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Tangent Line Calculator

Select the type of parabola equation, enter its values along with a point, and the calculator will instantly determine the tangent line equation at the specified point.

Choose Type

Enter a function f(x):

Enter a point (x₀):

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The Online Tangent Line Calculator is designed to determine the equation of a tangent line to a curve at a specified point. It works for explicit, implicit, parametric, and polar functions. The calculator can also identify horizontal and vertical tangent lines automatically.

What is a Tangent Line?

A tangent line is a straight line that touches a curve at only one point without intersecting it at that location. This touching point is known as the point of tangency. At that exact point, the tangent shares the same slope as the curve. Tangent Line Calculator

Tangent Line Formula

The general equation of a tangent line passing through a point \((x_1, y_1)\) with slope \(m\) is:

\[ y - y_1 = m(x - x_1) \]

Where:

\( (x_1, y_1) \) represents the tangency point
\( m \) represents the slope (derivative at that point)

Example 1: Tangent to a Parabola

Find the tangent line to the parabola \(x^2 = 16y\) at the point (4, 4).

Solution:

1. Differentiate implicitly with respect to y:

\[ 2x \frac{dx}{dy} = 16 \quad \Rightarrow \quad \frac{dx}{dy} = \frac{8}{x} \]

2. Evaluate slope at (4, 4):

\[ m = \frac{8}{4} = 2 \]

3. Form tangent equation:

\[ x - x_1 = m(y - y_1) \]

\[ x - 4 = 2(y - 4) \]

\[ x - 4 = 2y - 8 \quad \Rightarrow \quad x - 2y + 4 = 0 \]

The tangent line calculator can compute these steps instantly once the equation and point are entered.

Example 2: Tangent Line at a Point

Find the tangent line for \( f(x) = 3x^2 + 4x - 1 \) at \( x = 2 \).

Solution:

1. Evaluate the function at x = 2:

\[ f(2) = 3(2)^2 + 4(2) - 1 = 12 + 8 - 1 = 19 \]

2. Differentiate the function:

\[ f'(x) = 6x + 4 \quad \Rightarrow \quad f'(2) = 16 \]

3. Apply tangent formula:

\[ y - 19 = 16(x - 2) \]

\[ y = 16x - 32 + 19 \]

\[ y = 16x - 13 \]

How Tangent Line Calculator Works

Input:

Choose the curve type (explicit, implicit, parametric, or polar).
Enter the function expression.
Provide the coordinate point.
Press the calculate button.

Output:

Displays the entered function.
Computes the derivative step-by-step.
Shows slope at the given point.
Provides the final tangent line equation.

FAQs

Why is a tangent line important?

A tangent line reveals the instantaneous rate of change of a function at a specific point. It helps in understanding curve behavior and motion analysis.

Is tangent slope always equal to the derivative?

Yes. The derivative evaluated at a point gives the exact slope of the tangent line at that location.

References

Wikipedia: Tangent line to a curve

Krista King Math: Tangent Line at a Point

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