AP Calculus AB / Differentiation Composite, Implicit, and Inverse Functions / Question No. 31

31. The graph of $f (x)$ , shown below, consists of a semicircle and two-line segments. $f^{'} (1) =$

-1

- \frac{1}{\sqrt{3}}

\frac{1}{\sqrt{3}}

Answer is: option2

- \frac{1}{\sqrt{3}}

Solution:

Equation of the circle: $x^{2} + y^{2} = r^{2}$

Given:

x^{2} + y^{2} = 4

Substituting $x = 1$ :

1^{2} + y^{2} = 4

y^{2} = 3

y = \pm \sqrt{3}

Since $f (1) = - \sqrt{3}$ , we use implicit differentiation:

2 x + 2 y \frac{d y}{d x} = 0

Solving for $\frac{d y}{d x}$ :

\frac{d y}{d x} = - \frac{x}{y}

Substituting $x = 1$ and $y = - \sqrt{3}$ :

\frac{d y}{d x} = - \frac{1}{\sqrt{3}}

Final Answer:

f^{'} (1) = - \frac{1}{\sqrt{3}}

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