“A Little Calculus”

Posted on April 6, 2021 by Lin McMullin

A Little Calculus is an app for iPad and iPhones. While I don’t usually do product reviews, I think this one is so good that I am making an exception.

There are many good websites that illustrate calculus concepts. Many of them, however, do not allow teachers to enter their own examples; they must use the built-in ones. A Little Calculus is an exception.

A Little Calculus contains over 75 demonstrations of calculus and calculus related concepts, and also three graphing calculators (Cartesian, Parametric, and Polar). Each demonstration can be fully edited to any function and conditions by the user. The topics cover precalculus, limits and continuity, Differential Calculus, Integral Calculus, Sequences and Series, Parametric Curves, and Multivariable Calculus. A full list of the concepts included is at the end of this post. The app is available in the App Store; the price is (only) $0.99. Link.

Each topic has an information screen (circled I in the upper right of the screen) that includes (1) “How To” explaining how to input your example and how to use the features specific to that demonstration. (You should read this the first time you use each topic because some features may not be obvious.), (2) “Background” a textbook-like page that explains in detail the mathematics demonstrated, and (3) “Examples” discussing one or more examples of the concept.

The set-up screen (chevron upper right) allows the user to enter the example they want. This then changes to a pull-down screen. The screenshot below is the set-up screen for “The Area Under a Curve” demonstration. Note the Riemann rectangles available from the set-up screen.

The resulting graph is shown next for 3 right Riemann sum rectangles. By clicking on the “+” or “-“ in the bottom right, you can increase or decrease the number of rectangles one at a time; by holding the “+” sign the number increases rapidly to demonstrate n → ∞. The upper and lower limits may be specified on the set-up screen or be dragged from the graph. The current area of the rectangles is shown in the lower left.

The next illustration shows the “Disk Method” section. The lower graph shows the individual rectangles; the upper shows the 6 rectangles rotated around the line y = –1/2. The “+” button increases the number of rectangles.

This shot shows 150 rectangles; the upper graph has been rotated slightly – this is accomplished in real time by dragging your finger across the screen. All three-dimensional graphs may be moved in this manner.

I could go on …

The one slight drawback is that the input screen does not allow the user to enter parentheses. The parentheses are entered correctly by selecting what is to be enclosed and then tapping the next operation button. This leads to some strange looking expressions such as sin(x)² instead of (sin(x))². The expressions are interpreted correctly, but the strange look may confuse some students. It would be easier to type them yourself as usual. Not a big problem.

An effective way to use this app if for the teacher to project a demonstration and discuss the various ideas he or she wishes to discuss. Tapping the screen with two fingers toggles a pointer; this is useful for working on-line with students who cannot see where you are pointing otherwise. Students could use the app to investigate on their own.

UPDATE (August 18,2022) A new feature has been added: you may now save and recall your own example(s) instead of having to reenter them each time. Handy for preparing demonstrations ahead of time or having students prepare their own examples.

Also, two new topics have been added: Logistic growth and the derivative of an exponential function.

Here is a list of the other topics included:

Update 8-18-2022

Other Asymptotes

Posted on March 23, 2021 by Lin McMullin

A few days ago, a reader asked if you could find the location of horizontal asymptotes from the derivative of a function. The answer, alas, is no. You can determine that a function has a horizontal asymptote from its derivative, but not where it is. This is because a function and its many possible vertical translations have the same derivative but different horizontal asymptotes. To locate it precisely, we need more information, an initial condition (a point on the curve).

I have written a post about the relationship between vertical asymptotes and the derivative of the function here. Today’s post will discuss horizontal asymptotes and their derivatives.

Definition: a straight line associated with a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero and the slope of the curve at the point approaches the slope of the line. (Merriam-Webster dictionary). Thus, asymptotes can be vertical, horizontal, or slanted.

