Find the linear approximation. L(x) for the function f(x)=1/x at a=2. Solve by finding the equation of tangent line. Use l(x) to estimate 1/2.1

To find the linear approximation \( L(x) \) of the function \( f(x) = \frac{1}{x} \) at \( a = 2 \), we will determine the equation of the tangent line at \( x = 2 \) and use it to estimate \( f(2.1) \) by substituting \( x = 2.1 \) into \( L(x) \).

### Steps

1. **Find \( f(a) \):**
   \[
   f(2) = \frac{1}{2} = 0.5
   \]

2. **Compute \( f'(x) \):**
   The derivative of \( f(x) = \frac{1}{x} \) is
   \[
   f'(x) = -\frac{1}{x^2}
   \]

3. **Evaluate \( f'(a) \) at \( x = 2 \):**
   \[
   f'(2) = -\frac{1}{2^2} = -\frac{1}{4} = -0.25
   \]

4. **Write the equation of the tangent line (linear approximation \( L(x) \)):**
   The equation of the tangent line at \( x = 2 \) is given by
   \[
   L(x) = f(2) + f'(2)(x - 2)
   \]
   Substituting the values we found:
   \[
   L(x) = 0.5 - 0.25(x - 2)
   \]
   Simplifying, we get
   \[
   L(x) = 0.5 - 0.25x + 0.5 = 1 - 0.25x
   \]

5. **Estimate \( f(2.1) \) using \( L(x) \):**
   Substitute \( x = 2.1 \) into \( L(x) \):
   \[
   L(2.1) = 1 - 0.25 \cdot 2.1 = 1 - 0.525 = 0.475
   \]

### Answer

Thus, the linear approximation \( L(x) \) of \( f(x) = \frac{1}{x} \) at \( x = 2 \) is
\[
L(x) = 1 - 0.25x
\]
Using this, the estimated value of \( \frac{1}{2.1} \) is approximately \( 0.475 \).

Would you like a more detailed breakdown or have any questions?

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Here are five related questions to expand your understanding:

1. What would the linear approximation be for \( f(x) = \frac{1}{x} \) at \( a = 3 \)?
2. How does the accuracy of a linear approximation change as \( x \) moves further from \( a \)?
3. Could we use a similar approach to estimate \( f(2.05) \)? How close would it be?
4. What other methods could provide better approximations for values close to \( a \) than the linear one?
5. How does the slope \( f'(a) \) influence the approximation's accuracy?

**Tip:** Linear approximations work best for values of \( x \) near the point \( a \) where the approximation is calculated.

Find the linear approximation L(x) for the function f(x)=1/x at a=2. Solve by finding the equation of the tangent line. Use L(x) to estimate 1/2.1.

Learn how to find the linear approximation of the function f(x) = 1/x at x = 2 using the tangent line method. The solution includes steps for calculating the derivative, evaluating the function and its derivative at x = 2, and using the linear approximation to estimate f(2.1).

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