Reading and Sketching the Graphs of f(x), f′ (x), f″(x)

A skill that you will need to master is the ability to sketch the graphs of $f\left(x\right),f^{\prime }\left(x\right),\mathrm{and}{f}^{\prime \prime }\left(x\right)$ given a graph of one of the others, either the $f\left(x\right),f^{\prime }\left(x\right),\mathrm{or }{f}^{\prime \prime }\left(x\right)$ graph. There are two main skills that you need to have. First, the ability to move down a level, reading the derivative of the given graph. From the $f\left(x\right)\mathit{ to\; the }{f}^{\prime }\left(x\right),\mathrm{or\; from }{f}^{\prime }\left(x\right)\mathrm{ }\mathit{to\; the }{f}^{\prime \prime }\left(x\right)$. Second, the ability to move up a level, reading that graph’s data to learn about the graph before it. From the $f^{\prime }\left(x\right)\mathit{ to\; the }f\left(x\right),\mathrm{or\; from} f^{\prime \prime }\left(x\right)\mathit{ to\; the }{f}^{\prime }\left(x\right)$. These are some the most fundamental pieces of AP Calculus and these skills will show up throughout the entirety of the exam from mechanical problems to conceptual problems.

As you do all this work keep in mind that the everything boils down to plus (+), minus (-), and zero (0). Let’s remind ourselves of what $f\left(x\right)\mathrm{ and }{f}^{\prime }\left(x\right)$represent, and also learn some new information about what $f^{\prime \prime }\left(x\right)$represents.