differentiate the function. then find an equation of the tangent line at the indicated point on the graph of…

Explanation:

Step1: Recall tangent - line formula

The equation of a tangent line to a curve $w = g(z)$ at the point $(z_0,w_0)$ is given by $w - w_0=g^{\prime}(z_0)(z - z_0)$, where $g^{\prime}(z)$ is the derivative of $g(z)$ and $g^{\prime}(z_0)$ is the slope of the tangent line at $z = z_0$.

Step2: Identify the values of $z_0$, $w_0$ and $g^{\prime}(z_0)$

We are given that $(z_0,w_0)=(7,10)$ and $g^{\prime}(z)=-\frac{1}{2\sqrt{16 - z}}$. Substitute $z = 7$ into $g^{\prime}(z)$: [g^{\prime}(7)=-\frac{1}{2\sqrt{16 - 7}}=-\frac{1}{2\sqrt{9}}=-\frac{1}{6}]

Step3: Substitute into the tangent - line formula

Substitute $z_0 = 7$, $w_0 = 10$ and $g^{\prime}(7)=-\frac{1}{6}$ into $w - w_0=g^{\prime}(z_0)(z - z_0)$: [w-10=-\frac{1}{6}(z - 7)] [w-10=-\frac{1}{6}z+\frac{7}{6}] [w=-\frac{1}{6}z+\frac{7}{6}+10] [w=-\frac{1}{6}z+\frac{7 + 60}{6}] [w=-\frac{1}{6}z+\frac{67}{6}] [6w=-z + 67] [z+6w=67]

Answer:

$z + 6w=67$