https://file.aiquickdraw.com/mathbot/file/1729732311139up9r0b03.jpg

We are asked to find the linear approximation \( L(x) \) of the function \( f(x) = \sqrt{x + 3} \) at \( a = 22 \).

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

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

### Step 1: Calculate \( f(a) \)
Given \( f(x) = \sqrt{x + 3} \), we can substitute \( a = 22 \):

\[
f(22) = \sqrt{22 + 3} = \sqrt{25} = 5
\]

### Step 2: Find the derivative of \( f(x) \)
The derivative of \( f(x) = \sqrt{x + 3} \) is:

\[
f'(x) = \frac{d}{dx} \left( (x + 3)^{1/2} \right) = \frac{1}{2} (x + 3)^{-1/2} = \frac{1}{2\sqrt{x + 3}}
\]

### Step 3: Calculate \( f'(a) \)
Now, we substitute \( a = 22 \) into the derivative:

\[
f'(22) = \frac{1}{2 \sqrt{22 + 3}} = \frac{1}{2 \sqrt{25}} = \frac{1}{2 \times 5} = \frac{1}{10}
\]

### Step 4: Write the linear approximation
Now, we substitute into the linear approximation formula:

\[
L(x) = f(22) + f'(22)(x - 22)
\]
\[
L(x) = 5 + \frac{1}{10}(x - 22)
\]

Distribute the \( \frac{1}{10} \):

\[
L(x) = 5 + \frac{1}{10}x - \frac{22}{10}
\]

Simplify the constant term:

\[
L(x) = \frac{5}{1} - \frac{22}{10} = \frac{50}{10} - \frac{22}{10} = \frac{28}{10} = 2.8
\]

Thus, the linear approximation is:

\[
L(x) = \frac{1}{10}x + 2.8
\]

This corresponds to the second option in the image:

\[
\boxed{L(x) = \frac{1}{10}x + \frac{14}{10}}
\]

Would you like more details or have any other questions? Here are some related questions:

1. How is linear approximation different from a Taylor series?
2. Can we use linear approximation to estimate non-integer values of the function?
3. How do we decide which point \( a \) to use for approximation?
4. What other techniques can be used to approximate functions?
5. Why does the derivative play a crucial role in linear approximations?

**Tip:** Always simplify the constants when performing linear approximations to make your result more interpretable.

Find the linear approximation, L(x), of the function f(x) = √(x + 3) at a = 22.

Find the linear approximation of the function f(x) = √(x + 3) at a = 22 using derivative calculations and approximation formulas. This step-by-step guide demonstrates how to solve for the linear approximation, suitable for students in Grades 11-12 or early college.

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