∬Differential Calculus Unit 11 – Related Rates

Key Concepts

• Related rates problems involve two or more quantities that are changing with respect to time
• The rates of change of these quantities are related to each other through an equation or formula
• To solve related rates problems, differentiate the equation with respect to time and substitute known values
• The chain rule is often used to differentiate composite functions in related rates problems
• Implicit differentiation is employed when the dependent variable is not explicitly expressed as a function of the independent variable
• Related rates problems often involve geometric relationships (such as the Pythagorean theorem or the volume of a sphere)
• Units and unit conversions play a crucial role in setting up and solving related rates problems

Fundamental Principles

• Identify the quantities that are changing with respect to time and assign variables to them
• Determine the relationship between the changing quantities and express it as an equation
• Differentiate both sides of the equation with respect to time, applying the chain rule and implicit differentiation as needed
• Substitute known values (including the rates of change) into the differentiated equation
• Solve the resulting equation for the desired rate of change
• Interpret the result in the context of the problem and ensure that the units are consistent
• Sketch a diagram to visualize the problem and identify relevant relationships between quantities

Common Types of Related Rates Problems

• Problems involving the Pythagorean theorem (such as a ladder sliding down a wall or a boat sailing away from a lighthouse)
  • Example: A ladder is leaning against a wall, and the bottom of the ladder is sliding away from the wall at a constant rate. Find the rate at which the top of the ladder is sliding down the wall.
• Problems involving the volume or surface area of a shape that is changing over time (such as a expanding balloon or a filling water tank)
  • Example: A spherical balloon is being inflated with air. Given the rate at which the radius is increasing, find the rate at which the volume of the balloon is changing.
• Problems involving trigonometric functions (such as an angle of elevation or depression changing over time)
• Problems involving the distance between two moving objects (such as two cars traveling on perpendicular roads)
• Problems involving the area of a shape that is changing over time (such as a growing circle or a expanding rectangle)
• Problems involving the rate of change of a population or a substance (such as bacteria growth or radioactive decay)

Problem-Solving Strategies

• Read the problem carefully and identify the given information, the quantities that are changing, and the desired rate of change
• Assign variables to the changing quantities and label them in a diagram if applicable
• Write an equation that relates the changing quantities using the given information and relevant formulas
• Differentiate both sides of the equation with respect to time, applying the chain rule and implicit differentiation as needed
  • Remember to treat constants as constants and variables as variables during differentiation
• Substitute known values (including the rates of change) into the differentiated equation
  • Be careful with units and ensure that all rates are expressed in consistent units
• Solve the resulting equation for the desired rate of change
• Evaluate the result at the given time or value and interpret it in the context of the problem
• Check the reasonableness of the answer and verify that the units are consistent

Real-World Applications

• Optimization problems in engineering and design (such as minimizing the cost of materials or maximizing the efficiency of a process)
• Modeling the spread of diseases or the growth of populations in biology and epidemiology
• Analyzing the motion of objects in physics and mechanics (such as projectile motion or fluid dynamics)
• Calculating the rate of change of economic variables (such as inflation rates or GDP growth rates) in economics and finance
• Determining the rate of chemical reactions or the decay of radioactive substances in chemistry and nuclear physics
• Estimating the rate of change of environmental variables (such as temperature or sea level) in climate science and ecology
• Modeling the flow of traffic or the congestion of networks in transportation and communication systems

Common Mistakes and Pitfalls

• Failing to identify all the relevant changing quantities and their relationships
• Confusing the rates of change with the actual values of the quantities
• Forgetting to apply the chain rule when differentiating composite functions
• Neglecting to use implicit differentiation when the dependent variable is not explicitly expressed
• Making errors in algebraic manipulation or simplification during the solution process
• Using inconsistent units or forgetting to convert units when necessary
• Misinterpreting the meaning of the result or failing to consider the context of the problem
• Not checking the reasonableness of the answer or verifying that it makes sense in the given situation

Practice Problems and Solutions

• A circular oil slick is expanding on the surface of a lake. If the radius of the slick is increasing at a rate of 2 meters per minute, find the rate at which the area of the slick is increasing when the radius is 10 meters.
  • Solution: Let $r$ be the radius and $A$ be the area of the circular oil slick. The relationship between the area and radius of a circle is given by $A = \pi r^2$. Differentiating both sides with respect to time $t$, we get $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. We are given that $\frac{dr}{dt} = 2$ m/min and $r = 10$ m. Substituting these values, we find that $\frac{dA}{dt} = 2\pi (10)(2) = 40\pi$ m²/min. Therefore, the area of the oil slick is increasing at a rate of $40\pi$ square meters per minute when the radius is 10 meters.
• A 13-foot ladder is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of 2 feet per second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the wall?
• Water is being pumped into a conical tank at a rate of 10 cubic feet per minute. The tank has a height of 10 feet and a base radius of 5 feet. How fast is the water level rising when the water depth is 6 feet?
• Two cars start traveling from the same point. One car travels north at 60 miles per hour, and the other car travels east at 80 miles per hour. At what rate is the distance between the cars increasing after 2 hours?

Connections to Other Calculus Topics

• Derivatives: Related rates problems heavily rely on the concept of derivatives and the ability to differentiate functions with respect to time.
• Implicit differentiation: Many related rates problems involve equations where the dependent variable is not explicitly expressed as a function of the independent variable, requiring the use of implicit differentiation.
• Chain rule: The chain rule is frequently used in related rates problems to differentiate composite functions, especially when multiple variables are changing with respect to time.
• Optimization: Related rates problems are often connected to optimization problems, as both involve finding the maximum or minimum values of a function or the rate of change of a quantity.
• Integration: While related rates problems primarily focus on differentiation, the concepts learned can be applied to solve problems involving integration, such as finding the total change in a quantity over time.
• Differential equations: Related rates problems can be seen as a type of differential equation, where the rate of change of a quantity is expressed in terms of other variables and their rates of change.
• Multivariable calculus: The ideas and techniques used in related rates problems can be extended to solve problems involving functions of several variables and partial derivatives in multivariable calculus.