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9. Find the equation of the line tangent to y=secxatx=(pi )/(4) A y-(sqrt (2))/(2)=sqrt (2)(x-(pi )/(4)) B y-sqrt (2)=sqrt (2)(x-(pi )/(4)) C. y=sqrt (2)(x-(pi )/(4)) D. y-sqrt (2)=x-(pi )/(4) E y-sqrt (2)=-(x-(pi )/(4))

Pergunta

9. Find the equation of the line tangent to
y=secxatx=(pi )/(4)
A y-(sqrt (2))/(2)=sqrt (2)(x-(pi )/(4))
B y-sqrt (2)=sqrt (2)(x-(pi )/(4))
C. y=sqrt (2)(x-(pi )/(4))
D. y-sqrt (2)=x-(pi )/(4)
E y-sqrt (2)=-(x-(pi )/(4))

9. Find the equation of the line tangent to y=secxatx=(pi )/(4) A y-(sqrt (2))/(2)=sqrt (2)(x-(pi )/(4)) B y-sqrt (2)=sqrt (2)(x-(pi )/(4)) C. y=sqrt (2)(x-(pi )/(4)) D. y-sqrt (2)=x-(pi )/(4) E y-sqrt (2)=-(x-(pi )/(4))

Solução

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FernandoProfissional · Tutor por 6 anos

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To find the equation of the line tangent to the curve \( y = \sec(x) \) at \( x = \frac{\pi}{4} \), we need to follow these steps:<br /><br />1. **Find the point of tangency**:<br /> The point of tangency is given by \( \left( \frac{\pi}{4}, \sec\left(\frac{\pi}{4}\right) \right) \).<br /><br /> Since \( \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \), the point of tangency is \( \left( \frac{\pi}{4}, \sqrt{2} \right) \).<br /><br />2. **Find the derivative of \( y = \sec(x) \)**:<br /> The derivative of \( y = \sec(x) \) is \( y' = \sec(x) \tan(x) \).<br /><br />3. **Evaluate the derivative at \( x = \frac{\pi}{4} \)**:<br /> \[<br /> y'\left(\frac{\pi}{4}\right) = \sec\left(\frac{\pi}{4}\right) \tan\left(\frac{\pi}{4}\right)<br /> \]<br /> Since \( \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \) and \( \tan\left(\frac{\pi}{4}\right) = 1 \),<br /> \[<br /> y'\left(\frac{\pi}{4}\right) = \sqrt{2} \cdot 1 = \sqrt{2}<br /> \]<br /><br />4. **Use the point-slope form of the equation of a line**:<br /> The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency.<br /><br /> Here, \( m = \sqrt{2} \) and \( (x_1, y_1) = \left( \frac{\pi}{4}, \sqrt{2} \right) \).<br /><br /> So, the equation of the tangent line is:<br /> \[<br /> y - \sqrt{2} = \sqrt{2} \left( x - \frac{\pi}{4} \right)<br /> \]<br /><br />Therefore, the correct answer is:<br />\[ \boxed{B \; y - \sqrt{2} = \sqrt{2} \left( x - \frac{\pi}{4} \right)} \]
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