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

question 269

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.
-
$y = x$ $y = 2 x$

• A) Since \(\lim _ { x - \theta ^ { + } } f ^ { \prime } ( x ) = 1\) while \(\lim _ { x \rightarrow \theta ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )\) is not differentiable at \(x = 0\).
• B) Since \(\lim _ { x \rightarrow \theta ^ { + } } f ^ { \prime } ( x ) = 2\) while \(\lim _ { x \rightarrow \theta ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )\) is not differentiable at \(x = 0\).
• C) Since \(\lim _ { x - \theta ^ { + } } f ^ { \prime } ( x ) = - 2\) while \(\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )\) is not differentiable at \(x = 0\).
• D) Since \(\lim _ { x \rightarrow \theta ^ { + } } f ^ { \prime } ( x ) = 1\) while \(\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )\) is differentiable at \(x = 0\).

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Definitions:

Insects

A class of arthropods, characterized by a three-part body (head, thorax, abdomen), three pairs of legs, compound eyes, and two antennae.

Arthropod

An invertebrate that belongs to phylum Arthropoda; characterized by a hard exoskeleton; a segmented body; and paired, jointed appendages.

Parapodia

Paired, lateral extensions found in some annelids, such as polychaete worms, used for locomotion and respiration.

Oligochaetes

A subclass of earthworms characterized by having few bristles per segment, important for soil aeration and nutrient cycling.