Question

Explain how implicit differentiation can simplify the work in a related-rates problem.

Short answer

In summary, implicit differentiation is a powerful tool for dealing with related-rates problems because it allows us to differentiate equations containing multiple variables without isolating the dependent variable first. This technique simplifies the differentiation process and can be especially helpful in problems where isolating the dependent variable might be difficult.

Step-by-step solution

Introduction to related-rates problems

Related-rates problems are a type of Calculus problem where we are given information about quantities that are changing with respect to time, and we are asked to find the rate of change of some other related quantity. These problems typically involve the application of differentiation when we have a function that relates the quantities.

Introduction to implicit differentiation

Implicit differentiation is a technique used to differentiate equations with respect to one variable, even though the equation contains two or more variables. This technique allows us to differentiate both sides of the given equation without needing to isolate the dependent variable, which can simplify the differentiation process.

Example problem

To make this clear, let's consider a related-rates problem and see how implicit differentiation simplifies the work. Suppose we have a circular oil spill, and the oil is leaking out at a constant rate. The radius of the circle expands at a rate of dr/dt = 1 cm/min. We would like to know how fast the area of the circular oil spill is increasing. In this problem, the quantities of interest are the radius (r) and the area (A) of the circular spill, and the relationship between them is given by the formula: A = πr^2 We need to find the rate of change of the area with respect to time, or dA/dt.

Application of implicit differentiation

Since we have an equation that relates A and r, we can use implicit differentiation to differentiate both sides of the equation with respect to time (t): d(A)/dt = d(πr^2)/dt Now we can apply the chain rule to the right side of the equation to find the rate of change of r with respect to time (dr/dt): d(A)/dt = 2πr(dr/dt) Since we are given that dr/dt = 1 cm/min, we can substitute this value into the equation, which gives: d(A)/dt = 2πr(1) So, d(A)/dt = 2πr. Thus, by using implicit differentiation, we have found an expression for the rate of change of the area with respect to time in terms of the radius r, which can be evaluated for any desired radius.

Conclusion

Implicit differentiation is a useful technique for simplifying the work in related-rates problems, as it allows us to differentiate equations that involve multiple variables without the need to isolate the dependent variable first. This can make the differentiation process more straightforward, especially in problems where isolating the dependent variable might be challenging or cumbersome.

Key concepts

Related-Rates Problems

In mathematics, related-rates problems are a fascinating type of problem you'll encounter in calculus. They revolve around finding the rate at which one quantity changes in relation to another, often with respect to time.
For instance, how does the area of a circular oil spill change as its radius increases over time? These problems employ calculus to investigate how rates change and relate to one another. When dealing with such problems, it's typical to:

• Identify the quantities involved.
• Establish a relationship among these quantities using an equation.
• Differentiate with respect to the variable of interest, such as time.

Understanding related-rates can open doors to solving real-life problems where various factors influence each other dynamically.

Differentiation Techniques

Differentiation is a fundamental concept in calculus, used to describe how a function changes as its input changes. There are several techniques to differentiate functions, each suited to different situations.
In problems like the oil spill example, we may not only differentiate a standard function but use techniques like implicit differentiation. Differentiation techniques include:

• Implicit Differentiation: Useful when functions aren’t easily solved for one variable in terms of another. It helps differentiate without isolating variables.
• Product Rule: Used for functions that are the product of two differentiable functions.
• Quotient Rule: Used for functions that are the ratio of two differentiable functions.

Applying the right technique allows for smoother analysis and solution of calculus problems.

Chain Rule

The chain rule is a pivotal differentiation technique used in calculus for finding the derivative of composite functions.
It's the magic behind implicit differentiation, enabling us to differentiate functions nested within other functions.To apply the chain rule, you:

• Differentiate the outer function while keeping the inner function as it is.
• Multiply by the derivative of the inner function.

In our implicit differentiation example, for the oil spill area problem, the chain rule helped us differentiate the area of the circle (\(A = \pi r^2\)) as:\[d(A)/dt = 2\pi r (dr/dt).\]This application shows how the change in radius directly influences the change in area, underscoring the powerful simplicity of the chain rule.

Calculus

Calculus is an exciting area of mathematics that enables the exploration of concepts like change and motion. Central to calculus are differentiation and integration, which help in understanding and describing real-world phenomena.
Many processes in the natural and applied sciences, from the spread of diseases to the trajectory of rockets, are governed by calculus.
Its tools and methods allow us to:

• Determine instantaneous rates of change.
• Analyze the behavior and limits of functions.
• Optimize processes, revealing maxima or minima values.

In essence, calculus provides a powerful framework for solving problems involving dynamic changes, just as in the related-rates problems analyzed using implicit differentiation.