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 $36 .$
$$
y=x^{1 / 3}+(x-1)^{1 / 3}
$$

Key Concepts

Vertical Tangents

Vertical tangents are lines where the curve's slope becomes infinite, meaning as the function approaches a certain point, the derivative grows without bound. This indicates a sudden, steep change in the graph that could be visually identified or proven by limit analysis.

Limit Calculations

Limit calculations analyze the behavior of a function as it approaches a particular point. They are essential for confirming the existence of vertical tangents because they help determine if the derivative approaches infinity or negative infinity at those points, thus substantiating the visual evidence from the graph.

Graphing Curves

Graphing curves involves plotting the behavior of functions to understand key features such as slope, continuity, intercepts, and critical points like vertical tangents. This process integrates qualitative visual insights with quantitative analysis, particularly when dealing with functions that include complex or fractional exponents.

Fractional Exponents

Fractional exponents denote roots of numbers and are pivotal in understanding the behavior of functions with radical components. They can introduce nuances in the function's continuity and differentiability, particularly near points where the expression inside the fractional power is zero or changes sign, making the analysis of tangents and other features more delicate.

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