Question

Consider again the graph of f at the left. Label each of the following quantities to illustrate that

$f^{\mathrm{\prime }}\left(c\right)\approx \frac{f\left(z\right)-f\left(c\right)}{z-c}$

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

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

(c) the slopes $\frac{f\left(z\right)-f\left(c\right)}{z-c}$ and$f^{\mathrm{\prime }}\left(c\right)$

Short answer

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

$f^{\mathrm{\prime }}\left(c\right)\approx \frac{f\left(z\right)-f\left(c\right)}{z-c}$

Step-by-step solution

Step 1. Given information.

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

$f^{\mathrm{\prime }}\left(c\right)\approx \frac{f\left(z\right)-f\left(c\right)}{z-c}$

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

After locating the point, the graph is :

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

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

That is, the difference between z and c.

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

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

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

The ratio $\frac{f\left(z\right)-f\left(c\right)}{z-c}$represents the slope of the line passing through the points,

$(c,f(c\left)\right)\text{and}(z,f(z\left)\right)$

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