The Sign of the Derivative

SECANT LINES
Suppose that f(x) is increasing on the interval (a,b) and that the derivative of f exists at a point c in this interval. By definition,

f(c + h) - f(c)
f '(c) = lim
h-->0h
which is the limit of the slopes of secant lines cutting the graph of f(x) at (c,f(c)) and a second point. The limit is taken as the two points coalesce into (c,f(c)).

As long as the second point lies over the interval (a,b) the slope of every such secant line is positive. It follows that the limit, and hence the derivative, is non-negative. Why not POSITIVE? Similarly, if f(x) is decreasing on (a,b) then f '(x) 0 for x in (a,b).

Conversely, suppose that f(x) is differentiable at every point of an interval (a,b) and that f '(x) is positive at every point of this interval. Then f cannot be decreasing on any interval contained in (a,b), for by the above discussion f '(x) would then have to be negative or zero somewhere in (a,b). Does this imply that f is increasing? Not quite. But using a theorem called the Mean Value Theorem we can show that f(x) is actually increasing on (a,b). The precise statement, which allows for closed intervals as well, is: 

The First Derivative Test

Suppose that f is continuous on an interval J with endpoints a and b and that f is differentiable on the open interval (a,b) contained in J.

If f '(x) > 0 for all x in the interval (a,b), then f(x) is increasing on the interval J.

If f '(x) < 0 for all x in the interval (a,b), then f(x) is decreasing on the interval J.

The converse of this is not really a "test" but is worth repeating:

Suppose that f is continuous on an interval J with endpoints a and b and that f is differentiable on the open interval (a,b) contained in J.

If f is increasing on J then f '(x) 0 for all x in (a,b).

If f is decreasing on J then f '(x) 0 for all x in (a,b).

What happens if the derivative of a function is 0 on an entire interval? The answer is that the function must be constant! To prove this we again need the Mean Value Theorem, but all it says is "that which does not change is constant".

Constant Functions and The First Derivative

Let f(x) be continuous on an interval J with endpoints a and b and suppose that f is differentiable on the interval (a,b) contained in J.

  • If f(x) is constant on J then f '(x) = 0 for all x in (a,b).

  • If f '(x) = 0 for all x in (a,b) then f(x) is constant on J.

Applications

First derivatives arise in many contexts. Here are two examples and an equipment check which illustrate this.

Example 1: Suppose that Pork Chop is riding on a straight road and that her velocity is positive for the first hour and then negative for the next 20 minutes. What can you conclude?

Discussion: Velocity is the derivative of the distance function. In this case, distance presumably means the distance from some point on the straight road. When her velocity is positive, the distance from this point is increasing. When the velocity is negative, the distance is decreasing. In other words, she is getting closer to where she started. In addition to these conclusions, we might conclude that she turned around after one hour.

Example 2: If we differentiate f(x) = cos2(x) + sin2(x) we get

Dx(cos2(x) + sin2(x)) = 2 cos(x) (-sin(x)) + 2 sin(x) cos(x) = 0.

What is going on?

Discussion: The above calculation shows that f(x) = cos2(x) + sin2(x) has zero derivative. We conclude that f(x) must be a constant! This is not a surprise since

sin2(x) + cos2(x) = 1.

for all x.

Equipment Check:

  1. Suppose that m(t) gives the area of the lighted portion of the moon as seen from the campus of Oregon State University. How do you express the times at which the moon is waxing and at which the moon is waning? ANSWER

  2. Suppose the cost of spending h hours studying personal finance is given by $10*arctan(h) in terms of the cost of your time. Suppose that you will realize $(10 + h) in savings from each of the first 20 hours of study. After how many hours of study does your profit per hour of study start to go up?

    If you are stuck, try hint #1 and if you can't remember a derivative try hint #2. The "explain" button will pop up a complete solution.

    ANSWER: The profit goes up after hours. ANSWER

        EXPLAIN

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