Horizontal Asymptotes

Horizontal asymptotes are a form of end behavior: they appear as $\displaystyle x\to \infty$ or $\displaystyle x\to -\infty$ . Specifically, if a function has a horizontal asymptote(s), then $\displaystyle \underset{{x\to \pm \infty }}{\mathop{{\lim }}}\,{f}'\left( x \right)=0$ , At an asymptote, the curve must approach a horizontal line, therefore its slope must be approaching zero as $\displaystyle x\to \infty$ or $\displaystyle x\to -\infty$ . The converse is not true: If the derivative approaches zero, the function may not have a horizontal asymptote. A counterexample is $\displaystyle f\left( x \right)=\sqrt[3]{x}$ .

In the first four examples that follow, the derivative approaches 0 as $\displaystyle x\to \infty$ and/or $\displaystyle x\to -\infty$ .

Example 1: $\displaystyle {f}'\left( x \right)=\frac{1}{{1+{{x}^{2}}}}$ (Figure 1 in blue).This is the derivative of $\displaystyle f\left( x \right)={{\tan }^{{-1}}}(x),\ -\tfrac{\pi }{2}<f\left( x \right)<\tfrac{\pi }{2}$ . (Figure 1 in red). We see that the derivative approaches zero in both directions, this tells us that there may be horizontal asymptote(s). We must look at the function to find where they are: $\displaystyle y=\tfrac{\pi }{2}$ and $\displaystyle y=-\tfrac{\pi }{2}$

2021 Review Notes

Posted on March 16, 2021 by Lin McMullin

About this time of year, I have been posting notes on reviewing and on the ten types of problems that usually appear on the AP Calculus Exams AB and BC. Since the types do not change, I am posting all the links below. They are only slightly revised from last year. You can also find them under “AP Exam Review” on the black navigation bar above.

Each link provides a list of “What students should know” and links to other post and questions from past exams related to the type under consideration.

Note that the 10 Types are not the same as the 10 Units in the Fall 2020 Course and Exam Description. This is because many of the exam questions have parts from different units.

Here are the links to the various review posts:

AP Exam Review – General suggestions, hints, and information. Also, other resources for reviewing.
Type 1 questions – Rate and accumulation questions
Type 2 questions – Linear motion problems
Type 3 questions – Graph analysis problems
Type 4 questions – Area and volume problems
Type 5 questions – Table and Riemann sum questions
Type 6 questions – Differential equation questions
Type 7 questions – Miscellaneous
Type 8 questions – Parametric and vector questions (BC topic)
Type 9 questions – Polar equations (BC topic)
Type 10 questions – Sequences and Series (BC topic)

When assigning past exams questions for review (and you should assign past exam question), keep in mind that students can find the scoring standards online. Even though the AP program forbids this and makes every effort to prevent them from being posted, they are there. Students can “research” the solution. Keep this in mind when assigning questions from past exams. Here is a suggestion Practice Exams – A Modest Proposal

Extreme Polar Conditions

Posted on February 9, 2021 by Lin McMullin

Looking at the graphs of polar curves can be quite fascinating. Doing “calculus” kinds of things is different, yet the same as in Cartesian coordinates. A discussion on the AP Calculus Community here got me thinking about extreme values of polar functions.

The terms maximum and minimum here refers to the value of r(θ) which may be positive (when on the ray making an angle of θ with the polar axis), zero (at the pole), or negative (on the ray opposite to the one making an angle of θ with the polar axis). The distance from the pole to the point is |r(θ)|. As Mark Howell pointed out in the thread linked above, extreme values of r(θ) lie on a circle or circles centered at the pole with radius of |r(θ)|, and finding the slope of tangent lines at the extremes is relatively easy, requiring no calculus: the slope of the ray is y/x, so the slope of the tangent at the extreme is –x/y. As with Cartesian coordinates, at extreme values $\displaystyle dr/d\theta =0$ since r(θ) is change from increasing to decreasing at this point (or vice versa).

A quick look at the graph of simple polar functions seems to show obvious maximum values for r(θ), but a closer look reveals some complications.

