Monthly Archives: September 2015

Determining the Indeterminate

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When determining the limit of an expression the first step is to substitute the value the independent variable approaches into the expression. If a real number results, you are all set; that number is the limit.

When you do not get a real number often expression is one of the several indeterminate forms listed below:

Indeterminate Forms:

\displaystyle \frac{0}{0},\quad \quad \frac{\pm \infty }{\pm \infty },\quad \quad 0\cdot \infty ,\quad \quad \infty -\infty ,\quad \quad {{0}^{0}},\quad \quad {{1}^{\infty }},\quad \quad {{\infty }^{0}}

They are called indeterminate forms because by doing some algebraic manipulation to simplify (or sometimes complicate) the expression its value can be determined. The calculus technique called L’Hôpital’s Rule may be used in some situations.

Part of the reason they are called indeterminate forms is that different expressions with the same indeterminate form may result in different values.

Before continuing with the discussion of indeterminate forms, I should point out that there are also determinate forms: expressions similar to those above that always result in the same value.

Determinate Forms and the values they approach:

\displaystyle \frac{1}{\pm \infty }\to 0,\quad \infty +\infty \to \infty ,\quad \infty \cdot \infty \to \infty ,\quad {{0}^{\infty }}\to 0,\quad \frac{1}{{{0}^{\infty }}}={{0}^{-\infty }}\to \infty

Returning to indeterminate forms, textbooks contain examples and exercises that illustrate how to evaluate the indeterminate forms. Here are two examples illustrating a few of the techniques that can be used to evaluate them.

Example 1:  \displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{x} \right)}^{x}}=?.This is an example of the indeterminate form {{1}^{\infty }}. With exponents, logarithms may often be used to find the value.

\displaystyle \ln \left( ? \right)=\underset{x\to \infty }{\mathop{\lim }}\,\left( x\ln \left( 1+\frac{1}{x} \right) \right)=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\ln \left( 1+\frac{1}{x} \right)}{\frac{1}{x}}

The limit is now of the indeterminate form \frac{0}{0}, so L’Hôpital’s Rule may be used. Continuing

\displaystyle \ln \left( ? \right)=\underset{x\to \infty }{\mathop{\lim }}\,\left( x\ln \left( 1+\frac{1}{x} \right) \right)=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\ln \left( 1+\frac{1}{x} \right)}{\frac{1}{x}}=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\left( \frac{1}{1+\frac{1}{x}} \right)\left( -{{x}^{-2}} \right)}{-{{x}^{-2}}}=\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{1}{1+\frac{1}{x}} \right)=1

So then ln(?) = 1 and ? = e1 = e and therefore \displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{x} \right)}^{x}}=e

Example 2: \displaystyle \underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\sin \left( x+\frac{\pi }{6} \right)}{x}-\frac{\sin \left( \frac{\pi }{6} \right)}{x} \right) is an example of the indeterminate form \infty -\infty .

Write the expression with a common denominator:

\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\sin \left( x+\frac{\pi }{6} \right)}{x}-\frac{\sin \left( \frac{\pi }{6} \right)}{x} \right)=\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\sin \left( \frac{\pi }{6}+x \right)-\sin \left( \frac{\pi }{6} \right)}{x} \right)=\cos \left( \frac{\pi }{6} \right)=\frac{\sqrt{3}}{2}

The second limit above is the definition of the derivative of \sin \left( \frac{\pi }{6} \right). (L’Hôpital’s Rule may also be used with the second limit above.)

In fact, the definition of the derivative of all (any, every) functions, \displaystyle {f}'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}, gives indeterminate expressions of the form \frac{0}{0}.

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Matching Motion

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Particle motion 2

Here’s a little matching quiz. In the function column there is a list of properties of functions and in the motion column are a list of terms describing the motion of a particle. The two lists are very similar. Match the terms in the function list with the corresponding terms in the Linear Motion list (some may be used more than once). The answers are below. For more on this idea see my previous post Motion Problems: Same Thing, Different Context.

Function                                               Linear Motion
1. Value of a function at x                     A. acceleration
2. First derivative                                  B. “at rest”  
3. Second derivative                             C. farthest left 
4. Function is increasing                       D. farthest right
5. Function is decreasing                      E. moving to the left or down 
6. Absolute Maximum                          F. moving to the right or up
7. Absolute Minimum                            G. object changes direction
8. y ʹ = 0                                                H. position at time t
9. y ʹ changes sign                                I. speed 
10. Increasing & concave up                J. speed is decreasing
11. Increasing & concave down           K. speed is increasing
12. Decreasing & concave up              L. velocity
13. Decreasing & concave down             
14. Absolute value of velocity      

  

  
 
Answers:  1. H,   2. L,   3. A,   4. F,   5. E,   6. D,   7. C,   8. B,   9. G,   10. K,   11. J,   12. J,   13. K,   14. I

Using the Derivative to Graph the Function

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In my last post I showed how to use a Desmos graph to discover, by looking at the tangent line as it moved along the graph of the function, the properties of the derivative of a function. This post goes in the opposite direction. Now, instead of discovering the properties of the derivative from the graph of the function, we will use that knowledge to identify important information about the function from the graph of its derivative. We are not discovering anything here; rather we are putting our previous discoveries to use.

