Question

Consider again the graph of g(x) at the right. Label each of the following quantities to illustrate that

$g^{\text{'}}\left(c\right)\approx \frac{g\left(z\right)-g\left(c\right)}{z-c}$

(a) the locations c, z, g(c), and g(z)

(b) the distances z − c and g(z) − g(c)

(c) the slopes $\frac{g\left(z\right)-g\left(c\right)}{z-c}$and$g^{\text{'}}\left(c\right)$

Short answer

We have to label each of the quantities given in the question to illustrate that :

$g^{\text{'}}\left(c\right)\approx \frac{g\left(z\right)-g\left(c\right)}{z-c}$

Step-by-step solution

Part (a) Step 1. Given information.

We have to label each of the quantities given in the question to illustrate that :

$g^{\text{'}}\left(c\right)\approx \frac{g\left(z\right)-g\left(c\right)}{z-c}$

Part (a) Step 2. Locate c,z, g(c) and g(z)

After locating the point, the graph is :

Part (b) Step 1. Find the distance z-c and g(z)-gc)

The distance z-cis the difference between the x-coordinates of the first and second points.

That is, the difference between z and c.

The distance g(z)-g(c) is the difference between the y-coordinates of the first and second points.

That is, the difference between g(z) and g(c).

Part (c) Step 1. Find the slope and g'(c)

The ratio $\frac{g(c+h)-g\left(c\right)}{h}$represents the slope of the line passing through the points $(c,g(c\left)\right)\text{and}(z,g(z\left)\right)$

The slope$g^{\text{'}}\left(c\right)$is the slope of the tangent line to the graph at the point$(c,g(c\left)\right)$