The graph of $\displaystyle r\left( \theta \right)=\sin \left( {2\theta } \right)$ , shown below, appears to show 4 maximums. However, if we trace the graph, we find that these points are (1, π/4), (–1, 3π/4), (1, 5π/4) and (–1, 7π/4). Two of the values are maximums where r(θ) = 1 and two are minimums where r(θ) = –1.

Thus, points may be both maximums and minimums.

Polar Curves can be really fun. While working on the idea above, I explored some other curves. Try some yourself using Desmos, GeoGabra, or another graphing app with sliders. Shown below are members of the family of polar curves $\displaystyle r\left( \theta \right)=A\cos \left( {B\theta } \right)+C\sin \left( {D\theta } \right).$ The domain is extended to $\displaystyle 0\le \theta \le 20\pi$ . Notice:

How very slight changes in the parameters give very different looking graphs
Other values give far less “organized” curves
In the third graph, the maximums and minimums on the irregular part of the curve closest to the pole

Polar Equations for AP Calculus

Posted on January 26, 2021 by Lin McMullin

A recent thread on the AP Calculus Community bulletin boards concerned polar equations. One teacher observed that her students do not have a very solid understanding of polar graphs when they get to calculus. I expect this is a common problem. While ideally the polar coordinate system should be a major topic in pre-calculus courses, this is sometimes not the case. Some classes may even omit the topic entirely. Getting accustomed to a new coordinate scheme and a different way of graphing is a challenge. I remember not having that good an understanding myself when I entered college (where first-year calculus was a sophomore course). Seeing an animated version much later helped a lot.

This blog post will discuss the basics of polar equations and their graphs. It will not be as much as students should understand, but I hope the basics discussed here will be a help. There are also some suggestions for extending the study of polar function as the end.

Instead of using the Cartesian approach of giving every point in the plane a “name” by giving its distance from the y-axis and the x-axis as an ordered pair (x,y), polar coordinates name the point differently. Polar coordinates use the ordered pair (r, θ), where r, gives the distance of the point from the pole (the origin) as a function of θ, the angle that the ray from the pole (origin) to the point makes with the polar axis, (the positive half of the x-axis).

Start with this Desmos graph. It will help if you open it and follow along with the discussion below. The equation in the example is $\displaystyle r(\theta )=2+4\sin (\theta )$ You may change this to explore other graphs. (Because of the way Desmos graphs, you cannot have a slider for θ; the a-slider will move the line and the point on the graph. r(a) gives the value of r(θ).

Notice that as the angle changes the point at varying distance from the pole traces a curve.
Move the slider to π/6. Since sin(π/6) = 0.5, r(π/6) = 4. The red dot is at the point (4, π/6). Move the slider to other points to see how they work. For example, θ = π/2 gives the point (6,π/2).
When the slider gets to θ = 7π/6, r = 0 and the point is at the pole. After this the values of r are negative, and the point is now on the ray opposite to the ray pointing into the third and fourth quadrants. The dashed line turns red to remind you of this.
As we continue around, the point returns to the origin at θ = 11π/6, then values are again positive.
The graph returns to its starting point when θ = 2π. Note (2,0) is the same point as (2, 2π).
Even though this is the graph of a function, some points may be graphed more than once and the vertical line test does not apply.
If we continued around, the graph will retrace the same path. This often happens when the polar function contains trig functions with integer multiples of θ.
- This does not usually happen if no trig functions are involved – try the spiral r = θ.
- If you enter non-integer multiples of θ and extend the domain to large values, vastly different graphs will appear, often making nice designs. Try $\displaystyle r\left( \theta \right)=2+4\sin \left( {1.3\theta } \right)$ for $\displaystyle 0\le \theta \le 20\pi$ . This is an area for exploration (if you have time).

In pre-calculus courses several families of polar graphs are often studied and named. For example, there are cardioids, rose curves, spirals, limaçons, etc. The AP Exams do not refer to these names and students are not required to know the names. The exception is circles which have the following forms where R is the radius: θ=R, r = Rsin(θ) or r = Rsin(θ)

To change from polar to rectangular for use the equations $x=r\cos \left( \theta \right)$ and $y=r\sin \left( \theta \right)$ . This is simple right triangle trigonometry (draw a perpendicular from the point to the x-axis and from there to the pole).

