Lori Pearman

EMT 669

First and Second Derivatives

and
What They Tell About the Graphs of Functions

The slope at any point (x,f(x)) on the graph of a function f is given by the first derivative of f. For example, if f(x) = x^2 + 1, then the slope at any point on the graph of f is given by f'(x) = 2x. Thus, at the point (0,1), the slope is f'(0) = 2(0) = 0. (Simply plug the x-coordinate in for x.) Similarly, at the point (-1,2), the slope is f'(-1) = 2(-1) = -2. The following is a graph of the parabola f(x)= x^2 + 1 and the tangent lines (to the parabola) at the points (0,1) and (-1,2). The slopes of these tangent lines are 0 and -2 respectively.


Another example would be f(x) = 2[[radical]]x. Here, the slope at any point (x,f(x)) on the graph of f is given by f'(x) = 1/[[radical]]x. Thus, at the point (1,2), the slope is f'(1) = 1. However, at the point (0,0) the slope is undefined, since substituting x = 0 in f'(x) produces division by zero.

The graph below shows the graph of f(x) and the tangent line at (1,2).


Now let's look at second derivatives. For a function whose second derivative exists on an open interval I, we can test for concavity. If f''(x) > 0 for all x in I, the graph of the function is concave upward. If f''(x) < 0 for all x in I, the graph of the function is concave downward.

For example, let f(x) = (x2-1)/(x2-4). Then f'(x) = -6x/(x2-4)2, and f''(x) = 6(3x2+4)/(x2-4)3. The second derivative, f''(x), is discontinous at x = 2 and x = -2. Thus, the concavity can be tested in the intervals (-inf, -2), (-2, 2), and (2, inf). The interval (-2, 2) is the only interval which will produce negative values for the second derivative. Thus, the graph will be concave downward in the interval (-2, 2). The other two intervals produce positive values for the second derivative. Thus, the graph will be concave upward in those intervals.

The values for the second derivative are as follows:


          x-values    f''(x) values  

 
             -10        0.00206       
       -9        0.00325
          -8        0.00544   
               -7        0.00994       
               -6        0.02051       
               -5        0.05118       
               -4        0.18056       
               -3        1.48800       
               -2        #DIV/0!       
              -1      -1.55556      
               0       -0.37500      
               1       -1.55556      
                2        #DIV/0!       
                3        1.48800       
                4        0.18056       
                5        0.05118       
                6        0.02051       
                7        0.00994       
                8        0.00544       
                9        0.00325       
               10        0.00206
Now let's graph f(x) to show that the graph is indeed concave upwards on (-inf, -2) and (2, inf), but concave downwards on (-2, 2). The vertical lines

are not actually part of the graph. (They are aymptotic lines.)



Remark: The point (a,f(a)) is called a point of inflection if the concavity of f changes from upward to downward (or vice versa) at that point. If (a,f(a)) is a point of inflection of the graph of f, then either f''(a) = 0 or f'' is undefined at x = a.



The above graphs were copied from the software called "xFunctions". The graph on the left is a graph of f(x)=(x2-1)/(x2-4). The graph in the middle is of f'(x), and the one on the right is of f"(x). I chose a point on f(x) where the arrow is. The black line is the tangent to the graph at that point. This corresponds to the "+" sign in the graph of f'(x). Notice that the tangent line in the f(x) graph has a positive slope. Thus, f'(x) =y' is also positive at that point. In the graph on the right, f"(x) is positive where f(x) is concave upwards and negative where f(x) is concave downward.

The Second Derivative Test from calculus states that if f is a function such that f'(a) = 0 and the second dervative of f exists on an open interval containing a, then:

1. If f''(a) > 0, then f(a) is a relative minimum.

2. If f''(a) < 0, then f(a) is a relative maximum.

3. If f''(a) = 0, then the test fails.

Let's look at another example. Let f(x) = -x3+3x2-2. Then f'(x) = -3x2+6x = -3x(x-2), and f''(x) = -6x+6. The first thing we need to do is to find the critical points. (These are points with x- values which make the first derivative equal zero or make it undefined.) In this example, the critical points are (0,-2) and (2,2) since x=0 and x=2 make f'(x) = 0. Now we can apply the test.

Point Sign of f'' Conclusion:

(0,-2) f''(0)=6 > 0 relative minimum

(2,2) f''(2)=-6 < 0 relative maximum

We can also use first derivatives to give us a better idea of what the graph looks like. In our example, the first derivative is negative to the left of (0,-2), so the function must be decreasing there. It is positive between (0,-2) and (2,2), so f is increasing on that interval. And f' is negative once again to the right of (2,2), so f is decreasing again there. (See below.)

    x-values      f'(x) values     
      -5.0        -105.00          
      -4.5         -87.75           
      -4.0         -72.00           
      -3.5         -57.75           
      -3.0         -45.00           
      -2.5         -33.75           
      -2.0         -24.00           
      -1.5         -15.75           
      -1.0          -9.00            
      -0.5          -3.75            
         0.0           0.00             
         0.5           2.25             
         1.0           3.00             
         1.5           2.25             
         2.0           0.00             
        2.5          -3.75            
        3.0          -9.00            
       3.5         -15.75           
       4.0         -24.00           
       4.5         -33.75           
       5.0         -45.00


The above graph shows the x-values versus the first derivative values. Notice that f'(x) is positive between x=0 and x=2, showing that the actual graph of f(x) is increasing on that interval. Let's compare this graph with a graph a the original function f(x).


This graph seems to be increasing on (0, 2).

Let's look at one final example. Let f(x)=2x/(x2-x+1).

Then f'(x) = (-2x2+2)/(x2-x+1)2 , and f"(x) = [-4x(x2-x+1) -(4x-2)(-2x2+2)]/(x2-x+1)3.

Let's use Algebra Xpresser to graph f(x), f'(x), and f"(x) all on the same axis.


The red graph is of f(x), the green graph is f'(x), and the blue graph is f"(x). Let's compare the red graph with the green graph. When f'(x) is positive (that is, the green graph is above the x-axis), then the red graph, f(x), is increasing. The critical numbers are x=1 and x=-1 since f'(1)=0 and f'(-1)=0. The green graph is neither positive or negative at these values, so f(x) is neither increasing nor decreasing there. The slope of the tangents at the points (1, f(1)) and (-1, f(-1)) is zero, so f(1) is a relative maximum, and f(-1) is a relative minimum. That is, (-1,-2/3) is a relative minimum, and (1,2) is a relative maximum.

Informally, we can think of a relative maximum occuring on a "hill" on a graph and of a relative minimum occuring on a "valley". If the hill (or valley) is smooth and rounded, then the graph has a horizontal tangent line at the high point (or low point). On the other hand, if the hill (or valley) is sharp and) peaked, then the graph represents a function that is not differentiable at the high point (or low point). The hill and valley in the above (red) graph is smooth and rounded.

Now let's compare the blue, f"(x), graph with the red, f(x) graph. When f"(x) is positive (i.e. the blue graph is above the x-axis) then the red graph is concave upwards. And when f"(x) is negative, then the red graph is concave downwards. The points of inflection (points where the concavity changes) of the graph of f(x) occur when f"(x)=0 or f"(x) is undefined. Thus, when the blue graph crosses the x-axis, the red graph will change concavity.

In conclusion, the first and second derivatives of a function can tell us a lot about the graph of the function. From this essay, I have reinforced my knowledge of this subject. I have looked at derivatives in ways I had not before, and I was better able to actually see what was happening. I was also able to experiment with software I had not used before (namely, xFunctions ).

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