Join IDNLearn.com today and start getting the answers you've been searching for. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.