To change from rectangular to polar form use $r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}$ and $\displaystyle \theta =\arctan \left( {\tfrac{y}{x}} \right)$

AP Calculus Applications

There are two applications that are listed on the AP Calculus Course and Exam Description: using and interpreting the derivative of polar curves (Unit 9.7) and finding the area enclosed by a polar curve(s) (Units 9.8 and 9.9).

Since calculus is concerned with motion, AP Students should be able to analyze polar curves for how things are changing:

The rate of change of r away from or towards the pole is given by $\displaystyle \frac{{dr}}{{d\theta }}$
The rate of change of the point with respect to the x-direction is given by $\displaystyle \frac{{dx}}{{d\theta }}$ where $\displaystyle x=r\cos \left( \theta \right)$ from above.
The rate of change of the point with respect to the y-direction is given by $\displaystyle \frac{{dy}}{{d\theta }}$ where $\displaystyle y=r\sin \left( \theta \right)$ from above.
The slope of the tangent line at a point on the curve is $\displaystyle \frac{{dy}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}$ . See 2018 BC5 (b)

Area

$\displaystyle \underset{{\Delta \theta \to 0}}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{\infty }{{\tfrac{1}{2}}}{{r}_{i}}^{2}\Delta \theta =\tfrac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{r}^{2}}d\theta }}$

CAUTION: In using this formula, we need to be careful that the curve does not overlap itself. In the Desmos example, the smaller loop overlaps the larger loop; integrating from 0 to 2π counts the inner loop twice. Notice how this is handled by considering the limits of integration dividing the region into non-overlapping regions:

The area of the outer loop is $\displaystyle \tfrac{1}{2}\int_{{-\pi /6}}^{{7\pi /6}}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}\approx 35.525$
The area of the inner loop is $\displaystyle \tfrac{1}{2}\int_{{7\pi /6}}^{{11\pi /6}}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}\approx 2.174$
Integrating over the entire domain gives the sum of these two: $\displaystyle \tfrac{1}{2}\int_{0}^{{2\pi }}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}=12\pi \approx 37.699$ . This is not the correct area of either part.

This problem can be avoided by considering the geometry before setting up the integral: make sure the areas do not overlap. Restricting r to only non-negative values is often required by the fine print of the theorem in textbooks, but this restriction is not necessary when finding areas and makes it difficult to find, say, the area of the smaller inner loop of the example. Here is another example:

$\displaystyle r\left( \theta \right)=\cos \left( {3\theta } \right)$ . Between 0 and $\displaystyle 2\pi$ this curve traces the same path twice.

Infinite Sequences and Series – Unit 10

Posted on January 19, 2021 by Lin McMullin

Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

Topics 10.1 – 10.2

Topic 10.1: Defining Convergent and Divergent Series.

Topic 10. 2: Working with Geometric Series. Including the formula for the sum of a convergent geometric series.

Topics 10.3 – 10.9 Convergence Tests

The tests listed below are tested on the BC Calculus exam. Other methods are not tested. However, teachers may include additional methods.

Topic 10.3: The n^th Term Test for Divergence.

Topic 10.4: Integral Test for Convergence. See Good Question 14

Topic 10.5: Harmonic Series and p-Series. Harmonic series and alternating harmonic series, p-series.

Topic 10.6: Comparison Tests for Convergence. Comparison test and the Limit Comparison Test

Topic 10.7: Alternating Series Test for Convergence.

Topic 10.8: Ratio Test for Convergence.

Topic 10.9: Determining Absolute and Conditional Convergence. Absolute convergence implies conditional convergence.

Topics 10.10 – 10.12 Taylor Series and Error Bounds

Topic 10.10: Alternating Series Error Bound.

Topic 10.11: Finding Taylor Polynomial Approximations of a Function.

Topic 10.12: Lagrange Error Bound.

