Consider the function f(x)=(x-1)^1 / 3. (a) Compute the...

Consider the function $f(x)=(x-1)^{1 / 3}$. (a) Compute the slope of the secant lines between (1,0) and $(x, f(x))$ for $x=0.9,0.99,0.9999$ and for $x=1.1,1.01,1.0001$. (b) Discuss what the secant-line slopes in (a) suggest happens to the tangent line at 1 . (c) GU Plot the graph of $f$ near $x=1$ and verify your observation from (b).

Consider the function $f(x)=(x-1)^{1 / 3}$.
(a) Compute the slope of the secant lines between (1,0) and $(x, f(x))$ for $x=0.9,0.99,0.9999$ and for $x=1.1,1.01,1.0001$.
(b) Discuss what the secant-line slopes in (a) suggest happens to the tangent line at 1 .
(c) GU Plot the graph of $f$ near $x=1$ and verify your observation from (b).

Key Concepts

Secant Lines

Secant lines are straight lines that connect two points on the graph of a function. They represent the average rate of change of the function between those two points and are calculated using the difference quotient. Secant lines are central in understanding the behavior of a function as they provide an approximation of the slope of the tangent line when the points become very close to each other.

Difference Quotient

The difference quotient is an expression of the form (f(x) - f(a))/(x - a) and serves as a measure for the average rate of change of a function over an interval. As the interval shrinks (i.e., as x approaches a), the difference quotient tends to the instantaneous rate of change, which is defined as the derivative.

Limits

Limits are a fundamental concept in calculus used to describe the behavior of a function as its input approaches a specific value. In the context of finding the slope of a tangent line, limits help formalize the idea of letting the two points on the secant line come arbitrarily close, leading to the definition of the derivative.

Derivatives

The derivative of a function at a point is the limit of the difference quotient as the interval between the two points shrinks. It represents the instantaneous rate of change of the function and gives the slope of the tangent line at that particular point. Derivatives are crucial for understanding the local behavior of functions.

Vertical Tangents

A vertical tangent occurs at a point on a graph where the slope of the tangent line becomes infinite or undefined. This means that as the points used in the difference quotient approach the point of tangency, the difference quotient increases without bound. Identifying vertical tangents is important in the study of function behavior and differentiability.

Graphical Analysis

Graphical analysis involves interpreting and analyzing the graph of a function to understand its behavior at and around specific points. It is used to visually verify results obtained analytically, such as the presence of a vertical tangent, by observing how the steepness of the graph changes as it approaches the point of interest.

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