One of the uses of the derivative is to deduce properties of its antiderivative, i.e. the function of which it is the derivative. This is an important skill for students to be able to use. It is also a type of question that appears on every Advanced Placement Calculus exam in both the free-response sections and the multiple-choice sections. My previous post on this topic, Reading the Derivative’s Graph, is the most read post on this blog. This post expands on the concepts in that post and shows how to use the Desmos file to help students develop the skills necessary to answer this type of question.

Desmos is free. You and your students can set up their own account and save their own work there. There are also free Desmos apps for tablets and smart phones.

Click on the graph above. Here’s what you should see:

  • The first equation on the left side is f(x). This is the function whose antiderivative we will be trying to create. You may change this to any function you wish.
  • The second equation is F(x) the antiderivative of f(x). That is F ‘(x) = f(x). Desmos cannot compute an antiderivative so you will need to enter it yourself. Your students need not trust you on this: have them check your equation by differentiating. Of course, at this point they will not be ready to find the analytic form of the antiderivative except for the simplest functions. This comes later in the year.
  • Note the “+C” is included and later we will manipulate this with a slider.
  • The antiderivative is multiplied by \frac{\sqrt{a-x}}{\sqrt{a-x}}. This is a way of controlling the “a” slider and should always be multiplied by antiderivative. This is just a syntax trick to make the graph work and is not part of the antiderivative. When x is to the left of a, (x < a) the fraction is 1 and the graph will be seen; when x is to the right of a, (x > a) the expression is undefined, and nothing will graph. As you change “a” with the slider the graph of an antiderivative will be drawn.
  • The third equation x = a graphs a dashed vertical to help you line up the corresponding points on the two graphs.
  • The next two lines are the “a” and “C” sliders. Make C = 0 for now and move the “a” slider.

At this point students should know things about the relationship between a function and its first and second derivatives. This includes the things they discovered in the previous post such as when the function is increasing the derivative is non-negative, and when the tangent line is above the graph the slope (derivative) is decreasing and the graph of the function is concave down. All of these concepts are really “if, and only if” situations. So we now consider them in reverse and deduce properties of the antiderivative from the properties of the graph of the derivative.

Using the “a” slider starting at the left side of the graph ask the students what the derivative tells them about the function in this part of the domain. Is the function increasing or decreasing? Is it concave up or down? etc.  Go slowly from left to right asking what happens next, and why (that is, how do you know? What feature of the derivative tells you this?) This is preparing them to write the justifications required on the AP exams.

Certain points on the graph of the derivative are important. The zeros of the derivative and whether the derivative changes from positive to negative, negative to positive, or neither are important. Likewise, the extreme values of the derivative point to important features of the function – points of inflection.

Once you have a complete graph of the antiderivative move the “C” slider.

  • Remind students that the derivative of a constant is zero and therefore, the C does not show up in the derivative.
  • Discuss is why changing the antiderivative does not change the derivative.
  • Ask, how many functions have the same derivative?
  • Point out is that by changing “C” the graph of an antiderivative can be made to go through any (every, all) point in the plane.

Switch to a different function and its derivative to reinforce the concepts. Yo will have to enter both the derivative and the antiderivative.Do this as often as necessary. You could also give students or groups of students the graph of a derivative (on paper) and challenge them to sketch the antiderivative.

A disclaimer: A function and its derivative should not be graphed on the same axes, because the two have different units. Nevertheless, I have done it here, and it is commonly done everywhere, to compare the graphs of a function and its derivative so that the important features of the two can be lined up and easily compared.

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Tangents and Slopes

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Using the function to learn about its derivative. 
In this post we will look at a way of helping students discover the numerical and graphical properties of the derivative and how they can be determined from the graph of the function. These ideas can be used very early, when you are first relating the function and its derivative. (In my next post we will look at the problem the other way around – using the derivative to find out about the function.)
Click on the graph below. This will take you to a graph I posted on the Desmos website (Desmos.com). This is a free and really easy to use grapher, which you and your students can use. If you sign up for your own account, you can make and save graphs for your class or use some of those that are built-in (click on the three horizontal bars at the upper left of the screen). Your students may do this as well. There are also Desmos apps for smart phones and tablets.

