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Chapter 2: Q. 14 (page 165)

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

g'(c)鈮�g(z)鈭�g(c)z鈭�c

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

(b) the distances z 鈭� c and g(z) 鈭� g(c)

(c) the slopes g(z)鈭�g(c)z鈭�candg'(c)

Short Answer

Expert verified

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

g'(c)鈮�g(z)鈭�g(c)z鈭�c

Step by step solution

01

Part (a) Step 1. Given information.

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

g'(c)鈮�g(z)鈭�g(c)z鈭�c

02

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

After locating the point, the graph is :

03

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).

04

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

The ratio g(c+h)鈭�g(c)hrepresents the slope of the line passing through the points (c,g(c))and(z,g(z))

The slopeg'(c)is the slope of the tangent line to the graph at the point(c,g(c))

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