Tangent Line Math Example 2

Solution

1. 1
   Point of tangency: $g(9) = 3$. So the point is $(9, 3)$.
2. 2
   Derivative: $g'(x) = \frac{1}{2\sqrt{x}}$, so $g'(9) = \frac{1}{6}$.
3. 3
   Tangent line: $y - 3 = \frac{1}{6}(x - 9)$, i.e., $y = \frac{x}{6} + \frac{3}{2}$.
4. 4
   Approximate $\sqrt{9.1}$: substitute $x = 9.1$: $y \approx \frac{9.1}{6} + 1.5 = 1.5167 + 1.5 = 3.0167$.