when trying to find the intervals of where the derivative is positive is it the intervals of where the original graph is above the x axis?
no, it is where the graph is rising.
positive derivative means a positive slope.
It has nothing to do with where the graph is in relation to the x-axis. y can be negative, but still increasing when x increases.
Well, if we're talking about finding the intervals where the derivative is positive, it's actually a little different than just looking at where the original graph is above the x-axis. The derivative tells us how the original function is changing, so when the derivative is positive, it means the function is increasing.
So, to find the intervals where the derivative is positive, you're actually looking for where the function is increasing. This means you'll need to check where the original graph is going uphill, so to speak. It may be above or below the x-axis, depending on the shape of the graph.
Now, if you're trying to find intervals where the original graph is above the x-axis, that can be a separate question altogether. That's all about investigating the behavior of the function itself, not necessarily its derivative. But hey, nothing wrong with a little tangent, right?
Yes, when trying to find the intervals where the derivative is positive, you are essentially looking for the intervals where the original graph is above the x-axis. This is because the derivative represents the rate of change of the function, and a positive derivative indicates that the function is increasing in those intervals. In other words, if the original graph is above the x-axis, it means that the function is positive, and its derivative will be positive in those intervals.
When trying to find the intervals where the derivative is positive for a function, it is not necessarily the same as looking for the intervals where the original graph is above the x-axis.
To determine the intervals where the derivative is positive, you need to look for the regions where the original graph is increasing. Here's how you can do it:
1. Differentiate the original function to find its derivative.
2. Set the derivative equal to zero and solve for x to find the critical points.
3. Create a number line and plot the critical points on it.
4. Choose a test point from each interval between the critical points and evaluate the sign of the derivative at that point.
- If the derivative is positive, it means the original function is increasing.
- If the derivative is negative, it means the original function is decreasing.
5. Based on the signs of the derivative in each interval, you can determine the intervals where the derivative is positive.
It's important to note that if the original function is above the x-axis, it doesn't necessarily mean that the derivative is positive throughout those intervals. The derivative can be positive or negative depending on the behavior of the function.
So, to find the intervals where the derivative is positive, you must analyze the behavior of the derivative itself, not solely based on the position of the original function relative to the x-axis.