To find the slope of a curve at a specific point, you need to use calculus, specifically the concept of derivatives. The slope of a curve at a given point is defined as the derivative of the function at that point.
Here are the steps to determine the slope:
- Identify the Function: First, you need to have the equation of the curve, usually represented as a function, f(x).
- Differentiate the Function: Take the derivative of the function, which will give you a new function, f'(x), representing the slope of the curve at any point x.
- Plug in the Point: Substitute the x-coordinate of the given point into the derivative f'(x). This will yield the slope of the curve at that specific point.
Example: If you have a curve represented by the function f(x) = x², to find the slope at the point where x = 3:
- First, differentiate: f'(x) = 2x.
- Substitute x = 3 into the derivative: f'(3) = 2(3) = 6.
This means the slope of the curve at the point (3, 9) is 6. In other words, at x = 3, the curve is rising at a rate of 6 units vertically for every 1 unit it moves horizontally.