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Pnovotnak · Answer 1 · 2023-09-28T15:36:08+0000

Answer:

$y=-(2)/(√(4-\pi^2))x+Undefined$

Step-by-step explanation:

Given the function:

$f(x)=\cos^(-1)(x)$

First, determine the tangent point by evaluating f(x) at x=π/2:

$f((\pi)/(2))=\cos^(-1)((\pi)/(2))=Undefined$

Note: f(π/2) is undefined because the domain of arccosine is (-1,1).

The tangent point is:

$((\pi)/(2),Undefined)$

Next, find the slope of the tangent line. Begin by finding the derivative of f(x):

By the rules of derivative:

$f^(\prime)(x)=-(1)/(√(1-x^2))$

At x=π/2, find the value of the derivative:

$\begin{gathered} f^(\prime)((\pi)/(2))=-\frac{1}{\sqrt{1-((\pi)/(2))^2}} \\ =-\frac{1}{\sqrt{1-(\pi^2)/(4)}} \\ =-\frac{1}{\sqrt{(4-\pi^2)/(4)}} \\ Slope,m=-(2)/(√(4-\pi^2)) \end{gathered}$

Thus, the equation of the tangent line is:

$y=-(2)/(√(4-\pi^2))x+Undefined$

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