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

question 21

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.
-
$y = x ^ { 2 }$
$$y = \sqrt { x }$$

• A) Since \(\lim _ { x - z ^ { + } } f ^ { \prime } ( x ) = \frac { 1 } { 2 }\) while \(\lim _ { x - 3 ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )\) is not differentiable at \(x = 1\).
• B) Since \(\lim _ { x - z ^ { + } } f ^ { \prime } ( x ) = \frac { 1 } { 2 }\) while \(\lim _ { x - z ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )\) is not differentiable at \(x = 1\).
• C) Since \(\lim _ { x - z ^ { + } } f ^ { \prime } ( x ) = 2\) while \(\lim _ { x - z ^ { - } } f ^ { \prime } ( x ) = \frac { 1 } { 2 } , f ( x )\) is not differentiable at \(x = 1\).
• D) Since \(\lim _ { x - 3 ^ { + } } f ^ { \prime } ( x ) = 2\) while \(\lim _ { x - 7 ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )\) is differentiable at \(x = 1\).

---

Definitions:

Degrees of Freedom

The count of individual values or magnitudes that may be allocated to a statistical distribution without breaking any limitations.

Sample Size

The number of observations or individuals taken from a population to form a sample for the purpose of statistical analysis.

Variance Accounted

A measure of the proportion of the total variance in a dataset that is explained or accounted for by a statistical model or factor.

Correlation

A numerical indicator that reveals the degree to which two factors vary in tandem, showcasing the intensity and orientation of their association.