Category Archives: Graphing

Joining the Pieces of a Graph

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In this post we will consider how the shapes discussed in the previous two posts can join together. Continuity and the derivative at the point where two shapes join tell us what’s going on.

Graphs can change from one shape to another only at places where:

  • The first derivative changes sign. For this to happen, the first derivative has to be either zero or undefined.  The x-coordinate at such places is called a critical number; the point is called a critical point. The function may have a local extreme value (a maximum or minimum) at its critical values. Not all critical numbers are the location of an extreme value, but all extreme values occur at critical numbers.

  • The second derivative changes sign. Such places are called a point of inflection (or, outside of the USA, point of inflexion.)  As with the first derivative, the second derivative can change sign only where it is zero or undefined. (Also, in order for there to be a second derivative at a point, the first derivative cannot be undefined there.)

  • The function is not continuous. The separate pieces can easily be different shapes. This really falls under the first bullet above, but functions may be continuous and still fail to have a derivative at a critical number.

This suggests a procedure: First, find the critical numbers by finding where {f}'\left( x \right)=0 or is undefined and then determining if there is a change of sign of the first derivative at the critical number. This may be the location of an extreme value. Compare y = x2 and y = x3 at the origin.

Do the same for points of inflection: find where {{f}'}'\left( x \right)=0 or is undefined and determine if there is a sign change there. These places may be points of inflection. Compare y = x3 and y = x4 at the origin.

A word of caution: Some authors require a non-vertical tangent line at a point of inflection and/or that the derivative exists there. This eliminates functions like y = x1/3 which has no derivative at the origin and a vertical tangent line. I see no reason for this: if there is a point where the concavity changes, that’s a point of inflection. Still you should go with your textbook’s author. The AP exams avoid asking about this situation.

If the function is not continuous (and therefore not differentiable) at a point, then the shapes don’t join. You need to look separately on each side of the point where the function is not continuous. The missing point, the jump or step, or the vertical asymptote is the clue that there may be a change in the shape. There does not have to be a change in shape at all, but as with all discontinuities be sure to check what’s happening on both sides. .

If the function is continuous, but not differentiable at a point then the shape may, but does not have to change shape there. If this is the case, the graph is not locally linear. It may have a sharp point or just a little “kink” there. But the non-differentiability tells us that something interesting is happening there.

Next: Extreme Values

The Shapes of a Graph

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In my last post we discussed the five shapes of a graph. Hopefully, that activity, which is posted under the Resources tab above, helped your students discover that

  • A function is increasing and concave up, on any interval where its first derivative is positive and its second derivative is positive, like y = sin(x) on \left( \tfrac{3\pi }{2},2\pi \right).
  • A function is increasing and concave down, on any interval were its first derivative is positive and its second derivative is negative, like y = sin(x) on  \left( 0,\tfrac{\pi }{2} \right).
  • A function is decreasing and concave up, on any interval where its first derivative is negative and its second derivative is positive, like y = sin(x) on \left( \pi ,\tfrac{3\pi }{2} \right).
  • A function is decreasing and concave down, on any interval where its first derivative is negative and its second derivative is negative, like y = sin(x) on \left( \tfrac{\pi }{2},\pi \right).
  • To which we will add a function is linear where its first derivative is constant and its second derivative is zero.

Separating the increasing/decreasing behavior from the concavity:

  • On an interval where the first derivative is positive the graph is increasing, and on an interval where the first derivative is negative the function is decreasing.
  • On an interval where the second derivative is positive the function is concave, on an interval where the second derivative is negative the graph is concave down.

Be careful when presenting the ideas above.

None of them consider what happens if one of the other of the derivatives is zero or undefined. There is an important theorem which says, and we must be careful here, “If for all x in an interval, {f}'\left( x \right)>0 , then the function is increasing on that interval.”

True enough, but what about y = x3 on an interval containing the origin? Well, the theorem does not apply, since the derivative is not positive everywhere on the interval. The theorem says nothing about what happens when the derivative is zero, only what happens when it is positive.  In such cases we need to return to the definition of increasing (which incidentally does not mention derivatives), to determine that y = x3 is increasing on any interval containing the origin (any interval, anywhere, in fact).

Another thing to be careful of is this: Functions increase or decrease on intervals, not at points. If you are asked if a function is increasing or decreasing at a point, or “Is the velocity increasing when t = ….” interpret the question as asking, “Is there a small open interval containing the point, on which the function is increasing or decreasing.”

Using the derivative to give this kind of information about the graph is a big part of the calculus and one of the important uses of derivatives. We can determine information by working with the equation of the derivative. We can also work from the graph of the derivative. This is often easier since it is easy to tell from the graph when the derivative is positive or negative. From the graph of the derivative, we can also see where the derivative is increasing or decreasing, and this tells us the sign of the second derivative and hence about the concavity of the function. I will discuss this in a later post.

Next: Joining the Pieces

Concepts Related to Graphs

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This and the next several posts will be about graphing and specifically how the function and its first and second derivatives are related. Since I do not intend this to be a textbook, I will not be doing textbook stuff. Rather, I hope to add some big picture things about the concepts involved. Hope you find it useful.

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Graphs of functions come in some combination of five shapes:

• Linear,
• Increasing, concave up,
• Increasing, concave down,
• Decreasing, concave up, and
• Decreasing, concave down.

I have put together an activity, An Exploration of the Shape of a Graph, to help students learn the relationship between the shapes and the derivative. Leaving the linear sections aside, the idea of the activity is to call students’ attention to the other four shapes and the slope of the tangent line (the derivative) in the intervals where the graph has each shape.

The key is to keep focused on the tangent line. To this end I suggest drawing short tangent segments parallel to the graph, as illustrated in the figure below.

Air-graphing: Another way to help students internalize this idea is to draw the graph of a function on the board and then have the students hold their pen out in front of them so the pen looks like a segment tangent to the graph. Then move the pen along the graph paying attention to the slope.

Either way what you want students to be aware of is

(1) Where the slopes are positive and where they are negative. This will later be related to the intervals where the function increases and decreases, and

(2) How the slope itself is changing. Is the slope increasing or decreasing? This will later be related to the concavity.

I hope you and your students find this activity helpful.