Hello Mrs. Carlson! This is Jennifer Oh :)
When are you coming back?
It's hard to learn things thoroughly without you here to lead us.
I hope you can come back soon.
And, is there an easier way to sketch the graphs of f' and f''? Or do we just have to work our way through step by step?
How do you draw the sign charts of f' and f''? What numbers are the points for the sign charts: the zeros, undefined points?
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Since you are given the graph of f, the sign chart for f' is determined by the slope of the graph you are looking at. The sign chart for f'' is the concavity of that same graph.
From there, you need to remember that f'' is going to be the SLOPE of your graph for f'. That's about all you can do for the graph of f'--that is to identify if it's positive (above the axis), negative (below the x-axis), zero (because f had a horizontal tangent line there), and whether or not it will have increasing or decreasing slope. You cannot determine the concavity of the graph of f', because that would mean you would have to know what f''' is.
For the graph of f'', all you can determine is whether or not it's positive, negative, or zero. You cannot make a clear determination on whether it's increasing, decreasing, concave up, or concave down.
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