### Home > APCALC > Chapter 5 > Lesson 5.2.5 > Problem5-105

5-105.

You can use a tangent line to estimate the value of a function at points near a point of tangency. .

1. Sketch a graph of $y=\sqrt{x+1}$ and its tangent line at $x = 0$. Then write the equation of the tangent line.

2. Use the tangent line to approximate the value of $\sqrt { x + 1 }$ for $x = 0.1$ and $x = –0.1$.

One purpose of tangent lines:

Since evaluating square roots is more challenging than evaluating straight lines, and since the straight line (tangent line) is very close to the square root near its point of tangency, we will evaluate the tangent line at $x = 0.1$ and $x = −0.1$ in order to approximate the values of the square root graph.

3. Use your calculator to evaluate $\sqrt { 1.1 }$ and $\sqrt { 0.9 }$. Calculate the percentage error for your estimate in each case. How accurate was your approximation?

$\text{Percent Error}=\frac{\text{difference between (approximate value) and (actual value)}}{\text{actual value}}$

Use the eTool below to visualize the problem.
Click the link at right for the full version of the eTool: Calc 5-105 HW eTool