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Slope of a Tangent Line:
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The slope of a tangent line represents the instantaneous rate of change of a function at a specific point. It is calculated as the derivative of the function at that point, which geometrically corresponds to the steepness of the tangent line touching the curve.
The calculator uses the fundamental slope formula:
Where:
Explanation: The slope represents how much the function changes vertically (dy) for a given horizontal change (dx) at the point of tangency.
Details: Calculating the slope of a tangent line is essential in calculus for determining instantaneous rates of change, optimization problems, and understanding the behavior of functions at specific points.
Tips: Enter the change in y (dy) and change in x (dx) values. Ensure dx is not zero as division by zero is undefined. The result represents the slope at the point of interest.
Q1: What does a positive slope indicate?
A: A positive slope indicates that the function is increasing at that point - as x increases, y also increases.
Q2: What does a negative slope indicate?
A: A negative slope indicates that the function is decreasing at that point - as x increases, y decreases.
Q3: What does a slope of zero mean?
A: A slope of zero indicates a horizontal tangent line, meaning the function has a critical point (maximum, minimum, or inflection point) at that location.
Q4: How is this different from average slope?
A: The tangent slope represents instantaneous rate of change at a single point, while average slope represents the rate of change over an interval between two points.
Q5: Can the slope be undefined?
A: Yes, when dx = 0, the slope is undefined, which typically occurs at vertical tangent lines or points where the function is not differentiable.