Good Question 8 – or not?

Posted on January 5, 2016 by Lin McMullin

Today’s question is not a good question. It’s a bad question.

But sometimes a bad question can become a good one.

This one leads first to a discussion of units, then to all sorts of calculus.

Here’s the question a teacher sent me this week taken from his textbook:

The normal monthly rainfall at the Seattle-Tacoma airport can be approximated by the model $R=3.121+2.399\sin \left( 0.524t+1.377 \right)$ , where R is measured in inches and t is the time in months, t = 1 being January. Use integration to approximate the normal annual rainfall. Hint: Integrate over the interval [0,12].

Of course, with the hint it’s not difficult to know what to do and that makes it less than a good question right there. The answer is $\displaystyle \int_{0}^{12}{R(t)dt=37.4736}$ inches. You could quit here and go on to the next question, but …

Then a student asked. “If R is in inches shouldn’t be in units of the integral be inch-months, since the unit of an integral is the unit of the integrand times the units of the independent variable?” Well, yes, they should. So, what’s up with that?

Also, the teacher figured that the integral of a rate is an amount and our answer is an amount, so why isn’t the integrand a rate?

The only answer I could come up with is that the statement “R is measured in inches” is incorrect; R should be measured in inches /month. The opening phrase “normal monthly rainfall” also seems to point to the correct units for R being inches/month.

Problem solved; or maybe does this lead to a different concern?

The teacher pointed out that R(6) = 0.7658 inches is a reasonable answer for the amount of rain in June whereas $\displaystyle \int_{0}^{6}{R(t)dt=}20.4786$ is not.

If R is a rate, then the amount of rain that falls in June (t = 6) is given by $\displaystyle \int_{5}^{6}{R(t)dt}=0.9890$ .

From here on we will assume that R is a rate with units of inches/month. Here are the individual monthly rates calculated with a CAS.

The total amount of rainfall (second line above) appears be R(1) + R(2) + R(3) + … +R(12) = 37.4742. This is very close to the amount calculated by integration.

The slight difference of 0.0006 is not a round off error.

Remember, behind every definite integral there is a Riemann sum!

Again, the units are the problem. Why does the sum of the monthly rates seem to give the total amount? The reason is that the terms of the sequence above are actually the values of a right-side Riemann sum of the rate, R(t), over the interval [0,12] with 12 equal subdivisions of width 1 (month) each with the 1’s left out as 1’s often are. Therefore, their sum should come close to the total yearly rainfall, but it is really just an approximation of it.

The actual total for any month, n, is given by $\displaystyle \int_{n-1}^{n}{r(t)}dt$ . For example the amount of rain that falls in June is given by $\displaystyle \int_{5}^{6}{R(t)dt}=0.9890$ inches.

Here is the sequence of the actual monthly rainfall values in inches, and their sum.

This agrees with the integral. Why? Because one of the properties of integrals tell us that $\displaystyle \sum\limits_{n=1}^{12}{\int_{n-1}^{n}{r(t)dt}}=\int_{0}^{12}{r(t)dt}$ .

Another instructive thing with this integral is this: The function $R=3.121+2.399\sin \left( 0.524t+1.377 \right)$ is periodic with a period of $\frac{2\pi }{0.524}\approx 11.9908\approx 12$ . So the sine function takes on (almost) all its values in a year, as you would expect. Since the sine values all but cancel each other out

$\displaystyle \int_{0}^{12}{3.121+2.399\sin \left( 0.524t+1.377 \right)dt}\approx \int_{0}^{12}{3.121dt=3.121\left( 12-0 \right)=37.452}$ . Close!

The total rainfall divided by 12 is $\frac{37.452}{12}=3.121$ this must be close to the average rainfall each month. The average rainfall is $\displaystyle \frac{1}{12}\int_{0}^{12}{R\left( t \right)dt}=3.1228$ inches. Close, again!

So, there you have it. Is this a good question or not? We considered all these concepts while working not just with an equation but with numbers from a poorly stated problem:

Reading and interpreting words.
Unit analysis
Integration by technology
Realizing that a pretty good approximation is not correct, due again to units.
A Riemann sum approximation in a real situation that comes very close to the value by integration
Using a property of a periodic function to greatly simplify an integral
Finding average value two ways

So, it turned out to be a sunny day in Seattle.

November 2015

Posted on October 31, 2015 by Lin McMullin

Here is the list of “November Topics,” that is what AP classes usually consider from mid-November into December. There has been a lot of discussion about inverses this month at the AP Calculus Community. While not the most read on this blog, the series on inverses may be helpful in considering all the ins and outs of inverses.

