Introduction to Related Rates

Related rates is a crucial concept in Calculus I that involves determining the rate at which one quantity changes with respect to another quantity. This concept has numerous real-world applications, such as calculating the rate of change of volume, area, or distance in various scenarios.

Common Terms and Definitions

Rate of Change: The speed at which a quantity changes with respect to another quantity, often with respect to time.

Implicit Differentiation: A technique used to find the derivative of an equation that is not explicitly solved for one variable in terms of the other.

Chain Rule: A rule for finding the derivative of a composite function, which states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.

Optimization: The process of finding the maximum or minimum value of a function subject to certain constraints.

Problem-Solving Strategies for Related Rates

1. Read the problem carefully and identify the given information and the quantity to be determined.
2. Draw a diagram (if applicable) to visualize the problem and the relationships between the quantities involved.
3. Write an equation that relates the quantities in the problem, using variables to represent the unknown values.
4. Differentiate both sides of the equation with respect to time (or the appropriate variable) using implicit differentiation and the chain rule.
5. Substitute the known values and solve for the desired rate of change.
6. Interpret the result in the context of the problem and provide units when appropriate.

Common Applications of Related Rates

Volume Problems: Calculating the rate of change of the volume of a container as its dimensions change over time.

Distance Problems: Determining the rate at which the distance between two moving objects changes.

Optimization Problems: Finding the optimal dimensions or values that maximize or minimize a particular quantity, such as area or volume, subject to given constraints.

Trigonometric Problems: Solving related rates problems involving trigonometric functions, such as the height of a ladder leaning against a wall or the angle of elevation between an observer and a moving object.

Example Problem and Solution

A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 foot per second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?

Solution:

1. Let x be the distance between the bottom of the ladder and the wall, and let y be the height of the top of the ladder.
2. Using the Pythagorean theorem, we can write the equation: x^2 + y^2 = 10^2
3. Differentiate both sides of the equation with respect to time: 2x(dx/dt) + 2y(dy/dt) = 0
4. We know that dx/dt = 1 ft/s and x = 6 ft when we want to find dy/dt. Substitute these values into the differentiated equation: 2(6)(1) + 2y(dy/dt) = 0
5. Solve for dy/dt: dy/dt = -12 / (2y)
6. To find y when x = 6, use the Pythagorean theorem: 6^2 + y^2 = 10^2, so y = 8 ft
7. Substitute y = 8 into the expression for dy/dt: dy/dt = -12 / (2(8)) = -0.75 ft/s

Therefore, the top of the ladder is sliding down the wall at a rate of 0.75 feet per second when the bottom of the ladder is 6 feet from the wall.

Common Questions and Answers

What is the main idea behind related rates problems?

Related rates problems involve determining the rate at which one quantity changes with respect to another quantity, often using implicit differentiation and the chain rule.

How do you set up a related rates problem?

To set up a related rates problem, identify the given information and the quantity to be determined, draw a diagram (if applicable), write an equation relating the quantities, and differentiate both sides of the equation with respect to the appropriate variable.

What are some common applications of related rates in real life?

Related rates have numerous real-world applications, such as calculating the rate of change of volume, area, or distance in various scenarios, as well as solving optimization problems subject to given constraints.

Conclusion

Related rates is an essential concept in Calculus I that enables students to solve problems involving the rates of change of related quantities. By understanding the problem-solving strategies, common applications, and practicing with example problems, students can develop the skills necessary to tackle related rates problems with confidence.