I get mixed up when graphing derivatives. If I'm given the graph of f how do I get f' and f"?
in general, you cannot. You can, however, tell where f' is positive or negative or zero, or where f" is positive or negative or zero.
Okay, what are the rules for determining positive, negative, or zero?
oh, come on.
f' is the slope of f. So, if f' > 0, f is increasing.
f" indicates concavity. If f" > 0, the graph of f is concave up.
Have you not read any of your text?
Graphing derivatives can be a bit tricky, but with the right approach, it becomes manageable. To find the derivatives of a function given its graph, follow these steps:
1. Understand the concept: Derivatives represent the rate of change of a function at any given point. This means that the slope of the tangent line to the graph of a function is equal to its derivative at that point.
2. Focus on a specific point: Choose a point on the graph where you want to determine the derivative(s). Make sure you are confident about the coordinates of this point.
3. Find the slope: To find the derivative at the chosen point, draw a tangent line to the graph at that point. The slope of this tangent line will represent the value of the derivative at that specific point. Choose two points close to the chosen point, preferably one on each side, and determine their coordinates.
4. Calculate the slope: Use the slope formula, which is the difference in the y-coordinates divided by the difference in the x-coordinates of the two points you selected. This will give you the slope of the tangent line, which is the value of the derivative at that point.
5. Repeat for more points: If you want to find the derivatives at multiple points, repeat steps 3 and 4 for each point. This will give you a collection of slopes that correspond to the respective points.
6. Repeat the process for the second derivative: To find the second derivative, follow the same steps as above, but this time, calculate the slopes of the tangent lines to the first derivative graph. These slopes will represent the values of the second derivative.
Once you have determined the slopes or values of the derivatives at different points, you can plot them to obtain the graphs of f' and f". Note that the graph of f' will give you information about the rate at which f is changing, while the graph of f" will provide insights into the rate at which the rate of change of f is changing.