The four featured posts on the first page are the most popular from this month. Speed with 3563 hits this year and 6157 hits since it first appear is one of the most popular overall. “Open or Closed?” is another poplar post.

Thinking ahead into December, the first posts on integration are here and will continue into December. (As I’ve mentioned I try to post a few weeks ahead of where most people are now, so you have some time to read and plan.)

October 13, 2014 Extremes without Calculus

November 2, 2012 Open or Closed?

November 5, 2012 Inverses

November 7, 2012 Writing Inverses

November 9, 2012 The Range of the Inverse

November 12, 2012 The Calculus of Inverses

November 14, 2012 Inverses Graphically and Numerically

November 16, 2012 Motion Problems: Same Thing, Different Context.

November 19, 2012 Speed

April 17, 2013 The Ubiquitous Particle Motion Question

September 16, 2014 Matching Motion

November 21, 2012 Derivatives of Exponential Functions

November 26, 2012 Integration Itinerary

November 18, 2012 Antidifferentiation

November 30, 2012 The Old Pump

Good Question 7 – 2009 AB 3

Posted on October 5, 2015 by Lin McMullin

Another in my occasional series on Good Questions to teach from. This is the Mighty Cable Company question from the 2009 AB Calculus exam, number 3

This question presented students with a different situation than had been seen before. It is a pretty standard “in-out” question, except that what was going in and out was money. Students were told that the Mighty Cable Company sold its cable for $120 per meter. They were also told that the cost of the cable varied with its distance from the starting end of the cable. Specifically, the cost of producing a portion of the cable x meters from the end is $6\sqrt{x}$ dollars per meter. Profit was defined as the difference between the money the company received for selling the cable minus the cost of producing the cable.

Students had a great deal of trouble answering this question. (The mean was 1.92 out of a possible 9 points. Fully, 36.9% of students earned no point; only 0.02% earned all 9 points.) This was probably because they had difficulty in interpreting the question and translating it into the proper mathematical terms and symbols. Since economic problems are not often seen on AP Calculus exams, students needed to be able to use the clues in the stem:

The $120 per meter is a rate. This should be deduced from the units: dollars per meter.
The cost of producing the portion of cable x meters from one end cable is also a rate for the same reason. In economics this is called the marginal cost; the students did not need to know this term.
The profit is an amount that is a function of x, the length of the cable.

Part (a): Students were required to find the profit from the sale of a 25-meter cable. This is an amount. As always, when asked for an amount, integrate a rate. In this case integrate the difference between the rate at which the cable sells and the cost of producing it.

$\displaystyle P(25)=\int_{0}^{25}{\left( 120-6\sqrt{x} \right)dx}=\$2500$

$\displaystyle P(25)=120(25)-\int_{0}^{25}{6\sqrt{x}\ dx}=\$2500$

Part (b): Students were asked to explain the meaning of $\displaystyle \int_{25}^{30}{6\sqrt{x}dx}$ in the context of the problem. Since the answer is probably not immediately obvious, here is the reasoning involved.

This is the integral of a rate and therefore, gives the amount (of money) needed to manufacture the cable. This can be found by a unit analysis of the integrand: $\displaystyle \frac{\text{dollars}}{\text{meter}}\cdot \text{meters}=\text{dollars}$ .

Let C be the cost of production, so $\displaystyle \frac{dC}{dx}=6\sqrt{x}$ , and therefore, $\displaystyle \int_{25}^{30}{6\sqrt{x}dx}=C\left( 30 \right)-C\left( 25 \right)$ by the Fundamental Theorem of Calculus (FTC).

Therefore, $\displaystyle \int_{25}^{30}{6\sqrt{x}dx}$ is the difference in dollars between the cost of producing a cable 30 meters long, C(30), and the cost of producing a cable 25 meters long, C(25). (Another acceptable is that the integral is the cost in dollars of producing the last 5 meters of a 30 meter cable.)

Part (c): Students were asked to write an expression involving an integral that represents the profit on the sale of a cable k meters long.

Part (a) serves as a hint for this part of the question. Here the students should write the same expression as they wrote in (a) with the 25 replaced by k.

$\displaystyle P(k)=\int_{0}^{k}{\left( 120-6\sqrt{x} \right)dx}$

$\displaystyle P(k)=120k-\int_{0}^{k}{6\sqrt{x}\ dx}$

Part d: Students were required to find the maximum profit that can be earned by the sale of one cable and to justify their answer. Here they need to find when the rate of change of the profit (the marginal profit) changes from positive to negative.

