AP Calculus AB Exam Question

Consider the function $f(x) = \sqrt{x+2}$.

1. Find the linear approximation of $f(x)$ at $x=3$.
2. Use the linear approximation to estimate the value of $f(2.9)$.
3. Determine the differential $dy$ of $f$ when $x=3$.

Answer

1. To find the linear approximation of $f(x)$ at $x=3$, we need to find the equation of the tangent line to the graph of $f(x)$ at $x=3$.

First, we find the derivative of $f(x)$:

Next, we evaluate $f'(3)$:

The slope of the tangent line is equal to the value of the derivative at $x=3$. Let $y_1$ be the corresponding $y$-value of $f(3)$.

Thus, the equation of the tangent line can be written as:

Substituting the values we obtained, we get:

2. Now, let's use the linear approximation to estimate the value of $f(2.9)$. We substitute $x=2.9$ into the equation of the tangent line:

Simplifying the equation, we get:

Solving for $y$, we find:

Thus, the estimated value of $f(2.9)$ is approximately equal to $\sqrt{5} - \frac{0.1}{2\sqrt{5}}$.

3. The differential $dy$ of $f$ when $x=3$ is given by:

Substituting the value of $f'(3)$ we found earlier:

Since we are considering a specific point $x=3$, the differential $dx$ can be approximated as the change in $x$ from the given point. Let's say $dx = 0.1$. Then, the differential $dy$ can be calculated as:

Therefore, when $x=3$, the differential $dy$ is approximately equal to $\frac{0.1}{2\sqrt{5}}$.

(Note: This approximation is valid for small values of $dx$ near the point $x=3$).