it appears that the concavity column was answered based on the graph itself, since those results aren’t really consistent with the students’ answers about f”. in general it seems that the student had trouble remembering for the duration of the problem that the graph was of y=f'(x), not y=f(x). I might ask this student to add an additional column just about f'(x) itself and try filling those out first, in order to build on what seems to be good understanding of graph-reading (one column: f'(x) pos or negative – probe a bit on what is happening *at* “a”; another column: f(‘x) increasing or decreasing) and *then* have the student go back and fill in the f(x) and f”(x) columns with questions like “if you see that f'(x) [a function] is increasing here, what does that tell you about f”(x) [its’ derivative]?” or “what does the sign of f'(x) tell us about f(x)?” again calling attention to which graph is given and which are trying to be pictured/understood.
The student understands inc/dec on an interval, but not at a point.
It is pretty intuitive to ask if a function is increasing on an interval. It is confusing to ask if a function is increasing at a point. The natural reaction is, “what do you mean increasing, it is only one point, it isn’t increasing or decreasing”. I would re-state that to see if a function is increasing at a point, you need to imagine “zooming in” around that point and see if that section slopes steadily upward.
I don’t think they know how to get convavity. The usual mistake is to assume the concavity of f is the same as for the f ‘ graph, but they didn’t consistently apply that method. This is a tricky idea. I keep going back to: What does it mean for f ” to be positive? It means f ‘ is increasing. Or the same thing in terms of motion or a bucket of water or whatever. And, lots of practice matching and/or sketching.
They drew a curve that doesn’t match the table in any way. Seems likely they had a reason for picking b as an x-intercept – not sure why. I wonder if they mistakenly believe there is some sort of obvious connection between the shape of f & f ‘ and were not really paying attention to the table as the sketched. I would ask them to do the best they can just using line segments and looking at the inc/dec column. Then think about rounding out the line segments to get the right concavity.
I think that the student was trying to plot the derivative of the given function (rather than its antiderivative). With that interpretation, they did OK until somewhere between b and c.
That would explain why they seem convinced that point b is an x-intercept.
secretseasons, that’s a wonderful thing to notice, and I had missed it on my look at this problem. I completely agree that the graph the student sketched is of the derivative of the given graph. Maybe the problem is our notation, it’s way too easy to miss a little ‘ here or there, and so the names of all these graphs look really similar. Maybe the problem is the layout of the table, where it goes from f’ to f to f” so it seems natural to reverse the first two. Or maybe the student is just clueless about the difference between a derivative and an antiderivative.
It still leaves me with the mystery of how they got the concavity column, though.
Joshua – the concavity column looks to me to be an ~accurate description of the concavity of the given graph. In fact, the instructions at the top of that column don’t seem to specify *which* function’s concavity is in question.
Joshua – I agree that secret is on to something about the derivative, however that does not explain why there is no x-intercept at x = c here. What jumped out first to me is the symmetry in the graph that was sketched. My first thought was that I was pleased to see a sense of symmetry inherent in the sketch but the more I think about it the more I believe that this is just a bit of laziness/guessing on the part of the student. The column confusion that has been pointed out already just reinforces how difficult it is for students to flow from one representation to another (graphs to formulas to tables to written descriptions) and as much as we talk about the rule of four we need to emphasize it over and over again to help shepherd our students through this forest of ideas. To help diagnose this student’s work I would concentrate first on the graph and ask secret’s question and then prompt the student to try the sketch again independent of the table. Next, turn our attention to the table and try to sketch a graph that behaves according to the table. Examination of the differences might go a long way to curing this problem.
6 replies on “Finding f(x) from f'(x)”
it appears that the concavity column was answered based on the graph itself, since those results aren’t really consistent with the students’ answers about f”. in general it seems that the student had trouble remembering for the duration of the problem that the graph was of y=f'(x), not y=f(x). I might ask this student to add an additional column just about f'(x) itself and try filling those out first, in order to build on what seems to be good understanding of graph-reading (one column: f'(x) pos or negative – probe a bit on what is happening *at* “a”; another column: f(‘x) increasing or decreasing) and *then* have the student go back and fill in the f(x) and f”(x) columns with questions like “if you see that f'(x) [a function] is increasing here, what does that tell you about f”(x) [its’ derivative]?” or “what does the sign of f'(x) tell us about f(x)?” again calling attention to which graph is given and which are trying to be pictured/understood.
The student understands inc/dec on an interval, but not at a point.
It is pretty intuitive to ask if a function is increasing on an interval. It is confusing to ask if a function is increasing at a point. The natural reaction is, “what do you mean increasing, it is only one point, it isn’t increasing or decreasing”. I would re-state that to see if a function is increasing at a point, you need to imagine “zooming in” around that point and see if that section slopes steadily upward.
I don’t think they know how to get convavity. The usual mistake is to assume the concavity of f is the same as for the f ‘ graph, but they didn’t consistently apply that method. This is a tricky idea. I keep going back to: What does it mean for f ” to be positive? It means f ‘ is increasing. Or the same thing in terms of motion or a bucket of water or whatever. And, lots of practice matching and/or sketching.
They drew a curve that doesn’t match the table in any way. Seems likely they had a reason for picking b as an x-intercept – not sure why. I wonder if they mistakenly believe there is some sort of obvious connection between the shape of f & f ‘ and were not really paying attention to the table as the sketched. I would ask them to do the best they can just using line segments and looking at the inc/dec column. Then think about rounding out the line segments to get the right concavity.
I think that the student was trying to plot the derivative of the given function (rather than its antiderivative). With that interpretation, they did OK until somewhere between b and c.
That would explain why they seem convinced that point b is an x-intercept.
secretseasons, that’s a wonderful thing to notice, and I had missed it on my look at this problem. I completely agree that the graph the student sketched is of the derivative of the given graph. Maybe the problem is our notation, it’s way too easy to miss a little ‘ here or there, and so the names of all these graphs look really similar. Maybe the problem is the layout of the table, where it goes from f’ to f to f” so it seems natural to reverse the first two. Or maybe the student is just clueless about the difference between a derivative and an antiderivative.
It still leaves me with the mystery of how they got the concavity column, though.
Joshua – the concavity column looks to me to be an ~accurate description of the concavity of the given graph. In fact, the instructions at the top of that column don’t seem to specify *which* function’s concavity is in question.
Joshua – I agree that secret is on to something about the derivative, however that does not explain why there is no x-intercept at x = c here. What jumped out first to me is the symmetry in the graph that was sketched. My first thought was that I was pleased to see a sense of symmetry inherent in the sketch but the more I think about it the more I believe that this is just a bit of laziness/guessing on the part of the student. The column confusion that has been pointed out already just reinforces how difficult it is for students to flow from one representation to another (graphs to formulas to tables to written descriptions) and as much as we talk about the rule of four we need to emphasize it over and over again to help shepherd our students through this forest of ideas. To help diagnose this student’s work I would concentrate first on the graph and ask secret’s question and then prompt the student to try the sketch again independent of the table. Next, turn our attention to the table and try to sketch a graph that behaves according to the table. Examination of the differences might go a long way to curing this problem.