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The given problem asks to use linear approximation for the function \( f(x) = \sqrt{49 - x} \) at \( x = 0 \), and to approximate \( f(1) = 4 \cdot \sqrt{3} \) by a rational number.

### Step 1: Linear Approximation Formula
The formula for linear approximation is:
\[
f(x) \approx f(a) + f'(a)(x - a)
\]
where \( f'(a) \) is the derivative of \( f(x) \) at \( x = a \).

### Step 2: Function \( f(x) = \sqrt{49 - x} \)
We are given \( f(x) = \sqrt{49 - x} \). Let's calculate the derivative:

\[
f'(x) = \frac{d}{dx} \left( \sqrt{49 - x} \right) = \frac{1}{2\sqrt{49 - x}} \cdot (-1) = -\frac{1}{2\sqrt{49 - x}}
\]

### Step 3: Evaluate at \( x = 0 \)
Now we evaluate \( f(x) \) and \( f'(x) \) at \( x = 0 \):

\[
f(0) = \sqrt{49 - 0} = \sqrt{49} = 7
\]

\[
f'(0) = -\frac{1}{2\sqrt{49}} = -\frac{1}{2 \cdot 7} = -\frac{1}{14}
\]

### Step 4: Linear Approximation for \( f(1) \)
Now, use the linear approximation formula at \( x = 0 \) to approximate \( f(1) \):

\[
f(1) \approx f(0) + f'(0)(1 - 0) = 7 + \left( -\frac{1}{14} \right)(1) = 7 - \frac{1}{14} = \frac{98}{14} - \frac{1}{14} = \frac{97}{14}
\]

Thus, \( f(1) \approx \frac{97}{14} \).

### Step 5: Approximation of \( 4 \cdot \sqrt{3} \)
We are asked to approximate \( 4 \cdot \sqrt{3} \) by a rational number. Using the approximation \( \sqrt{3} \approx 1.732 \), we get:

\[
4 \cdot \sqrt{3} \approx 4 \cdot 1.732 = 6.928
\]

Hence, the rational approximation is close to \( \frac{97}{14} \), as calculated earlier.

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Do you want to go over these steps in more detail or have any further questions?

### Related Questions:
1. How can linear approximation be applied to other functions?
2. What are the limitations of linear approximation?
3. Why does linear approximation work well near \( x = 0 \)?
4. How do we improve the accuracy of this approximation?
5. What is the difference between linear approximation and a Taylor series?

### Tip:
Linear approximation is most accurate near the point where the function is approximated, so always check how far you are extrapolating!

Use linear approximation of the function f(x) = sqrt(49 - x) at the point x = 0 to find an approximation of f(1) = 4 * sqrt(3) by a rational number.

Learn how to use linear approximation for the function f(x) = sqrt(49 - x) at x = 0 to approximate f(1) = 4 * sqrt(3). This problem involves calculus concepts like derivatives and square roots, ideal for Grades 10-12 students.

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