Compare the Right-Hand and Left-Hand Derivatives to Determine Whether or Not

question 398

Multiple Choice

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point
whose coordinates are given.
-
$$y = \frac { 1 } { x } \quad y = - 1$$

• A) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = - 1\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0\), \(f ( x )\) is not differentiable at \(x = - 1\).
• B) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )\) is not differentiable at \(x = - 1\).
• C) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )\) is not differentiable at \(x = - 1\).
• D) Since \(\lim _ { x \rightarrow 1 ^ { + } } + f ^ { \prime } ( x ) = 0\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0 , f ( x )\) is differentiable at \(x = - 1\).

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Definitions:

Absorption Costing

A method in accounting that integrates every manufacturing expense – direct materials, labor, and all overheads (variable and fixed) – into the product’s price.

Net Operating Income

A company's total profit from operational activities, excluding expenses related to interest and taxes.

Period Cost

Expenses that are not directly tied to production activities and are expensed in the period in which they occur.

Contribution Margin

The difference between the sales revenue of a product and its variable costs, used to cover fixed costs and contribute to profit.