AP Calculus AB Exam Question

Find the linear approximation of the function $f(x) = \sqrt{x + 3}$ at $x = 2$, and use it to estimate $f(2.1)$.

Step-by-Step Detailed Explanation

To find the linear approximation, we start by finding the equation of the tangent line at the given point. Recall that the equation of a line can be written in the form $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept.

1. Find the slope $m$: To find the slope $m$, we need to find the derivative of the function $f(x) = \sqrt{x + 3}$. Using the power rule, we have

Plugging in $x = 2$ to calculate the slope at that point, we get

2. Find the $y$-intercept $b$: To find the $y$-intercept $b$, we need the value of $f(x)$ at $x = 2$. Plugging $x = 2$ into the original function, we get

Therefore, $b = \sqrt{5}$.

3. Write the equation of the tangent line: Using the slope-intercept form, the equation of the tangent line is

Now that we have the equation of the tangent line, we can estimate $f(2.1)$ using the linear approximation.

4. Estimate $f(2.1)$: Plugging in $x = 2.1$ into the equation of the tangent line, we get

Therefore, the linear approximation of $f(x) = \sqrt{x + 3}$ at $x = 2$ gives an estimate of $f(2.1)$ as approximately $\boxed{2.714}$.