The slope of a graph at a specific point can be found using derivatives. Here's how:
Understanding Derivatives and Slope
Derivatives, denoted as f'(x), represent the instantaneous rate of change of a function f(x). Geometrically, this rate of change corresponds to the slope of the tangent line at any given point on the graph of f(x). The reference states, "If f'(x) is the derivative of f(x) , input the x value of the point to f'(x)".
Steps to Find the Slope
To find the slope at a specific point:
- Find the derivative: Determine the derivative of the function, f'(x).
- Identify the x-value: Determine the x-coordinate of the specific point on the graph where you wish to find the slope.
- Evaluate the derivative: Substitute the identified x-value into the derivative, f'(x). The result will be the slope of the graph at that particular point.
Example
Let's illustrate this with the example from the reference where f(x) = x2:
| Step | Description | Example |
|---|---|---|
| 1. Function | Original function | f(x) = x2 |
| 2. Derivative | Calculate the derivative of the function | f'(x) = 2x |
| 3. Point | Specific point of interest where x = 3. | (3, 9) |
| 4. Substitution | Input the x-value into the derivative function | f'(3) = 2 * 3 |
| 5. Slope | Calculation of slope | f'(3) = 6 |
Therefore, the slope of the function f(x) = x2 at the point (3, 9) is 6. As the reference states, "So at (3,9) the function is sloping upwards at 6 units".
Practical Insights
- Positive Slope: A positive derivative (f'(x) > 0) indicates the function is increasing, or sloping upwards, at that point.
- Negative Slope: A negative derivative (f'(x) < 0) indicates the function is decreasing, or sloping downwards, at that point.
- Zero Slope: A derivative of zero (f'(x) = 0) signifies a horizontal tangent, typically at a maximum or minimum point on the graph.
Table Summary
| Concept | Description |
|---|---|
| Derivative f'(x) | Represents the instantaneous rate of change or slope of a function f(x) at a point. |
| Finding the Slope | Substitute the x-coordinate of a point into f'(x) to obtain the slope at that point. |
By using these steps, you can accurately determine the slope of a graph at any given point using derivatives.
