1. The problem is to understand and apply the concept of linear approximation using derivatives. 2. Linear approximation uses the tangent line at a point to approximate the value of a function near that point. 3. The formula for linear approximation at $x=a$ is: $$L(x) = f(a) + f'(a)(x - a)$$ where $f'(a)$ is the derivative of $f$ at $a$. 4. This means we approximate $f(x)$ by the value of the tangent line at $x=a$. 5. To use this, first find $f(a)$ and $f'(a)$. 6. Then plug these into the formula to get $L(x)$. 7. For example, if $f(x) = \\sqrt{x}$ and we want to approximate near $a=4$: - $f(4) = 2$ - $f'(x) = \\frac{1}{2\\sqrt{x}}$, so $f'(4) = \\frac{1}{4}$ - Linear approximation: $$L(x) = 2 + \\frac{1}{4}(x - 4)$$ 8. This linear function approximates $\\sqrt{x}$ near $x=4$. This method is useful for estimating values of functions that are difficult to compute exactly.