(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}{x+1}$ at (0,0)$

(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}{x+1}$ at (0,0)$

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Key Concepts

Horizontal Tangent Lines

A horizontal tangent line occurs at a point on a function where the derivative is zero, indicating that the function has no instantaneous rate of change at that point. Identifying horizontal tangents involves setting the derivative equal to zero, which can also help in finding local extrema.

Graphical Representation of Functions and Tangents

Graphing the function along with its tangent lines involves plotting the original function and then overlaying the tangent line(s) at the given point(s). This visual representation aids in understanding the behavior of the function, its rate of change, and the points where the function has horizontal or other special tangent lines.

Derivative

The derivative of a function at a point is a fundamental concept in calculus that measures the rate at which the function value changes with respect to changes in the independent variable. It provides the slope of the tangent line to the function’s graph at that point, capturing the idea of instantaneous change.

Point-Slope Form

The point-slope form of a linear equation is used to express the equation of a straight line when a point on the line and its slope are known. This form is particularly useful in tangent line problems, where the slope is given by the derivative at the point of tangency.

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