Enter the function, the point of tangency, and the function's derivative at that point to find the equation of the tangent line.
Tangent Line Equation:
Enter values to calculate.
Understanding the Equation of a Tangent Line
The equation of a tangent line to a curve at a specific point is a fundamental concept in calculus. It represents the straight line that "just touches" the curve at that single point, sharing the same slope as the curve at that exact location.
The Math Behind It
To find the equation of a tangent line, we use the point-slope form of a linear equation, which is:
y - y₀ = m(x - x₀)
(x₀, y₀): This is the point on the curve where the tangent line touches.
m: This is the slope of the tangent line at that point. In calculus, the slope of the tangent line to a function f(x) at x = x₀ is given by the value of its derivative, f'(x₀).
Therefore, the equation of the tangent line to f(x) at the point (x₀, y₀) is:
y - y₀ = f'(x₀)(x - x₀)
This calculator assumes you have already found the derivative of your function and evaluated it at the specific x-coordinate x₀ to get f'(x₀). If you need to calculate the derivative itself, that typically involves applying differentiation rules to the function f(x).
How This Calculator Works
This calculator simplifies the process of finding the tangent line equation once you have the necessary components:
Function f(x): While not directly used in the final point-slope calculation, it's provided for context.
Point of Tangency (x₀, y₀): You input the x and y coordinates of the point where the tangent line touches the curve.
Derivative at the Point f'(x₀): You input the value of the derivative of the function evaluated at x₀. This is the crucial slope value.
The calculator then plugs these values into the point-slope formula and rearranges it into the standard slope-intercept form (y = mx + b) if possible, or keeps it in the point-slope form for clarity.
Use Cases
Approximation: Tangent lines provide a linear approximation of a function near a point, useful in numerical methods.
Optimization: Finding where the derivative (slope) is zero (i.e., a horizontal tangent line) is key to finding local maximums and minimums of a function.
Physics and Engineering: Understanding instantaneous rates of change, such as velocity (derivative of position) or acceleration.
Geometry: Analyzing the curvature and behavior of curves.
Example: If you have the function f(x) = x², and you want the tangent line at the point (2, 4). The derivative is f'(x) = 2x. Evaluating the derivative at x₀ = 2 gives f'(2) = 2 * 2 = 4. So, x₀ = 2, y₀ = 4, and m = 4. The equation is y - 4 = 4(x - 2), which simplifies to y = 4x - 4.
function calculateTangentLine() {
var x0 = parseFloat(document.getElementById("xValue").value);
var y0 = parseFloat(document.getElementById("yValue").value);
var derivativeAtX0 = parseFloat(document.getElementById("derivativeAtX").value);
var functionInput = document.getElementById("functionInput").value; // Not used in calculation but kept for context
var tangentLineEquationElement = document.getElementById("tangentLineEquation");
if (isNaN(x0) || isNaN(y0) || isNaN(derivativeAtX0)) {
tangentLineEquationElement.textContent = "Please enter valid numbers for all inputs.";
tangentLineEquationElement.style.color = "#dc3545";
return;
}
var m = derivativeAtX0;
var equationString = "";
if (m === 0) {
// Horizontal tangent line: y = y0
equationString = "y = " + y0.toFixed(4);
} else {
// General case: y – y0 = m(x – x0)
// Rearrange to slope-intercept form: y = mx + (y0 – m*x0)
var b = y0 – m * x0;
var formattedM = m.toFixed(4);
var formattedB = b.toFixed(4);
if (b >= 0) {
equationString = "y = " + formattedM + "x + " + formattedB;
} else {
equationString = "y = " + formattedM + "x – " + Math.abs(b).toFixed(4);
}
// Optionally, also show point-slope form
// equationString = "y – " + y0.toFixed(4) + " = " + m.toFixed(4) + "(x – " + x0.toFixed(4) + ")";
}
tangentLineEquationElement.textContent = equationString;
tangentLineEquationElement.style.color = "#0056b3"; // Reset color on successful calculation
}