The First Derivative Test

                                                                                                                 Briya Kirksey
We have now started to learn about the derivative application #3 which is Curve Sketching. You can find this section in our calculus book labeled as "Connecting the Graphs of f ', f '', and f''' ". The first part of this section is "The First Derivative Test".

                           The First Derivative Test 

     What are we looking for?

• We are taking a functions equation and determining where it is  increasing and/ or decreasing, and its relative extrema (relative maximum/minimum) on its graph. 

   

  How do we do this?

  1.   Take the derivative of the given function (if its not already given). 
  2.   Set the derivative equal to zero. Why? - Because that is where the slopes of the functions graph changes signs (positive to negative or vise versa). The values we get by doing this is what we know as critical points. 
  3. Make an x-axis number line labeling all of the critical points on it. 
  4. Choose numbers on the number before, after, and in between the critical points, known as testing points. Example: If critical points are x= -1, 0, 3 , then my testing points could be x= -1.5, -.5, 2, 4. 
  5. Plug each of the testing points into the derivative. Why?- To determine the behavior of the functions graph in between its critical points. 
  6. The answers produced by doing step number 5 will give you the x intervals in which the graph is increasing/decreasing and which x values there may be a max/min.  

                 Tip: 

  • When plugging in testing points into a functions derivative, pay attention to the values sign to make it easier in determining whether the slope will be positive or negative.
   Example:  y' = x (x-3)^2   testing point : -1
  
   Without doing any math, one can already tell that the slope will end up being negative. The value in the parenthesis is -4, but the fact that it is then squared makes it positive. However the x value  (-1) being multiplied to the positive value makes the slope  negative.   

  Concavity : Refers to the way Graphs look

  This is the worksheet and notes used to learn about Concavity. Concave up= a u-shape facing up. Concave down= u -shape facing downward. Inflation point= point on graph where cure changes; where y '' = 0.