When you click on the graph a file called “function => derivative” should open. This is what you should see:
On the left is a list of equations. Those with a colored circle to its left are turned on; click the circle to toggle the graphs on and off. Here’s what they do:

  • f(x) is the equation of the function we will start with. Later you may change this to whatever function you are investigating. You will not have to change any of the others when you change the function. This one should be turned on.
  • g(x) is the derivative of f(x). Leave this turned off.
  • h(x) is a special function. The expression at the end, \frac{\sqrt{a-x}}{\sqrt{a-x}}, is what makes the slider work. This is a syntax trick and not part of the derivative. If x is to the right of a (i.e. x < a) the expression is equal to one and the derivative will graph. If x is to the right of a, (i.e. a < x) then the expression is undefined, and nothing will graph. Initially, turned off.
  • k(x) is the graph of a segment tangent to f at x = a. Click on the equation if you want to see how the segment is drawn by restricting the domain of x. Initially, turned on.
  • The next equation, x = a, is the equation of the dashed vertical line. This is included so that, later, we can see precisely how points on the graph are located above one another. Initially, turned off.
  • (a, f(a)) is the point of tangency. Initially, turned on.
  • The last box is the slider for the variable a. Its domain is shown at the ends. This may be changed by clicking on one end or clicking on the “gear” icon at the top of the list.

The icons at the top right of the graph let you zoom in and out or set the viewing window. You may click on the wrench icon and make other adjustments. “Projector Mode” makes the graph thicker and may help students see better when you project the graph.
Okay, you are now Desmos experts! Really, it’s that easy. The “?” at the top right has a few more instructions and you can download the user’s guide, a brief 13 pages.

Investigating the derivative
Do this before proving all the associated theorems. Let the class discover the relations between the graph of the function and its derivative. Prove them, or explain why they are so, later. You may want to spread all this over several days, perhaps dealing with where the function is increasing or decreasing and extreme values the first day and working with concavity the second.
1. Begin with just f(x), k(x), and the point (a, f(a)) turned on (click the circle to the left of the equation to toggle graphs on and off). Use the slider to move the tangent segment along the graph.
Draw the class’s attention to important things but let them formulate the observations in words. Ask your class a series of questions about what they see. Things like:

  • When the function is increasing, is the slope of the tangent segment positive or negative? When the function is decreasing, is the slope of the tangent segment positive or negative? Why?
  • What happens with the derivative when the function changes from increasing to decreasing or vice versa?
  • Notice that sometimes the tangent segment lies above the function and sometimes below. What does the function look like when the tangent is above (below)?
  • In the box for the slider delete the number and type a = 0; this moves the slider to the origin. Can you see what its slope is here? You can type in other numbers such as pi/6, or pi/2 and read the slope there. (If the slider disappears when you do this, type in a = 0 and it will come back.)
  • If you can project on a white board or are using a Smartboard, mark the points (a, slope at a) and see if you can graph the derivative.

2. Next turn on h(x) and x = a. Move the slider.

  • What are you seeing? As you move the slider the dashed vertical line moves to show you where you are. The graph of the derivative is drawn to the left of the dashed line.
  • Once again question the class about what they observe. Notice such things as on intervals where the function increases, the derivative is greater than or equal to zero, etc. Review all the things you discovered in part 1. Remember often students don’t associate things such as “the derivative is negative” with the “graph of the derivative lies below the x-axis.“
  • How does the concavity relate to the graph of the derivative?

3. Now change the starting function to something else.

  • First, just add a constant to f(x). If you really want to get fancy type f(x) = sin(x) + b. A slider for b will appear. Discuss why this transformation does not change the graph of the derivative one bit.
  • Some good examples are f(x) = cos(x), f(x) = x+2sin(x), a third- or fourth-degree polynomial (find a good example in your textbook and see below). Repeat all of steps 1 and 2 above.

Give the students a new function and see if they can sketch the (approximate) graph of the derivative themselves.

For further exploration click the graph below. This is similar to the first one; however, the function is a fourth degree polynomial with variable coefficients. Use the various sliders at the bottom to adjust the graph to an interesting shape. Make p = 0 to graph a cubic and make both p and q zero to graph a parabola. (This might make a good lesson in an advanced math or Algebra 2 class.)

A disclaimer: A function and its derivative should not be graphed on the same axes, because the two have different units. Nevertheless, I have done it here, and it is commonly done everywhere to compare the graphs of a function and its derivatives so that the important features of the two can be lined up and compared easily.

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