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Second Derivatives and Beyond

Second Derivatives and Beyond

Finding Points

There are three types of points to find on the function, and the great thing is that you already know how to find all of them.

  • Intercepts: To find the y-intercept of a function, plug x = 0 into the function and see what we find. To find the x-intercepts, also known as roots, we set the function equal to zero and solve. Sometimes we'll also find vertical asymptotes in this step (find where f is undefined).
      
  • Critical points: To find the critical points, set '(x) = 0 (or undefined) and solve.
      
  • Inflection points: To find the inflection points, set "(x) = 0 (or undefined), solve, and check each solution to see if it's a real inflection point.

In summary, we're finding where f, ', and " are zero or undefined. These will mostly be dots, but there may be asymptotes or holes where f is undefined.

Here's the only thing you need to do that we didn't do earlier: after finding the x-value of a CP or IP, plug that x-value back into the original function f to find the corresponding y-value. In order to graph a point, we need to know both coordinates.

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