Calculus Week #5
So to cut it rather short, this week we learned and reviewed a few things and then near the end, we got a study guide and multiple choice and free response take home test.
I don't know if anyone else has looked at this stuff but it's pretty intense if you ask me.
One thing I want to cover in this blog is a few relationships between the graphs of the original, first derivative, and second derivative that I don't think everyone grasped exactly.
If I were to give you the graph of f(x) and ask "What are the zeros of f'(x)," what would you do?
Well what you should do is first recognize what we have been doing. For the longest time now we have been focusing on two tests: the First Derivative Test and the Second Derivative Test. In the first derivative test (what this question is asking about) we take the derivative and set it equal to 0. The 0 is important because that is the key to how we solve this question. It asks for the zeros of the first derivative, right? So when you do the first derivative test, you set it equal to 0 to solve for critical values or possible max's and min's, correct? So for the problem I introduced, you would take a look at the max's and min's of the original and those x-values would be where your zeroes of the derivative function would be. (By the way, the zeroes, the x-intercepts, and roots are all the same: it's where the function crosses the x-axis.)
Likewise, suppose I gave you the graph of the original and asked "What are the zeros of f''(x)," what would you do? Well looking back to what we did before, we would look at the second derivative. When we take the second derivative and set equal to 0, what are we finding? We are finding possible points of inflection, right? Points of inflection are simply where the graph changes concavity, right? So if you look at the original graph and identify where the concavity changes (like an umbrella to a bowl), at that x-value is a zero of the second derivative.
Use both of these explanations in your multiple choice to be able to rule out answers rather easily.
And a question for some of you to perhaps answer:
If I show you a graph of the first derivative, what are the max's and min's of this graph in reference to the graph of the second derivative?
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the way i think of this is lets say the graph of the first derivative shows x^3, well we know that the second derivative will be x^2, so knowing that there will only be one max or one min, it all depends on which way the graph is facing, up or down. Hope this helps
ReplyDeleteWell mainly you have to know the graph of the second derivative or the equation of the graph for your first derivative then use the second derivative test to find your max and mins.
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