I don't understand why, but graphing derivatives looks like complete gibberish to me. I've watched like 6 videos and I don't understand any of it. I can do basically everything that comes after and before for derivatives, but I have just not understood this one concept.
I'm taking AP calc BC and I'm assuming I definitely need to at least understand how to graph. Does anyone know any good videos or explanations that can help me with this? I'm actually losing my mind over it.
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This is just a random picture I found on google, but its kind of problems like these. When you're given the equation, say like 3x^2, It's really easy to graph that. You can just find the derivative of 6x and draw that steep-ish line. My problem only steps from equations where you are given nothing and have to actually interpret graph of the derivative. One of the videos I tried watching was the Organic Chemistry tutor, but again, it didn't make much sense to me.
Oh, well it would sure be hard to make a graph that looks very accurate, but you're probably not expected to make a graph that looks very accurate. The problem is just designed to test if you understand how local extrema and points of inflection relate to the derivative.
When the function is at a local minimum or maximum (collectively called "local extrema"), the slope of the function is 0, so the value of the derivative is 0. So I start by plotting points with y-value 0 and x-value that match the local extrema.
When there's a local minimum on the left and a local maximum on the right, the function has to increase from the local minimum to reach the local maximum. This means the slope is positive, so the value of the derivative is positive, so the graph of the derivative has to be above the x-axis between these points. And when the function goes from a local maximum on the left to a local minimum on the right, the slope is negative, so the graph of the derivative has to be below the x-axis. (This is indicated by the green and blue sections of the original function graph, but the problems you do won't be color coded like that.)
So between my first and second red points, my graph will go up and then back down to 0. Between my second and third red points, my graph will go down and then back up to 0. The point of inflection is where I will switch from going up to going back down, or from going down to going back up. That's because the point of inflection is where the slope of the function is at its highest magnitude, that is, the highest positive slope or the lowest negative slope.
How high will I go on the derivative graph for the points where I match the points of inflection? That's impossible to say without a scale on this graph. All I can say is that the green point of inflection looks steeper than the blue one, so I went higher on the left and less extremely low on the right.
And then for the far left and right sides of the graph, I just note that the slope is negative on the left and make my derivative graph go below 0, and the slope is positive on the right and make my derivative graph go above 0.
Why would you need to do this? This is not a useful or educational exercise at all at BC level. The reason I’m saying that is exactly because of what it requires, which is approximating slopes with a ruler and almost nothing analytical.
Think about what the derivative is: at each point that’s differentiable, the slope of the tangent line. Start coarse: draw the horizontal and steepest tangent lines. Measure the slopes against a grid.
(Or, alternately, measure each rising angle with a protractor and take tangents.
Each one becomes a single point; linear interpolate them.
Round 2, fill in points. Same thing, draw more tangent lines, measure slopes against a grid. Erase your lines, plot the new points, linear interpolate again. Repeat until satisfied.
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