Linear Approximation 012360

1. **State the problem:** We want to estimate the value of $\frac{1}{101}$ using a linear approximation or differential. 2. **Recall the function and point:** Consider the function $f(x) = \frac{1}{x}$. We want to approximate $f(101)$ near a point where the function is easy to evaluate, such as $x=100$. 3. **Formula for linear approximation:** The linear approximation of $f(x)$ near $x=a$ is given by: $$ f(x) \approx f(a) + f'(a)(x - a) $$ where $f'(x)$ is the derivative of $f(x)$. 4. **Calculate $f(a)$ and $f'(a)$:** - $f(100) = \frac{1}{100} = 0.01$ - $f'(x) = -\frac{1}{x^2}$, so $f'(100) = -\frac{1}{100^2} = -\frac{1}{10000} = -0.0001$ 5. **Apply the linear approximation:** $$ f(101) \approx f(100) + f'(100)(101 - 100) = 0.01 + (-0.0001)(1) = 0.01 - 0.0001 = 0.0099 $$ 6. **Interpretation:** Using linear approximation, $\frac{1}{101} \approx 0.0099$. 7. **Check with exact value:** The exact value is approximately $0.00990099$, so the approximation is very close. **Final answer:** $$ \frac{1}{101} \approx 0.0099 $$