Using the FTC to differentiate either of the answers in part (c) or by starting fresh from the given information:

$\displaystyle \frac{dP}{dx}=120-6\sqrt{x}$

$\displaystyle \frac{dP}{dx}=0$ when x = 400 and P(400)= $16,000.

Justification: The maximum profit on the sale of one cable is $16,000 for a cable 400 meters long. For $0<x<400,{P} '(x)>0$ and for $x>400,{P} '(x)<0$ therefore, the maximum profit occurs at x = 400. (The First Derivative Test).

Once students are familiar with in-out questions, this is a good question to challenge them with. The actual calculus is not that difficult or unusual but concentrating on the translation of the unfamiliar context into symbols and calculus ideas is different. Show them how to read the hints in the problem such as the units.

Steel cable or steel wire rope as it is called also has some interesting geometry in its construction. You can find many good illustrations of this, such as the ones below, by Googling “steel wire rope.”

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Good Question 6: 2000 AB 4

Posted on August 25, 2015 by Lin McMullin

Another of my favorite questions from past AP exams is from 2000 question AB 4. If memory serves it is the first of what became known as an “In-out” question. An “In-out” question has two rates that are working in opposite ways, one filling a tank and the other draining it.

In subsequent years we saw a question with people entering and leaving an amusement park (2002 AB2/BC2), sand moving on and off a beach (2005 AB 2), another tank (2007 AB2), an oil leak being cleaned up (2008 AB 3), snow falling and being plowed (2010 AB 1), gravel being processed (2013 AB1/BC1), and most recently water again flowing in and out of a pipe (2015 AB1/BC1). The in-between years saw rates in one direction only but featured many of the same concepts.

The questions give rates and ask about how the quantity is changing. As such, they may be approached as differential equation initial value problems, but there is an easier way. This easier way is that a differential equation that gives the derivative as a function of a single variable, t, with an initial point $\left( {{t}_{0}},y\left( {{t}_{0}} \right) \right)$ always has a solution of the form

$y\left( t \right)=y\left( {{t}_{0}} \right)+\int_{{{t}_{0}}}^{t}{{y}'\left( x \right)dx}$ .

This is sometimes called the “accumulation equation.” The integral of a rate of change ${y}'\left( t \right)$ gives the net amount of change over the interval of integration $[{{t}_{0}},t]$ . When this is added to the initial amount the result is an expression that gives the amount at any time t.

In a motion context, this same idea is that the position at any time t, is the initial position plus the displacement:

$\displaystyle s\left( t \right)=s\left( {{t}_{0}} \right)+\int_{{{t}_{0}}}^{t}{v\left( x \right)dx}$ where $v\left( t \right)={s}'\left( t \right)$

The scoring standard gave both forms of the solution. The ease of the accumulation form over the differential equation solution was evident and subsequent standards only showed this one.

2000 AB 4

The question concerned a tank that initially contained 30 gallons of water. We are told that water is being pumped into the tank at a constant rate of 8 gallons per minute and the water is leaking out at the rate of $\sqrt{t+1}$ gallons per minute.

Part a asked students to compute the amount of water that leaked out in the first three minutes. There were two solutions given. The second solves the problem as an initial value differential equation:

Let L(t) be the amount that leaks out in t minutes then

$\displaystyle \frac{dL}{dt}=\sqrt{t+1}$

$L\left( t \right)=\frac{2}{3}{{\left( t+1 \right)}^{3/2}}+C$

$L\left( 0 \right)=\frac{2}{3}{{\left( 0+1 \right)}^{3/2}}+C=0$ since nothing has leaked out yet, so C = -2/3

$L\left( t \right)=\frac{2}{3}{{\left( t+1 \right)}^{3/2}}-\frac{2}{3}$

$L\left( 3 \right)=\frac{14}{3}$

The first method, using the accumulation idea takes a single line:

$\displaystyle L\left( 3 \right)=\int_{0}^{3}{\sqrt{t+1}dt}=\left. \frac{2}{3}{{\left( t+1 \right)}^{3/2}} \right|_{0}^{3}=\frac{2}{3}{{\left( 4 \right)}^{3/2}}-\frac{2}{3}{{\left( 1 \right)}^{3/2}}=\frac{14}{3}$

I think you’ll agree this is easier and more direct.

