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Chapter 2: Q. 13 (page 165)

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

f鈥�(c)鈮�f(z)鈭�f(c)z鈭�c

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

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

(c) the slopes f(z)鈭�f(c)z鈭�c andf鈥�(c)

Short Answer

Expert verified

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

f鈥�(c)鈮�f(z)鈭�f(c)z鈭�c

Step by step solution

01

Step 1. Given information.

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

f鈥�(c)鈮�f(z)鈭�f(c)z鈭�c

02

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

After locating the point, the graph is :

03

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).

04

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

The ratio f(z)鈭�f(c)z鈭�crepresents the slope of the line passing through the points,

(c,f(c))and(z,f(z))

f'(c) is the slope of the tangent line to the graph at the point (c,f(c))

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