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.
$$f(x)=3 \cos x \quad \text { at } x=\pi / 2$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Tangent Line

A tangent line represents the instantaneous rate of change of a function at a specific point. It is the line that just touches the curve at that point without intersecting it nearby, providing an approximation of the function’s behavior over a very short interval.

Secant Line

A secant line is a line that connects two distinct points on a curve, and its slope represents the average rate of change of the function between those points. By selecting points increasingly close to the point of interest, the slope of the secant line approximates the slope of the tangent line.

Derivative

The derivative is a fundamental concept that captures the instantaneous rate of change of a function at a given point. It is defined as the limit of the slopes of secant lines as the two points converge, effectively becoming the slope of the tangent line.

Limit

The concept of a limit underpins calculus by describing the value that a function or sequence approaches as the input approaches a particular point. In the context of derivatives, limits are used to formalize the process of finding the instantaneous rate of change from average rates over shrinking intervals.

Transcript

Please give Ace some feedback