(a) Find the slope of the tangent line for each function fat...

(a) Find the slope of the tangent line for each function fat the given point. (b) Find an equation of the tangent line to the graph of each function $f$ at the given point. (c) Find the points, if any, where the graph of the function has a horizontal tangent line. (d) Graph each function, the tangent line found in $(b),$ and any tangent lines found in $(\mathrm{c})$ on the same set of axes. $$f(x)=\frac{x^{3}}{x+1}$ at $\left(1, \frac{1}{2}\right)$$

(a) Find the slope of the tangent line for each function fat the given point.
(b) Find an equation of the tangent line to the graph of each function $f$ at the given point.
(c) Find the points, if any, where the graph of the function has a horizontal tangent line.
(d) Graph each function, the tangent line found in $(b),$ and any tangent lines found in $(\mathrm{c})$ on the same set of axes.
$$f(x)=\frac{x^{3}}{x+1}$ at $\left(1, \frac{1}{2}\right)$$

Key Concepts

Graphical Analysis

Graphing involves plotting both the function and its tangent lines to visually understand their relationships and behaviors. In this context, graphing helps in verifying the accuracy of the analytical solutions, such as the point of tangency and the horizontal tangents, by providing a visual representation of the function’s behavior and how it is interacted by its tangent lines.

Derivative

The derivative of a function measures its instantaneous rate of change and is central to finding the slope of the tangent line at any given point. In the context of this problem, the derivative provides the slope of the curve at a specified point, which is essential for determining the tangent line’s equation and for identifying points where the slope (rate of change) is zero.

Quotient Rule

The Quotient Rule is a method used to differentiate functions that are expressed as one function divided by another. Since the given function is a ratio of polynomials, the Quotient Rule is required to correctly compute its derivative, ensuring accuracy in finding slopes for the tangent line and for solving other related problems.

Tangent Line Equation

The tangent line to a curve at a specific point touches the curve without crossing it locally and has the same slope as the curve at that point. Knowing the slope (from the derivative) and the point of tangency allows us to write the linear equation of the tangent line, which approximates the function near the point of contact.

Horizontal Tangent Lines

Horizontal tangent lines occur at points where the derivative of the function is zero. These points indicate that the instantaneous rate of change of the function is zero, often corresponding to local extrema or inflection points. Identifying these points is useful in understanding the overall behavior and critical aspects of the function’s graph.

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