AP Calculus AB Exam Question:

Let $f(x) = \sqrt{x}$, where $x$ is a positive number.

(a) Find the linear approximation of $f(x)$ near $x = 4$.

(b) Use the linear approximation from part (a) to estimate $\sqrt{4.2}$.

Answer:

(a) To find the linear approximation of $f(x)$ near $x = 4$, we will use the concept of tangent lines and the differential of $f(x)$.

Recall that the equation of the tangent line to a curve at a given point $(a, f(a))$ is given by:

Taking the derivative of $f(x)$, we have:

Now, let's find the equation of the tangent line to $f(x)$ at $x = 4$. In this case, $a = 4$ and $f(a) = f(4) = \sqrt{4} = 2$. Plugging these values into the equation of the tangent line, we have:

Simplifying further, we get:

Therefore, the linear approximation of $f(x)$ near $x = 4$ is $L(x) = 2 + \frac{1}{4}(x - 4)$.

(b) To estimate $\sqrt{4.2}$ using the linear approximation above, we need to evaluate $L(4.2)$.

Plugging $x = 4.2$ into the linear approximation equation, we have:

Simplifying, we get:

Therefore, $\sqrt{4.2}$ is approximately equal to 2.05 using the linear approximation.