Problem 41 Use $d y$ to approximate \(\De... [FREE SOLUTION]

Chapter 3: Problem 41

Use $d y$ to approximate $\Delta y$ when $x$ changes as indicated. $$y=\sqrt{3 x-2} ; \text { from } x=2 \text { to } x=2.03$$

Short Answer

Expert verified

The approximate change $ \Delta y $ is 0.0225.

Step by step solution

Identify the Function and Derivative

We are given the function $ y = \sqrt{3x - 2} $. To approximate $ \Delta y $ using differentials, we first need to find the derivative of $ y $ with respect to $ x $. The derivative is determined by differentiating the function $ y $ using the chain rule. Therefore, $ \frac{dy}{dx} = \frac{1}{2\sqrt{3x - 2}} \cdot 3 = \frac{3}{2\sqrt{3x - 2}} $.

Set Up the Differential dy

The differential $ dy $ is given by $ dy = \frac{dy}{dx} \cdot dx $. With $ \frac{dy}{dx} = \frac{3}{2\sqrt{3x - 2}} $, we can express $ dy $ as $ dy = \frac{3}{2\sqrt{3x - 2}} \cdot \Delta x $, where $ \Delta x = 2.03 - 2 = 0.03 $.

Evaluate the Function and Derivative at x=2

First, calculate $ y $ at $ x = 2 $ by substituting $ x = 2 $ into the function: $ y = \sqrt{3(2) - 2} = \sqrt{4} = 2 $. Next, calculate the derivative $ \frac{dy}{dx} $ at $ x = 2 $: $ \frac{3}{2\sqrt{4}} = \frac{3}{4} $.

Calculate dy Using the Differential Formula

Substitute into the differential formula: $ dy = \frac{3}{4} \cdot 0.03 $. Simplifying gives $ dy = 0.0225 $. This approximation $ dy $ is used to estimate $ \Delta y $ when $ x $ changes from 2 to 2.03.

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Key Concepts

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

Chain Rule

The Chain Rule is a core concept in calculus, essential for differentiating functions that are composed of other functions. When you have a function inside another function, like in our case with $ y = \sqrt{3x - 2} $, the Chain Rule helps you find the derivative. It is like peeling layers of an onion. Here is how it works:

First, identify the outer and the inner functions. For $ y $, the outer function is $ \, \sqrt{u} \, $ where $ u = 3x - 2 $.
Next, differentiate the outer function with respect to the inner function, $ u $. For $ \, \sqrt{u} \, $, the derivative is $ \frac{1}{2\sqrt{u}} $.
Then, differentiate the inner function $ u = 3x - 2 $ with respect to $ x $, which is simply $ 3 $.
Multiply these derivatives together to get $ \frac{dy}{dx} = \frac{3}{2\sqrt{3x - 2}} $.

This rule simplifies the process of finding derivatives when functions have layers, and makes differential calculus much more manageable.

Derivative Calculation

In calculus, the calculation of a derivative is paramount as it represents the rate at which a function changes. To calculate the derivative of $ y = \sqrt{3x - 2} $, we utilized the Chain Rule. It's crucial to explicitly understand each step. Begin by identifying our function structure:

The function is a composition of the square root and a linear function.
Apply the Chain Rule to differentiate each part, leading to the calculation $ \frac{dy}{dx} = \frac{3}{2\sqrt{3x - 2}} $.

Derivatives like this one are useful for understanding not only the slope or gradient of a curve at any point but also to provide insight into how small changes in $ x $ will affect changes in $ y $. This understanding is further applied in differentials and approximations.

Differential Approximation

Differential approximations offer a simplified method to estimate the changes in a function value, using its derivative. For small changes in $ x $, the change in $ y $, or $ \Delta y $, can be approximated using the differential $ dy $. Here's how it's done:

Calculate $ \Delta x $, which is the change in $ x $. In this example, it is $ 2.03 - 2 = 0.03 $.
Find the derivative $ \frac{dy}{dx} $ at the initial $ x $ value, which is 2 in this case. We found $ \frac{3}{4} $.
Use the formula $ dy = \frac{dy}{dx} \cdot \Delta x $ to find $ dy $. Substituting our values, $ dy = \frac{3}{4} \cdot 0.03 = 0.0225 $.

While $ dy $ is an approximation of $ \Delta y $, it effectively provides a linear estimate, aiding in solving real-world problems where exact calculations are impossible to determine immediately.

Function Evaluation

Function evaluation involves substituting a variable with a specific value to determine the corresponding function output. This step is essential to apply differential approximation accurately. Let's break down the process for $ y = \sqrt{3x - 2} $:

First, substitute $ x = 2 $ into the function to find $ y $. You get $ y = \sqrt{4} = 2 $.
Now, evaluate the derivative $ \frac{dy}{dx} $ at the same $ x $ value to ensure it matches the function鈥檚 behavior as $ x $ changes. We computed it to be $ \frac{3}{4} $.

These evaluations give you precise starting points (initial $ y $ value) and measure how steep the function is at that point (derivative), both critical for approximations. Using this technique simplifies complex functions, making them easier to handle in practical scenarios.

91影视

Use \(d y\) to approximate \(\Delta y\) when \(x\) changes as indicated. $$y=\sqrt{3 x-2} ; \text { from } x=2 \text { to } x=2.03$$

Short Answer

Step by step solution

Identify the Function and Derivative

Set Up the Differential dy

Evaluate the Function and Derivative at x=2

Calculate dy Using the Differential Formula

Key Concepts

Chain Rule

Derivative Calculation

Differential Approximation

Function Evaluation

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