Answer :
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]
