/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Find an equation of the line tan... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 33

Find an equation of the line tangent to the graph of \(f\) at the given point. $$f(x)=\cos ^{-1} x^{2} ;(1 / \sqrt{2}, \pi / 3)$$

Short Answer

Expert verified
Answer: \(y = -\sqrt{2}x +1 + \frac{\pi}{3}\)

Step by step solution

01

Find the derivative of the given function

Differentiate \(f(x)=\cos^{-1}x^2\) with respect to \(x\). Using the chain rule, we have: $$f'(x)=\frac{d}{dx}(\cos^{-1}x^2)=-\frac{1}{\sqrt{1-x^4}}\cdot(2x)$$
02

Evaluate the derivative at the given point

Find the slope of the tangent line at the point \((1/\sqrt{2}, \pi/3)\) by substituting the x-coordinates into the derivative expression: $$f'(1/\sqrt{2})=-\frac{1}{\sqrt{1-(1/\sqrt{2})^4}}\cdot(2\cdot(1/\sqrt{2}))$$ $$f'(1/\sqrt{2})=-\frac{1}{\sqrt{1-1/2}}\cdot(2\cdot(1/\sqrt{2}))$$ $$f'(1/\sqrt{2})=-\frac{1}{\sqrt{1/2}}\cdot(\sqrt{2})$$ $$f'(1/\sqrt{2})=-\sqrt{2}$$
03

Use the point-slope form to find the equation of the tangent line

Now that we have the slope and the point \((1/\sqrt{2}, \pi/3)\), we can use the point-slope form to find the equation of the tangent line: $$y-\frac{\pi}{3}=-\sqrt{2}(x-\frac{1}{\sqrt{2}})$$
04

Simplify the equation of the tangent line

To simplify the equation further, distribute the slope and simplify the results: $$y-\frac{\pi}{3}=-\sqrt{2}x+\sqrt{2}\cdot\frac{1}{\sqrt{2}}$$ $$y-\frac{\pi}{3}=-\sqrt{2}x+1$$ $$y = -\sqrt{2}x +1 + \frac{\pi}{3}$$ The equation of the tangent line to the graph of \(f(x)=\cos^{-1}x^2\) at the point \((1/\sqrt{2}, \pi/3)\) is: $$y = -\sqrt{2}x +1 + \frac{\pi}{3}$$

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