Part b asked how much water was in the tank at t = 3 minutes. We have 30 gallons to start plus 8(3) gallons pumped in and 14/3 gallons leaked out gives 30 + 24 – 14/3 = 148/3 gallons.

This part, worth only 1 point, was a sort of hint for the next part of the question.

Part c asked students to write an expression for the total number of gallons in the tank at time t.

Following part b the accumulation approach gives either

$\displaystyle A\left( t \right)=30+8t-\int_{0}^{t}{\sqrt{x+1}dx}$ or

$\displaystyle A\left( t \right)=30+\int_{0}^{t}{\left( 8-\sqrt{x+1} \right)dx}$ .

The first form is not a simplification of the second, but rather the second form is treating the difference of the two rates, in minus out, as the rate to be integrated.

The differential equation approach is much longer and looks like this:

$\displaystyle \frac{dA}{dt}=8-\sqrt{t+1}$

$A\left( t \right)=8t-\frac{2}{3}{{\left( t+1 \right)}^{3/2}}+C$

$A\left( 0 \right)=30=8(0)-\frac{2}{3}{{\left( 0+1 \right)}^{3/2}}+C$ , so $C=\frac{92}{3}$

$A\left( t \right)=8t-\frac{2}{3}{{\left( t+1 \right)}^{3/2}}+\frac{92}{3}$

Again, this is much longer. In recent years when asking student to write an expression such as this, the directions included a phrase such as “write an equation involving one or more integrals that gives ….” This pretty much leads students away from the longer differential equation initial value problem approach.

Part d required students to find the time when in the interval $0\le t\le 120$ minutes the amount of water in the tank was a maximum and to justify their answer. The usual method is to find the derivative of the amount, A(t), set it equal to zero, and then solve for the time.

${A}'\left( t \right)=8-\sqrt{t+1}$

Notice that this is the same regardless of which of the three forms of the expression for A(t) you start with. Thus, an excellent example of the Fundamental Theorem of Calculus used to find the derivative of a function defined by an integral. Or you could just start here without reference to the forms above: the overall rate in the rate in minus the rate out.

${A}'\left( t \right)=0$ when t = 63

This is a maximum by the First Derivative Test since for 0 < t < 63 the derivative of A is positive and for 63 < t <120 the derivative of A is negative.

There is an additional idea on this part of the question in the Teaching Suggestions below.

I like this question because it is a nice real (as real as you can hope for on an exam) situation and for the way the students are led through the problem. I also like the way it can be used to compare the two methods of solution. Then the way they both lead to the same derivative in part d is nice as well. I use this one a lot when working with teachers in workshops and summer institutes for these very reasons.

Teaching Suggestions

Certainly, have your students work through the problem using both methods. They need to learn how to solve an initial value problem (IVP) and this is good practice. Additionally, it may help them see how and when to use one method or the other.
Be sure the students understand why the three forms of A(t) in part c give the same derivative in part d. This makes an important connection with the Fundamental theorem of Calculus.
Like many good AP questions part d can be answered without reference to the other parts. The question starts with more water being pumped in than leaking out. This will continue until the rate at which the water leaks out overtakes the rate at which it is being pumped in. At that instant the rate “in” equals the rate “out” so you could start with $8=\sqrt{t+1}$ . After finding that t = 63, the answer may be justified by stating that before this time more water is being pumped in than is leaking out and after this time the rate at which water leaks out is greater than the rate at which it is pumped in, so the maximum must occur at t = 63.
And as always, consider the graph of the rates.

I used this question as the basis of a lesson in the current AP Calculus Curriculum Module entitled Integration, Problem Solving and Multiple Representations © 2013 by the College Board. The lesson gives a Socratic type approach to this question with a number of questions for each part intended to help the teacher not only work through this problem but to bring out related ideas and concepts that are not in the basic question. The module is currently available at AP sponsored workshops and AP Summer Institutes. Eventually, it will be posted at AP Central on the AB and BC Calculus Home Pages.

Good Question 4: 2008 AB 10

Posted on July 15, 2015 by Lin McMullin

Continuing my occasional series on Good Questions, today’s Good Question is a multiple-choice question from the 2008 AB Calculus exam, number 10. As an exam question it is only so-so, but it has a lot of potential for having a discussion of relative accuracy of Riemann sums in relation to the definite integral they approximate. The key to doing this is to look at the graph. The question relates the numerical and the graphical aspect of Riemann sums, two parts of the Rule of Four.

