Slopes of tangent lines For the following functions, make a...

Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.
$$f(x)=x^{3}-x \text { at } x=1$$

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Key Concepts

Secant Line

A secant line is a line that intersects a curve at two distinct points. It represents the average rate of change of the function between those two points and serves as the starting point for understanding more refined approximations of change.

Difference Quotient

The difference quotient is a formula that computes the average rate of change of a function over an interval, generally expressed as [f(a+h) - f(a)]/h. It is a fundamental concept that leads to the definition of the derivative as h approaches zero.

Limit Process

The limit process in calculus involves finding the value that a function approaches as the input gets arbitrarily close to a specified point. This is crucial when transitioning from the secant line’s average rate of change to the tangent line’s instantaneous rate of change.

Tangent Line

The tangent line is a line that touches a curve at a single point and has the same instantaneous direction (slope) as the curve at that point. It is directly linked to the derivative of the function, providing a local linear approximation.

Derivative

The derivative is a measure of the instantaneous rate of change of a function with respect to its variable. It is defined as the limit of the difference quotient as the interval shrinks to zero and provides the exact slope of the tangent line at a given point.

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