Question:

Let $f(x) = \sqrt{1 + x}$ be a function defined on the interval $[0, 2]$.

(a) Use linear approximation to estimate the value of $f(1.5)$.

(b) Determine the differential of $f(x)$ at $x = 1$.

Answer:

(a) We can use linear approximation to estimate the value of $f(1.5)$. To do this, we'll find the equation of the tangent line to the function at $x = 1$, and then use that line to estimate $f(1.5)$.

Let's find the slope of the tangent line:

We evaluate $f'(1)$ as follows:

Now, we have the slope and a point on the line $(1, f(1)) = (1, \sqrt{2})$. We can find the equation of the tangent line using the point-slope form:

We can rearrange this equation to get:

Now, we can estimate $f(1.5)$ by evaluating the equation of the tangent line at $x = 1.5$:

Therefore, the estimate of $f(1.5)$ using linear approximation is $\sqrt{2} + \frac{1}{8}$.

(b) To find the differential of $f(x)$ at $x = 1$, we'll use the differential form:

Here, $dx$ represents a small change in $x$. We'll evaluate this at $x = 1$:

So, the differential of $f(x)$ at $x = 1$ is $\frac{1}{4}dx$.