Questions: Consider the natural log function called log(x). Use Taylor's Theorem to find a linear approximation to the function f(x)=log (x), expanded around x=2. Then use that to estimate log (2.8). Choose the number here that is closest to your estimate of log (2.8). 1.10 1.20 1.30 1.40 1.50

Consider the natural log function called log(x). Use Taylor's Theorem to find a linear approximation to the function f(x)=log (x), expanded around x=2. Then use that to estimate log (2.8). Choose the number here that is closest to your estimate of log (2.8).
1.10
1.20
1.30
1.40
1.50

Transcript text: Consider the natural log function called log(x). Use Taylor's Theorem to find a linear approximation to the function $f(x)=\log (x)$, expanded around $x=2$. Then use that to estimate $\log (2.8)$. Choose the number here that is closest to your estimate of $\log (2.8)$. 1.10 1.20 1.30 1.40 1.50

Solution

Solution Steps

Step 1: Calculate $ f(2) $

We start by evaluating the function $ f(x) = \log(x) $ at the point $ x = 2 $: \[ f(2) = \log(2) \]

Step 2: Calculate $ f'(2) $

Next, we find the derivative of the function $ f(x) $: \[ f'(x) = \frac{1}{x} \] Now, we evaluate the derivative at $ x = 2 $: \[ f'(2) = \frac{1}{2} \]

Step 3: Apply the Linear Approximation Formula

Using the first-order Taylor expansion around $ a = 2 $, we can express the linear approximation for $ f(x) $: \[ f(x) \approx f(2) + f'(2)(x - 2) \] Substituting the values we calculated: \[ f(x) \approx \log(2) + \frac{1}{2}(x - 2) \]

Step 4: Estimate $ \log(2.8) $

Now, we substitute $ x = 2.8 $ into the linear approximation: \[ f(2.8) \approx \log(2) + \frac{1}{2}(2.8 - 2) \] Calculating the expression gives us: \[ f(2.8) \approx \log(2) + \frac{1}{2}(0.8) = \log(2) + 0.4 \]

Step 5: Final Calculation

Using the approximate value of $ \log(2) \approx 0.693147 $, we can compute: \[ f(2.8) \approx 0.693147 + 0.4 \approx 1.093147 \]

Thus, the estimate for $ \log(2.8) $ is approximately $ 1.093147 $.