To compute \( \Delta y \) and \( dy \), we first need to understand the function given:
\[ y = \frac{1}{x} \]
Given the values:
- \( x = 1 \)
- \( \Delta x = 1 \)
- Therefore, \( x + \Delta x = 1 + 1 = 2 \)
Step 1: Calculate \( \Delta y \)
The change in \( y \) (which is \( \Delta y \)) can be computed as follows:
\[ \Delta y = y(x + \Delta x) - y(x) \]
Calculating \( y(1) \):
\[ y(1) = \frac{1}{1} = 1 \]
Calculating \( y(2) \):
\[ y(2) = \frac{1}{2} = 0.5 \]
Now we can find \( \Delta y \):
\[ \Delta y = y(2) - y(1) = 0.5 - 1 = -0.5 \]
Step 2: Calculate \( dy \)
The differential \( dy \) can be computed using the derivative of \( y \):
- Find the derivative \( dy/dx \):
\[ \frac{dy}{dx} = -\frac{1}{x^2} \]
Evaluating the derivative at \( x = 1 \):
\[ \frac{dy}{dx} \bigg|_{x=1} = -\frac{1}{1^2} = -1 \]
- Now, calculate \( dy \):
\[ dy = \frac{dy}{dx} \cdot \Delta x \]
Substituting \( \Delta x = 1 \):
\[ dy = -1 \cdot 1 = -1 \]
Final Results:
Thus, we have:
\[ \Delta y = -0.500 \qquad \text{(rounded to three decimal places)} \] \[ dy = -1.000 \qquad \text{(rounded to three decimal places)} \]
So, the answers are:
\[ \Delta y = -0.500 \] \[ dy = -1.000 \]
1 answer