Tangent Line Calculator
Example: "1,0,-2" represents x² - 2
Tangent Line Calculator
Select a function type and enter the required parameters, then click Calculate to find the tangent line.
Example: Polynomial f(x) = x² + 2x - 3
At x = 2, the tangent line is y = 6x - 7
Example: Trigonometric f(x) = sin(x)
At x = π/4, the tangent line is y = 0.7071x + 0.0
Example: Exponential f(x) = e^x
At x = 0, the tangent line is y = x + 1
What is a Tangent Line?
A tangent line to a curve at a specific point is a straight line that touches the curve at that point and has the same slope as the curve at that point. It represents the instantaneous rate of change of the function at the given point.
The tangent line is a fundamental concept in calculus and is closely related to the derivative of a function. The slope of the tangent line at a point is equal to the derivative of the function evaluated at that point.
How to Find the Tangent Line
To find the equation of the tangent line to a function f(x) at a point x = a, follow these steps:
- Find the derivative of the function f'(x).
- Evaluate the derivative at x = a to find the slope of the tangent line: m = f'(a).
- Find the y-coordinate of the point on the curve: b = f(a).
- Use the point-slope form of the line equation: y - b = m(x - a).
- Simplify to the slope-intercept form: y = mx + (b - ma).
Applications of Tangent Lines
Tangent lines have numerous applications in mathematics, physics, engineering, and other fields:
- Linear approximation of functions near a point
- Optimization problems to find maximum and minimum values
- Physics for calculating instantaneous velocity and acceleration
- Engineering for analyzing curves and surfaces
- Computer graphics for rendering smooth curves
- Economics for marginal analysis