Find an equation of the tangent line at the indicated point on the graph of the function.
-$$w = g ( z ) = 4 + \sqrt { 6 - z } , ( z , w ) = ( 5,5 )$$
A) $w = - \frac { 1 } { 2 } z - \frac { 15 } { 2 }$
B) $w = \frac { 1 } { 2 } z + \frac { 15 } { 2 }$
C) $w = - \frac { 1 } { 2 } z + \frac { 15 } { 2 }$
D) $w = \frac { 1 } { 2 } z - \frac { 15 } { 2 }$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line at the point (5,5), we first need to find the derivative of $g(z)$ to get the slope of the tangent line. The derivative of $g(z) = 4 + \sqrt{6 - z}$ with respect to $z$ is $g'(z) = -\frac{1}{2\sqrt{6-z}}$. Evaluating this derivative at $z = 5$ gives $g'(5) = -\frac{1}{2}$, which is the slope of the tangent line. Using the point-slope form of a line, $w - w_1 = m(z - z_1)$, where $m$ is the slope and $(z_1, w_1)$ is the point (5,5), we get $w - 5 = -\frac{1}{2}(z - 5)$. Simplifying this equation gives $w = -\frac{1}{2}z + \frac{15}{2}$.

Find an Equation of the Tangent Line at the Indicated w=g(z)=4+6−z,(z,w)=(5,5)w = g ( z ) = 4 + \sqrt { 6 - z } , ( z , w ) = ( 5,5 )w=g(z)=4+6−z​,(z,w)=(5,5)

Find an Equation of the Tangent Line at the Indicated $w = g ( z ) = 4 + \sqrt { 6 - z } , ( z , w ) = ( 5,5 )$