the unknown
• Steps to solve a related rates problem

  1️⃣ Identify given information and unknown quantity

  2️⃣ Draw a diagram to visualize the problem

  3️⃣ Write an equation relating the quantities

  4️⃣ Differentiate the equation with respect to time

  5️⃣ Substitute known rates to solve for the unknown
• To identify related quantities, it is important to recognize that the rate of change of one depends on the rates of change of the others
• What is an example of related quantities in a growing sphere?

  Radius, surface area, volume
• Steps to set up equations involving related quantities

  1️⃣ Write an equation relating the quantities

  2️⃣ Differentiate the equation with respect to time

  3️⃣ Substitute the known rates of change
• After differentiating an equation with respect to time, the next step is to substitute the known rates of change and solve for the unknown
• Steps to set up related rates equations

  1️⃣ Write an equation relating the quantities

  2️⃣ Differentiate the equation with respect to time

  3️⃣ Substitute the known rates of change
• To differentiate equations with respect to time, the chain rule must be applied to each term
• What is the fundamental principle of related rates?

  Change in one affects others
• Related rates refer to the relationship between two or more changing quantities
• The rate of change of one quantity in a related rates problem depends on the rates of change of other quantities.


  True
• When solving a related rates problem, a diagram helps visualize the problem and identify the relevant variables
• What is the second step in setting up equations for related quantities?

  Write an equation
• What is the derivative of the area of a circle with respect to time?

  $\frac{dA}{dt} =$$2\pi r \frac{dr}{dt}$
• The derivative of the volume of a sphere with respect to time is $\frac{dV}{dt} =$$4\pi r^{2} \frac{dr}{dt}$
• To solve for the unknown rate, known rates of change are substituted into the related rates equation
• Steps to solve a related rates problem

  1️⃣ Identify given information and unknown quantity

  2️⃣ Draw a diagram

  3️⃣ Write an equation

  4️⃣ Differentiate the equation

  5️⃣ Substitute known rates and solve
• After differentiating, you substitute known rates to solve for the unknown rate
• In a growing sphere scenario, the radius, surface area, and volume are all related quantities
• To solve a related rates problem, the equation must be differentiated with respect to time
• To solve a related rates problem, the first step is to identify the given information and the unknown quantity
• The second step in solving a related rates problem is to draw a diagram
• What is the first step to identify related quantities in a given scenario?

  Identify changing quantities
• Related quantities are two or more changing quantities where the rate of change of one depends on the rates of change of the others

  True
• To set up an equation involving related quantities, the first step is to write an equation that relates the given and unknown quantities
• What is the purpose of differentiating an equation with respect to time in related rates problems?

  Express rates of change
• Writing an equation relating the quantities is the first step in setting up related rates equations


  True
• What is the related rates equation obtained by differentiating $V =$$\frac{4}{3}\pi r^{3}$ with respect to time?


  $\frac{dV}{dt} =$$4\pi r^{2} \frac{dr}{dt}$
• When differentiating an equation with respect to time, the chain rule is always necessary


  True
• Steps to solve a related rates problem

  1️⃣ Identify given information and unknown quantity

  2️⃣ Draw a diagram to visualize the problem

  3️⃣ Write an equation relating the quantities

  4️⃣ Differentiate the equation with respect to time

  5️⃣ Substitute known rates to solve for the unknown
• Steps to solve a related rates problem

  1️⃣ Identify the given information and the unknown quantity

  2️⃣ Draw a diagram to visualize the problem

  3️⃣ Write an equation that relates the quantities

  4️⃣ Differentiate the equation with respect to time

  5️⃣ Substitute the known rates of change to solve for the unknown
• What is the first step in solving a related rates problem?

  Identify the given information
• If a balloon is inflated, the volume, radius, and surface area are related quantities.


  True
• Differentiating the equation in a related rates problem is done with respect to time
• The chain rule is essential for differentiating equations in related rates problems.


  True