To tackle the problem, we need to use the concept of linear approximation to estimate [tex]\( \Delta y \)[/tex] when [tex]\( y = \sin(3x) \)[/tex] and [tex]\( \Delta x = 0.3 \)[/tex] at [tex]\( x = 0 \)[/tex].
Here are the detailed steps to solve this problem:
1. Expression for [tex]\( y \)[/tex] and its derivative:
[tex]\[ y = \sin(3x) \][/tex]
To use linear approximation, we need the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{d}{dx} [\sin(3x)] = 3 \cos(3x) \][/tex]
2. Initial values:
[tex]\[ x_{\text{initial}} = 0 \][/tex]
[tex]\[ \Delta x = 0.3 \][/tex]
3. Linear approximation formula:
The linear approximation formula around [tex]\( x = x_{\text{initial}} \)[/tex] is given by:
[tex]\[ \Delta y \approx \left. \frac{dy}{dx} \right|_{x=x_{\text{initial}}} \Delta x \][/tex]
4. Evaluate the derivative at [tex]\( x = 0 \)[/tex]:
[tex]\[ \left. \frac{dy}{dx} \right|_{x=0} = 3 \cos(3 \cdot 0) = 3 \cos(0) = 3 \cdot 1 = 3 \][/tex]
5. Estimate [tex]\( \Delta y \)[/tex] using the linear approximation:
[tex]\[ \Delta y \approx 3 \cdot \Delta x = 3 \cdot 0.3 = 0.9 \][/tex]
Therefore, the linear approximation estimate for [tex]\( \Delta y \)[/tex] is:
[tex]\[ \Delta y \approx 0.9 \][/tex]
6. Calculate the exact change in [tex]\( y \)[/tex]:
We need to find the exact value of [tex]\( \Delta y \)[/tex]:
[tex]\[ \Delta y_{\text{exact}} = y(x_{\text{initial}} + \Delta x) - y(x_{\text{initial}}) \][/tex]
Here, [tex]\( x_{\text{initial}} = 0 \)[/tex] and [tex]\( \Delta x = 0.3 \)[/tex], so:
[tex]\[ y(0.3) = \sin(3 \cdot 0.3) = \sin(0.9) \][/tex]
[tex]\[ y(0) = \sin(3 \cdot 0) = \sin(0) = 0 \][/tex]
Hence,
[tex]\[ \Delta y_{\text{exact}} = \sin(0.9) - \sin(0) = \sin(0.9) \][/tex]
7. Plugging in the value:
Using the computed numerical result (which we treat as an exact calculation):
[tex]\[ \Delta y_{\text{exact}} \approx 0.776772\][/tex]
8. Calculate the percentage error:
The percentage error is calculated as:
[tex]\[ \text{Percentage error} = \left| \frac{\Delta y_{\text{exact}} - \Delta y_{\text{approx}}}{\Delta y_{\text{exact}}} \right| \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{Percentage error} = \left| \frac{0.776772 - 0.9}{0.776772} \right| \times 100 \approx 14.89\% \][/tex]
Thus, the final results are:
[tex]\[ \Delta y \approx 0.9 \][/tex]
[tex]\[ \text{Percentage error} \approx 14.89\% \][/tex]
