### Home > APCALC > Chapter 7 > Lesson 7.3.4 > Problem7-145

7-145.

Write the equation of a line tangent to the graph of $f(x) = \arcsin(x)$ at $x =\frac { 1 } { 2 }$.

1. Use the tangent line to approximate the value of $f(0.52)$.

Because $y = \arcsin(0.52)$ is hard to evaluate, the tangent line at $x = 0.52$ will give a decent approximation of $f(0.52)$.

2. Is your approximation an overestimate or an underestimate? Use the second derivative to justify your answer.

$\text{The sign of the 2nd-derivative at }x=\frac{1}{2} \text{ reveals the concavity of }f(x) \text{ at }x=\frac{1}{2},$

and that will indicate if the tangent line is above or below the graph.