Question

differentiate the function. then find an equation of the tangent line at the indicated point on the graph of the fun $w = g(z)=7+sqrt{16 - z},\\ (z,w)=(7,10)$. the derivative of the function $w = g(z)=7+sqrt{16 - z}$ is $-\frac{1}{2sqrt{16 - z}}$. (type an exact answer, using radicals as needed.) an equation of the tangent line is . (type an equation.)

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:

$z + 6w=67$