Q. 12 - Calculus

Question

For the graph of g(x) appearing next at the right, label each of the following quantities to illustrate that 

$g^{\mathrm{\prime }}\left(c\right)\approx \frac{g(c+h)-g\left(c\right)}{h}$

(a) the locations c, c + h, g(c), and g(c + h) 

(b) the distances h and g(c + h) − g(c) 

(c) the slopes $\frac{g(c+h)-g\left(c\right)}{h}$ and $g^{\mathrm{\prime }}\left(c\right)$

Step-by-Step Solution

Answer

Part (a) Step 1. Given information.

For the graph of g(x) appearing next at the right, we have to label each of the following quantities given in the question to illustrate that 

$g^{\mathrm{\prime }}\left(c\right)\approx \frac{g(c+h)-g\left(c\right)}{h}$

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

After locating the point, the graph is :

Part (b) Step 1. Find the distance h and g(c+h)-g(c)

The objective is to determine the distances h and g(c+h)-g(c).

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

That is, the difference between c and c+h.

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

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

Part (c) Step 1. 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 $(c,g(c\left)\right)\text{ and }(c+h,g(c+h\left)\right)$

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