Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve this problem, we will use the concepts of linear approximation and percentage error.
### Step 1: Define the function and compute its derivative
The function given is [tex]\( y = \sin(4x) \)[/tex]. To use linear approximation, we need to compute the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
The derivative of [tex]\( y = \sin(4x) \)[/tex] is:
[tex]\[ \frac{dy}{dx} = 4 \cos(4x) \][/tex]
### Step 2: Evaluate the derivative at [tex]\( x = 0 \)[/tex]
Now, let's evaluate the derivative [tex]\( \frac{dy}{dx} \)[/tex] at the point [tex]\( x = 0 \)[/tex]:
[tex]\[ \left. \frac{dy}{dx} \right|_{x=0} = 4 \cos(4 \cdot 0) = 4 \cos(0) = 4 \cdot 1 = 4 \][/tex]
### Step 3: Use linear approximation to estimate [tex]\( \Delta y \)[/tex]
With the derivative computed, we can use linear approximation to estimate [tex]\( \Delta y \)[/tex]. The formula for the linear approximation is:
[tex]\[ \Delta y \approx \frac{dy}{dx} \Delta x \][/tex]
Given [tex]\( \Delta x = 0.4 \)[/tex]:
[tex]\[ \Delta y \approx 4 \cdot 0.4 = 1.6 \][/tex]
So, the estimated change in [tex]\( y \)[/tex] using linear approximation is:
[tex]\[ \Delta y \approx 1.6 \][/tex]
### Step 4: Calculate the actual change in [tex]\( y \)[/tex] to find the true [tex]\( \Delta y \)[/tex]
To compute the true [tex]\( \Delta y \)[/tex], we need the actual values of [tex]\( y \)[/tex] at [tex]\( x = 0 \)[/tex] and [tex]\( x = 0.4 \)[/tex].
1. Calculate [tex]\( y \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ y(0) = \sin(4 \cdot 0) = \sin(0) = 0 \][/tex]
2. Calculate [tex]\( y \)[/tex] at [tex]\( x = 0.4 \)[/tex]:
[tex]\[ y(0.4) = \sin(4 \cdot 0.4) = \sin(1.6) \][/tex]
Evaluating [tex]\( \sin(1.6) \)[/tex] gives approximately:
[tex]\[ y(0.4) \approx 0.9995736030415051 \][/tex]
The true change in [tex]\( y \)[/tex] is:
[tex]\[ \Delta y_{\text{true}} = y(0.4) - y(0) \][/tex]
[tex]\[ \Delta y_{\text{true}} = 0.9995736030415051 - 0 = 0.9995736030415051 \][/tex]
### Step 5: Calculate the percentage error
To find the percentage error between the estimated [tex]\( \Delta y \)[/tex] and the true [tex]\( \Delta y \)[/tex], we use the formula:
[tex]\[ \text{Percentage error} = \left| \frac{\Delta y_{\text{estimated}} - \Delta y_{\text{true}}}{\Delta y_{\text{true}}} \right| \times 100\% \][/tex]
Substitute the values:
[tex]\[ \text{Percentage error} = \left| \frac{1.6 - 0.9995736030415051}{0.9995736030415051} \right| \times 100\% \][/tex]
[tex]\[ \text{Percentage error} \approx 60.0682526160671\% \][/tex]
So, the solutions to the problem are:
[tex]\[ \Delta y \approx 1.6 \][/tex]
[tex]\[ \text{Percentage error} \approx 60.07\% \][/tex]
### Step 1: Define the function and compute its derivative
The function given is [tex]\( y = \sin(4x) \)[/tex]. To use linear approximation, we need to compute the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
The derivative of [tex]\( y = \sin(4x) \)[/tex] is:
[tex]\[ \frac{dy}{dx} = 4 \cos(4x) \][/tex]
### Step 2: Evaluate the derivative at [tex]\( x = 0 \)[/tex]
Now, let's evaluate the derivative [tex]\( \frac{dy}{dx} \)[/tex] at the point [tex]\( x = 0 \)[/tex]:
[tex]\[ \left. \frac{dy}{dx} \right|_{x=0} = 4 \cos(4 \cdot 0) = 4 \cos(0) = 4 \cdot 1 = 4 \][/tex]
### Step 3: Use linear approximation to estimate [tex]\( \Delta y \)[/tex]
With the derivative computed, we can use linear approximation to estimate [tex]\( \Delta y \)[/tex]. The formula for the linear approximation is:
[tex]\[ \Delta y \approx \frac{dy}{dx} \Delta x \][/tex]
Given [tex]\( \Delta x = 0.4 \)[/tex]:
[tex]\[ \Delta y \approx 4 \cdot 0.4 = 1.6 \][/tex]
So, the estimated change in [tex]\( y \)[/tex] using linear approximation is:
[tex]\[ \Delta y \approx 1.6 \][/tex]
### Step 4: Calculate the actual change in [tex]\( y \)[/tex] to find the true [tex]\( \Delta y \)[/tex]
To compute the true [tex]\( \Delta y \)[/tex], we need the actual values of [tex]\( y \)[/tex] at [tex]\( x = 0 \)[/tex] and [tex]\( x = 0.4 \)[/tex].
1. Calculate [tex]\( y \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ y(0) = \sin(4 \cdot 0) = \sin(0) = 0 \][/tex]
2. Calculate [tex]\( y \)[/tex] at [tex]\( x = 0.4 \)[/tex]:
[tex]\[ y(0.4) = \sin(4 \cdot 0.4) = \sin(1.6) \][/tex]
Evaluating [tex]\( \sin(1.6) \)[/tex] gives approximately:
[tex]\[ y(0.4) \approx 0.9995736030415051 \][/tex]
The true change in [tex]\( y \)[/tex] is:
[tex]\[ \Delta y_{\text{true}} = y(0.4) - y(0) \][/tex]
[tex]\[ \Delta y_{\text{true}} = 0.9995736030415051 - 0 = 0.9995736030415051 \][/tex]
### Step 5: Calculate the percentage error
To find the percentage error between the estimated [tex]\( \Delta y \)[/tex] and the true [tex]\( \Delta y \)[/tex], we use the formula:
[tex]\[ \text{Percentage error} = \left| \frac{\Delta y_{\text{estimated}} - \Delta y_{\text{true}}}{\Delta y_{\text{true}}} \right| \times 100\% \][/tex]
Substitute the values:
[tex]\[ \text{Percentage error} = \left| \frac{1.6 - 0.9995736030415051}{0.9995736030415051} \right| \times 100\% \][/tex]
[tex]\[ \text{Percentage error} \approx 60.0682526160671\% \][/tex]
So, the solutions to the problem are:
[tex]\[ \Delta y \approx 1.6 \][/tex]
[tex]\[ \text{Percentage error} \approx 60.07\% \][/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.