a. Find the local linear approximation of the function f(x) = √(1+x)at x knot= 0, and use
it to approximate √0.9 and √1.1.

The local linear approximation of a function \( f(x) \) at a point \( x_0 \) is given by the formula:

\[
L(x) = f(x_0) + f'(x_0)(x - x_0)
\]

### Step 1: Find \( f(x) \) and \( f'(x) \)

Given the function \( f(x) = \sqrt{1 + x} \), we first need to compute its derivative \( f'(x) \).

\[
f(x) = (1 + x)^{1/2}
\]
\[
f'(x) = \frac{1}{2}(1 + x)^{-1/2} \cdot 1 = \frac{1}{2\sqrt{1 + x}}
\]

### Step 2: Evaluate \( f(x_0) \) and \( f'(x_0) \) at \( x_0 = 0 \)

\[
f(0) = \sqrt{1 + 0} = 1
\]
\[
f'(0) = \frac{1}{2\sqrt{1 + 0}} = \frac{1}{2}
\]

### Step 3: Write the local linear approximation \( L(x) \)

Using the formula for the linear approximation:

\[
L(x) = f(0) + f'(0)(x - 0)
\]
\[
L(x) = 1 + \frac{1}{2}(x)
\]
\[
L(x) = 1 + \frac{x}{2}
\]

### Step 4: Approximate \( \sqrt{0.9} \) and \( \sqrt{1.1} \)

- To approximate \( \sqrt{0.9} \), let \( x = -0.1 \):
  \[
  L(-0.1) = 1 + \frac{-0.1}{2} = 1 - 0.05 = 0.95
  \]

- To approximate \( \sqrt{1.1} \), let \( x = 0.1 \):
  \[
  L(0.1) = 1 + \frac{0.1}{2} = 1 + 0.05 = 1.05
  \]

### Final Answer:
- \( \sqrt{0.9} \approx 0.95 \)
- \( \sqrt{1.1} \approx 1.05 \)

Would you like more details or have any questions?

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Here are 5 related questions:
1. How does the local linear approximation differ from a Taylor series expansion?
2. What is the error between the approximation and the exact value for \( \sqrt{0.9} \)?
3. Can we apply the same method to approximate \( \sqrt{0.8} \) or \( \sqrt{1.2} \)?
4. How does the choice of \( x_0 \) affect the accuracy of the approximation?
5. What are other applications of linear approximation in calculus?

**Tip:** Local linear approximations are particularly accurate when the point of interest is close to the point where the approximation is made.

Find the local linear approximation of the function f(x) = √(1+x) at x = 0, and use it to approximate √0.9 and √1.1.

Learn how to find the local linear approximation of the function f(x) = √(1 + x) at x = 0 and use it to approximate √0.9 and √1.1. This problem involves key calculus concepts such as derivatives and linear approximation. Suitable for students in Grades 11-12 or early college.

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