Tangent Line Calculator

How to Use This Calculator

1. Enter the function f(x) (e.g. x^2+sin(x)).
2. Enter the point x₀.
3. Get the tangent line equation, slope m=f'(x₀), and y₀=f(x₀).
4. Use the Normal Line tab for the perpendicular line.
5. Use Linear Approximation to estimate f(x) near x₀ and check the error.

Formula

Example

Frequently Asked Questions

• What is the tangent line to a curve? ▼
  The tangent line at point (x₀, f(x₀)) has slope m=f'(x₀) and equation y−f(x₀)=f'(x₀)(x−x₀). It is the best linear approximation to the curve near x₀.
• What is the normal line? ▼
  The normal line is perpendicular to the tangent line at the same point. Its slope is −1/m (negative reciprocal of the tangent slope), so its equation is y−f(x₀) = (−1/m)(x−x₀).
• What is linear approximation? ▼
  Linear approximation (or linearization) uses the tangent line L(x) = f(x₀) + f'(x₀)(x−x₀) to estimate f(x) for x near x₀. For example, sin(0.1) ≈ 0 + 1·(0.1) = 0.1 (exact: 0.0998).
• How do I find the x-intercept of the tangent line? ▼
  Set y=0: 0 = f(x₀) + f'(x₀)(x−x₀), so x = x₀ − f(x₀)/f'(x₀). This is one step of Newton's method for root-finding.
• What is the y-intercept of the tangent line? ▼
  Set x=0: y = f(x₀) − f'(x₀)·x₀. The tangent line in slope-intercept form is y = f'(x₀)·x + [f(x₀)−f'(x₀)·x₀].

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