Topics 10.13 – 10.15 Power Series

Topic 10.13: Radius and Interval of Convergence of a Power Series. The Ratio Test is used almost exclusively to find the radius of convergence. Term-by-term differentiation and integration of a power series gives a series with the same center and radius of convergence. The interval may be different at the endpoints.

Topic 10.14: Finding the Taylor and Maclaurin Series of a Function. Students should memorize the Maclaurin series for $\displaystyle \frac{1}{{1-x}}$ , sin(x), cos(x), and e^x.

Topic 10.15: Representing Functions as Power Series. Finding the power series of a function by, differentiation, integration, algebraic processes, substitution, or properties of geometric series.

Timing

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics:

Before sequences

Amortization Using finite series to find your mortgage payment. (Suitable for pre-calculus as well as calculus)

A Lesson on Sequences An investigation, which could be used as early as Algebra 1, showing how irrational numbers are the limit of a sequence of approximations. Also, an introduction to the Completeness Axiom.

Everyday Series

Convergence Tests

Reference Chart

Which Convergence Test Should I Use? Part 1 Pretty much anyone you want!

Which Convergence Test Should I Use? Part 2 Specific hints and a discussion of the usefulness of absolute convergence

Good Question 14 on the Integral Test

Sequences and Series

Graphing Taylor Polynomials Graphing calculator hints

Introducing Power Series 1

Introducing Power Series 2

Introducing Power Series 3

New Series from Old 1 substitution (Be sure to look at example 3)

New Series from Old 2 Differentiation

New Series from Old 3 Series for rational functions using long division and geometric series

Geometric Series – Far Out An instructive “mistake.”

A Curiosity An unusual Maclaurin Series

Synthetic Summer Fun Synthetic division and calculus including finding the (finite)Taylor series of a polynomial.

Error Bounds

Error Bounds Error bounds in general and the alternating Series error bound, and the Lagrange error bound

The Lagrange Highway The Lagrange error bound.

What’s the “Best” Error Bound?

Review Notes

Type 10: Sequences and Series Questions

Parametric Equations, Polar Coordinates, and Vector-Valued Functions – Unit 9

Posted on January 12, 2021 by Lin McMullin

Unit 9 includes all the topics listed in the title. These are BC only topics (CED – 2019 p. 163 – 176). These topics account for about 11 – 12% of questions on the BC exam.

Comments on Prerequisites: In BC Calculus the work with parametric, vector, and polar equations is somewhat limited. I always hoped that students had studied these topics in detail in their precalculus classes and had more precalculus knowledge and experience with them than is required for the BC exam. This will help them in calculus, so see that they are included in your precalculus classes.

Topics 9.1 – 9.3 Parametric Equations

Topic 9.1: Defining and Differentiation Parametric Equations. Finding dy/dx in terms of dy/dt and dx/dt

Topic 9.2: Second Derivatives of Parametric Equations. Finding the second derivative. See Implicit Differentiation of Parametric Equations this discusses the second derivative.

Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations.

Topics 9.4 – 9.6 Vector-Valued Functions and Motion in the plane

Topic 9.4 : Defining and Differentiating Vector-Valued Functions. Finding the second derivative. See this A Vector’s Derivatives which includes a note on second derivatives.

Topic 9.5: Integrating Vector-Valued Functions

Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions. Position, Velocity, acceleration, speed, total distance traveled, and displacement extended to motion in the plane.

Topics 9.7 – 9.9 Polar Equation and Area in Polar Form.

Topic 9.7: Defining Polar Coordinate and Differentiation in Polar Form. The derivatives and their meaning.

Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves. Students should know how to find the intersections of polar curves to use for the limits of integration.

Timing

The suggested time for Unit 9 is about 10 – 11 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics :

Parametric Equations

Vector Valued Functions

Polar Form

Teaching Calculus

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

Author Archives: Lin McMullin

“A Little Calculus”

Other Asymptotes

2021 Review Notes

Extreme Polar Conditions

Polar Equations for AP Calculus