The question presented the graph of a function f, shown below and asked which of five answer choices has the least value. The choices were $\displaystyle \int_{a}^{b}{f\left( x \right)}dx$ (which I will call I), the left Riemann sum approximation of the integral, L, the right Riemann sum approximation, R, the Midpoint Riemann sum approximation, M, and a Trapezoidal sum approximation, T. Each of the four approximations were to have 4 subintervals of equal length.

There are important things in the stem – namely that the graph is strictly decreasing and concave downward, and one unimportant thing – the number of subdivisions. As long as the graph is strictly monotonic and does not change concavity the number of subdivisions does not matter; the relative size of the five quantities will be the same. Therefore, to see which is least we can look at one subdivision covering the entire interval. That saves a lot of trouble and is worth discussing with your class. Usually, we let the number of subdivisions go off to infinity; here we go the other way.

Looking at a single interval from 1 to 3, it is easy to see by drawing or picturing the rectangle that the least Riemann sum will be R, the right Riemann sum.

So that answers the question, but there is a lot more you can do with the situation. The first that comes to mind is to have your students to put the five values in order from least to greatest. Stop here and try it for yourself.

R is the smallest and L is the largest. Since the top of a trapezoid between the endpoints of the function on the interval lies below the graph of the function, T is less than I.

So far we have R < T < I < L, but where does the midpoint Riemann sum fit in, and why?

Consider the figure above. C is the midpoint of segment AD.The area of the region between AD and the x-axis is the midpoint approximation. Segment BE is tangent to f(x) at C. Notice that $\Delta ABC\cong \Delta DEC$ (Why?) and therefore, the area of the region between segment BE and the x-axis is the same as the area between segment AD and the x-axis. The midpoint approximation is the same as the area a trapezoid whose side is tangent to the graph at the midpoint of the interval and extending to the sides of the interval. So the midpoint approximation of the integral is greater than the integral. (The midpoint rectangle as the same area as the “midpoint trapezoid” and distinguishes it from the endpoint trapezoid).

Then R < T < I < M < L.

At least in this case.

In this case, the function was strictly decreasing and concave down. Have your class investigate other combinations of increasing and decreasing functions that are concave up and concave down. Ask your students, individually or in small groups, to investigate these different cases, and discover and justify that:

There are four cases in all.
The left sum is greatest, and the right sum is least when the function is strictly decreasing.
The left sum is least, and the right sum is greatest when the function is strictly increasing.
When the function is concave down, the endpoint trapezoid lies below the graph of the function and the midpoint trapezoid lies above the graph, therefore T < I < M.
When the function is concave up, the endpoint trapezoid lies above the graph of the function and the midpoint trapezoid lies below the graph, therefore M < I < T.
Consider cases where the function is below the x-axis.

Foreshadowing the FTC

Posted on December 15, 2014 by Lin McMullin

This is an example to help prepare students to tackle the Fundamental Theorem of Calculus (FTC). Use it after the lesson on Riemann sums and the definition of the definite integral, but before the FTC derivation.

Consider the area, A, between the graph of $f\left( x \right)=\cos \left( x \right)$ and the x-axis on the interval $\left[ 0,\tfrac{\pi }{2} \right]$ . Set up a Riemann sum using the general partition:

$0={{x}_{0}}<{{x}_{1}}<{{x}_{2}}<{{x}_{3}}<\cdots <{{x}_{n-2}}<{{x}_{n-1}}<{{x}_{n}}=\tfrac{\pi }{2}$

$\displaystyle A=\int_{0}^{{\scriptstyle{}^{\pi }\!\!\diagup\!\!{}_{2}\;}}{\cos \left( x \right)dx}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\cos \left( {{c}_{i}} \right)\left( {{x}_{i}}-{{x}_{i-1}} \right)}$

Since we can choose the value of ${{c}_{i}}$ any way we want, let’s take the same intervals and use ${{c}_{i}}$ the number guaranteed by the Mean Value Theorem for the function $F\left( x \right)=\sin \left( x \right)$ on the each sub-interval interval. That is, on each sub-interval at $x={{c}_{i}}$

$\left. \frac{d}{dx}\sin \left( x \right) \right|_{x={{c}_{i}}}^{{}}=\cos \left( {{c}_{i}} \right)=\frac{\sin \left( {{x}_{i}} \right)-\sin \left( {{x}_{i-1}} \right)}{\left( {{x}_{i}}-{{x}_{i-1}} \right)}$

