Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
The tangent problem involves finding the slope of the tangent line to a curve at a given point. This is foundational for understanding derivatives in calculus.
5 Must Know Facts For Your Next Test
The tangent line to a curve at a point is the line that just touches the curve at that point and has the same slope as the curve does at that point.
The slope of this tangent line can be found using the limit definition: $m = \lim_{{h \to 0}} \frac{{f(a+h) - f(a)}}{h}$ where $f$ is the function and $a$ is the point of tangency.
A key step in solving tangent problems involves simplifying complex fractions or expressions before taking limits.
Understanding how to differentiate basic functions like polynomials, trigonometric functions, and exponentials helps solve tangent problems more quickly.
The concept of instantaneous rate of change, which is central to calculus, originates from solving tangent problems.
Review Questions
Related terms
Derivative: The derivative represents the rate at which a function is changing at any given point and can be interpreted as the slope of the tangent line at that point.
Secant Line: A secant line intersects a curve at two or more points, and its slope can be considered an average rate of change between those points.
Instantaneous Rate of Change: This measures how fast a quantity changes instantaneously, equivalent to finding the derivative or slope of the tangent line at a specific point on a curve.