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Tangent Line Equation:
where \( m = \frac{dy}{dx} \) at \( x_0 \)
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A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. It represents the instantaneous rate of change of the function at that specific point, which is the derivative at that point.
The calculator uses the point-slope form of a line equation:
Where:
Explanation: The equation gives the straight line that passes through (\( x_0 \), \( y_0 \)) with slope \( m \), which matches the curve's slope at that point.
Details: Tangent lines are fundamental in calculus for understanding derivatives, approximating functions, and solving optimization problems. They have applications in physics, engineering, and economics.
Tips: Enter the point of tangency (\( x_0 \), \( y_0 \)) and the slope \( m \) (which you can find by calculating the derivative at \( x_0 \)). The calculator will provide both the point-slope and slope-intercept forms.
Q1: How do I find the slope for the tangent line?
A: The slope \( m \) is the derivative of the function evaluated at \( x_0 \). You'll need to calculate this separately based on your function.
Q2: Can this calculator find the derivative for me?
A: No, this calculator only constructs the tangent line equation once you provide the slope (derivative value).
Q3: What's the difference between tangent and secant lines?
A: A secant line connects two points on a curve, while a tangent line touches at exactly one point with matching slope.
Q4: Can a tangent line intersect the curve at other points?
A: Yes, for many functions the tangent line will intersect the curve elsewhere, but it only "touches" at the point of tangency.
Q5: How is this used in real-world applications?
A: Tangent lines are used for linear approximations, calculating velocities, determining marginal costs in economics, and more.