Then, substituting into the Riemann sum above

$\displaystyle A=\underset{\Delta x\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\cos \left( {{c}_{i}} \right)\left( {{x}_{i}}-{{x}_{i-1}} \right)}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\frac{\sin \left( {{x}_{i}} \right)-\sin \left( {{x}_{i-1}} \right)}{\left( {{x}_{i}}-{{x}_{i-1}} \right)}\left( {{x}_{i}}-{{x}_{i-1}} \right)}$

$\displaystyle A=\underset{\Delta x\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\left( \sin \left( {{x}_{i}} \right)-\sin \left( {{x}_{i-1}} \right) \right)}$

Now writing out the terms we have a telescoping series:

$\displaystyle A=\left( \sin \left( {{x}_{1}} \right)-\sin \left( {{x}_{0}} \right) \right)+\left( \sin \left( {{x}_{2}} \right)-\sin \left( {{x}_{1}} \right) \right)+\left( \sin \left( {{x}_{3}} \right)-\sin \left( {{x}_{2}} \right) \right)+$

$\displaystyle \cdots +\left( \sin \left( {{x}_{n-1}} \right)-\sin \left( {{x}_{n-2}} \right) \right)+\left( \sin \left( {{x}_{n}} \right)-\sin \left( {{x}_{n-1}} \right) \right)$

$\displaystyle A=\sin \left( {{x}_{n}} \right)-\sin \left( {{x}_{0}} \right)$

$\displaystyle A=\sin \left( \tfrac{\pi }{2} \right)-\sin \left( 0 \right)=1$

As you can see, this is really just the derivation of the FTC applied to a particular function. Now the students should have a better idea of what’s going on when you solve the problem in general, i.e. when you prove the FTC.

Arbitrary Ranges

Posted on February 9, 2014 by Lin McMullin

In my last post I discussed the idea that the ranges of the inverse trigonometric functions are chosen somewhat arbitrarily. For good reasons, the ranges always include the first quadrant and the adjoining quadrant (II or IV) where the function is negative. If possible, the range is also chosen to be continuous. Still the choices are arbitrary.

I discussed the range of the inverse tangent function in relation to the value of the improper integral $\int_{0}^{\infty }{\frac{1}{1+{{x}^{2}}}dx}$ . I noted that if we used some other continuous range for the inverse tangent that the result of this or any other definite integral of this function gives the same value. Thus a range for the inverse tangent of $\left( -\tfrac{\pi }{2},\tfrac{\pi }{2} \right),\left( \tfrac{\pi }{2},\tfrac{3\pi }{2} \right),\left( \tfrac{3\pi }{2},\tfrac{5\pi }{2} \right)$ , etc. will give the same result.

For antiderivatives involving the inverse tangent or inverse cotangent this is true, but what about the other inverse trigonometric functions?

When evaluating the difference between two values as one does when evaluating a definite integral any range which results in a graph “parallel” to the graph over the commonly accepted range gives the same value.

However, for an integral requiring the inverse sine, if we use the range $\left[ \tfrac{\pi }{2},\tfrac{3\pi }{2} \right]$ ,

$\displaystyle \int_{0}^{1/2}{\frac{1}{\sqrt{1-{{x}^{2}}}}dx}=\left. {{\sin }^{-1}}\left( x \right) \right|_{0}^{1/2}$

$={{\sin }^{-1}}\left( \tfrac{1}{2} \right)-{{\sin }^{-1}}\left( 0 \right)=\tfrac{5\pi }{6}-\pi =-\tfrac{\pi }{6}$

Indicating that a region above the x-axis has a negative area!

So $\left[ \tfrac{\pi }{2},\tfrac{3\pi }{2} \right]$ is not a good choice. We could use other ranges for the inverse sine function but they would have to be such that they result in inverse sine graphs “parallel” to the usual graph. So we could use $\left( \tfrac{3\pi }{2},\tfrac{5\pi }{2} \right)$ or $\left( \tfrac{7\pi }{2},\tfrac{9\pi }{2} \right)$ , but not $\left( \tfrac{5\pi }{2},\tfrac{7\pi }{2} \right)$ .

The same problem arises with the inverse cosine, the inverse secant, and the inverse cosecant.

It is best to stick with the commonly accepted ranges. Still, going off on tangents often helps sharpen a student’s understanding.

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Category Archives: Integration Theory

Good Question 8 – or not?

November 2015

Good Question 7 – 2009 AB 3

Good Question 6: 2000 AB 4

Good Question 4: 2008 AB 10

Foreshadowing the FTC

Arbitrary Ranges

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