Question:

For a certain function $f(x)$, the linear approximation at $x = 2$ is given by $L(x) = 4 - 2x$.

1. Use the linear approximation to estimate the value of $f(2.1)$.
2. Use differentials to approximate the value of $f(2.1)$.
3. Calculate the actual value of $f(2.1)$.
4. Determine the absolute error in the approximation obtained using the linear approximation.
5. Determine the percentage error in the approximation obtained using the linear approximation.

Answer:

To estimate the value of $f(2.1)$ using linear approximation at $x = 2$, we can use the equation $L(x) = 4 - 2x$.

1. Estimating $f(2.1)$ using the linear approximation:

To estimate $f(2.1)$ using the linear approximation, substitute $x = 2.1$ into the equation:

$L(2.1) = 4 - 2(2.1) = 4 - 4.2 = -0.2$

Therefore, the estimate of $f(2.1)$ using the linear approximation is $-0.2$.

2. Approximating $f(2.1)$ using differentials:

To approximate $f(2.1)$ using differentials, we can use the differential form $df \approx f'(x) \cdot dx$.

First, we need to find $dx$, which is the change in $x$ from $x = 2$ to $x = 2.1$:

$dx = 2.1 - 2 = 0.1$

Next, we need to find $f'(x)$, which represents the derivative of $f(x)$ with respect to $x$. However, we are not given this information in the question.

Therefore, we cannot proceed with finding the approximation using differentials.

3. Calculating the actual value of $f(2.1)$:

Without further information about the function $f(x)$, we are unable to calculate the exact value of $f(2.1)$ for this question.

4. Calculation of the absolute error:

The absolute error is the absolute difference between the actual value of $f(2.1)$ and the estimated value obtained using the linear approximation.

Since we do not have the actual value of $f(2.1)$, we cannot calculate the absolute error.

5. Calculation of the percentage error:

The percentage error is the absolute error expressed as a percentage of the actual value.

Since we do not have the actual value of $f(2.1)$, we cannot calculate the percentage error.

In conclusion, using the given linear approximation $L(x) = 4 - 2x$, we estimated the value of $f(2.1)$ to be $-0.2$. However, without additional information on the function $f(x)$, we cannot calculate the actual value, absolute error, or percentage error.