For each function f and value x=c, use a sequence of...

For each function $f$ and value $x=c$, use a sequence of approximations to estimate $f^{\prime}(c)$. Illustrate your work with an appropriate sequence of graphs of secant lines.
$$
f(x)=|x-1|, c=3
$$

Key Concepts

Piecewise Functions and Differentiability

For functions defined piecewise, such as those involving absolute values, different rules apply in different parts of the domain. Understanding the behavior of the function in each interval is crucial. In the case of the function f(x) = |x-1|, it simplifies to a linear expression on either side of the point where the expression inside the absolute value changes sign, ensuring differentiability at points away from that transition. At x = 3, the function behaves linearly, making the computation of the derivative straightforward.

Derivative as a Limit

The derivative at a point is defined as the limit of the difference quotient as the interval approaches zero. This concept involves calculating the limit of [f(c + h) - f(c)]/h as h tends to zero, providing the instantaneous rate of change or the slope of the tangent line at that point.

Difference Quotient

This expression, [f(c + h) - f(c)]/h, represents the average rate of change of the function over an interval. By taking values of h that get smaller and smaller, the difference quotient approaches the precise slope of the tangent line at c, which is the derivative at that point.

Secant Lines

Secant lines are lines that pass through two points on the graph of a function. By choosing one of the points to be the point of interest, c, and the other to be c+h, the slope of the secant line gives an approximation of the function's derivative. As h becomes smaller, the secant line approaches the tangent line at c.

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