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Chapter 4: Problem 4

Write the indicated related-rates equation. $$ y=9 x^{3}+12 x^{2}+4 x+3 ; \text { relate } \frac{d y}{d t} \text { and } \frac{d x}{d t} $$

Short Answer

Expert verified
\( \frac{dy}{dt} = (27x^2 + 24x + 4) \frac{dx}{dt} \).

Step by step solution

01

Differentiate Both Sides with Respect to Time

To find the relationship between \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \), we need to differentiate the equation \( y = 9x^3 + 12x^2 + 4x + 3 \) with respect to time \( t \). Note that this requires the use of the chain rule. Differentiate each term separately.
02

Apply the Chain Rule to Each Term

For the term \( 9x^3 \), use the chain rule to get \( 27x^2 \cdot \frac{dx}{dt} \).For \( 12x^2 \), differentiate to get \( 24x \cdot \frac{dx}{dt} \).For \( 4x \), it becomes \( 4 \cdot \frac{dx}{dt} \).The constant \( 3 \) differentiates to \( 0 \).
03

Write the Full Derivative Equation

Combine the results from Step 2 to form the related-rates equation: \( \frac{dy}{dt} = 27x^2 \frac{dx}{dt} + 24x \frac{dx}{dt} + 4 \frac{dx}{dt} \).
04

Simplify the Equation

Since each term on the right side includes \( \frac{dx}{dt} \), we can factor it out:\( \frac{dy}{dt} = (27x^2 + 24x + 4) \frac{dx}{dt} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is often seen as one of the cornerstones of modern mathematics. It helps us understand changes and motion. When we talk about a related-rates problem, we're looking to find out how one rate of change affects another. This often happens with functions that describe changing quantities. In this case, we want to find out how the rate of change of \( y \) is related to the rate of change of \( x \).
Calculus gives us the tools to explore these relationships.
  • It helps us model real-world situations where quantities change over time.
  • We can use calculus to make predictions about future changes.
  • The fundamental concepts include limits, derivatives, and integrals.
These are essential for understanding how related rates operate.
Differentiation
Differentiation is the process of finding a derivative, which measures how a function changes as its input changes. In the context of our problem, differentiation is used to observe how \( y \) changes with respect to \( t \), given that \( y \) is defined in terms of \( x \).
When differentiating with respect to time, we consider the rate of change of both \( y \) and \( x \). This is crucial in related-rate problems because each function component's change affects the whole equation. Differentiation has several key uses:
  • Calculating the slope of a curve at a given point.
  • Understanding how small changes in one variable affect another variable.
  • Solving problems involving motion, areas, volumes, and other dynamic systems.
Chain Rule
The chain rule is a powerful tool in differentiation, especially when dealing with composite functions. For related rates, it allows us to differentiate a function of a function. Here, we apply the chain rule to each term of \( y = 9x^3 + 12x^2 + 4x + 3 \) because \( y \) changes with \( x \), which in turn changes with time \( t \).
The chain rule basically states:
\[ \text{If } \; z = f(g(x)), \; \text{then} \; \frac{dz}{dx} = f'(g(x)) \cdot g'(x). \]
This rule is instrumental in breaking down the differentiation into manageable parts.
  • It helps find the derivative of a function that's inside another function.
  • It simplifies complex differentiation into simpler steps.
  • It's crucial for finding the rate of change when variables are interdependent.
Derivative
A derivative represents the rate at which one quantity changes with respect to another. In our equation, \( \frac{dy}{dt} \) represents how \( y \) changes over time, while \( \frac{dx}{dt} \) represents how \( x \) changes over time.
Derivatives are fundamental in calculus for analyzing functions and are used to:
  • Determine the instantaneous rate of change of one variable concerning another.
  • Find points on a curve where the slope is zero (critical points).
  • Help in optimizing functions, finding maxima, and minima.
In related rates, deriving with respect to time allows us to see the dynamic relationship between interconnected quantities.

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Most popular questions from this chapter

Generic Profit If the marginal profit is negative for the sale of a certain number of units of a product, is the company that is marketing the item losing money on the sale? Explain.

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