Graph the curves in Exercises 37-46 . a. Where do the graphs...

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{5 / 3}-5 x^{2 / 3}
$$

Key Concepts

Differentiation of Power Functions

Differentiating power functions using the power rule helps determine the slope of the curve at any given point. When dealing with exponents that are fractions, it is important to carefully assess the derivative, particularly near points where the function may not be smooth or where the derivative may not be finite.

Limit Calculations for Derivative Behavior

Limit calculations are used to rigorously analyze the behavior of the derivative as it approaches points of interest. By computing the limit of the derivative near a suspected vertical tangent, one can confirm if the slope indeed becomes infinite, providing a mathematical basis for identifying vertical tangents.

Vertical Tangents

A vertical tangent occurs when the slope of the tangent line to the curve becomes infinite, meaning the derivative approaches positive or negative infinity. This concept is important for analyzing the geometric behavior of curves, as it points to locations where the rate of change increases without bound.

Graphing Functions

Graphing functions involves plotting the behavior of the curve by identifying key characteristics such as intercepts, asymptotes, and regions of increase or decrease. For power functions in particular, understanding how fractional exponents affect the shape of the graph is essential to accurately visualize the function.

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