1. Identify the function and its derivative:
- The function is \( f(x) = \frac{1}{x} \).
- The derivative of \( f(x) \) is \( f'(x) = -\frac{1}{x^2} \).
2. Find the slope of the tangent line at \( x = 2 \):
- Evaluate \( f'(2) = -\frac{1}{2^2} = -\frac{1}{4} \).
3. Determine the point on the curve at \( x = 2 \):
- \( f(2) = \frac{1}{2} \), so the point is \( (2, \frac{1}{2}) \).
4. Use the point-slope form of the line equation:
- The point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point.
- Substitute \( m = -\frac{1}{4} \), \( x_1 = 2 \), and \( y_1 = \frac{1}{2} \):
\[
y - \frac{1}{2} = -\frac{1}{4}(x - 2)
\]
5. Simplify the equation:
- Distribute \( -\frac{1}{4} \):
\[
y - \frac{1}{2} = -\frac{1}{4}x + \frac{1}{2}
\]
- Add \( \frac{1}{2} \) to both sides:
\[
y = -\frac{1}{4}x + 1
\]
6. Final answer:
- The equation of the tangent line is \( y = -\frac{1}{4}x + 1 \).
Parent Tip: Review the logic above to help your child master the concept of projectile motion worksheet with answers.
