After presenting the Fundamental Theorem of Calculus to my calculus students, I make a point of doing the following example in class:

$\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx$

Hopefully my students are able to produce the correct answer:

$\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx = \displaystyle \left[ \frac{x^3}{12} \right]^4_0$

$= \displaystyle \frac{(4)^3}{12} - \frac{(0)^3}{12}$

$= \displaystyle \frac{64}{12}$

$= \displaystyle \frac{16}{3}$

Then I tell my students that they’ve probably known the solution of this one since they were kids… and I show them the classic video “Unpack Your Adjectives” from Schoolhouse Rock. They’ll watch this video with no small amount of confusion (“How is this possibly connected to calculus?”)… until I reach the 1:15 mark of the video below, when I’ll pause and discuss this children’s cartoon. This never fails to get an enthusiastic response from my students.

If you have no idea what I’m talking about, be sure to watch the first 75 seconds of the video below. I think you’ll be amused.

Inverse Functions: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the different definitions on inverse functions that appear in Precalculus and Calculus.

Square Roots, nth Roots, and Rational Exponents

Part 1: Simplifying $\sqrt{x^2}$

Part 2: The difference between $\sqrt{t}$ and solving $x^2 = t$

Part 3: Definition of an inverse function and the horizontal line test

Part 4: Why extraneous solutions may occur when solving algebra problems involving a square root

Part 5: Defining $\sqrt{x}$

Part 6: Consequences of the definition of $\sqrt{x}$ : simplifying $\sqrt{x^2}$

Part 7: Defining $\sqrt[n]{x}$ if $n$ is odd or even

Part 8: Rational exponents if the denominator of the exponent is odd or even

Arcsine

Part 9: There are infinitely many solutions to $\sin x = 0.8$

Part 10: Defining arcsine with domain $[-\pi/2,\pi/2]$

Part 11: Pedagogical thoughts on teaching arcsine.

Part 12: Solving SSA triangles: impossible case

Part 13: Solving SSA triangles: one way of getting a unique solution

Part 14: Solving SSA triangles: another way of getting a unique solution

Part 15: Solving SSA triangles: continuation of Part 14

Part 16: Solving SSA triangles: ambiguous case of two solutions

Part 17: Summary of rules for solving SSA triangles

Arccosine

Part 18: Definition for arccosine with domain $[0,\pi]$

Part 19: The Law of Cosines and solving SSS triangles

Part 20: Identifying impossible triangles with the Law of Cosines

Part 21: The Law of Cosines provides an unambiguous angle, unlike the Law of Sines

Part 22: Finding the angle between two vectors

Part 23: A proof for why the formula in Part 22 works

Arctangent

Part 18: Definition for arctangent with domain $(-\pi/2,\pi/2)$

Part 24: Finding the angle between two lines

Part 25: A proof for why the formula in Part 24 works.

Arcsecant

Part 26: Defining arcsecant using $[0,\pi/2) \cup (\pi/2,\pi]$

Part 27: Issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 28: More issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 29: Defining arcsecant using $[0,\pi/2) \cup [pi,3\pi/2)$

Logarithm

Part 30: Logarithms and complex numbers

Symphonic Equations: Waves and Tubes

This excellent and engaging video describes how sine and cosine functions can be applied to music.

Different definitions of e: Index

Part 1: Justification for the formula for discrete compound interest

Part 2: Pedagogical thoughts on justifying the discrete compound interest formula for students.

Part 3: Application of the discrete compound interest formula as compounding becomes more frequent.

Part 4: Informal definition of $e$ based on a limit of the compound interest formula.

Part 5: Justification for the formula for continuous compound interest.

Part 6: A second derivation of the formula for continuous compound interest by solving a differential equation.

Part 7: A formal justification of the formula from Part 4 using the definition of a derivative.

Part 8: A formal justification of the formula from Part 4 using L’Hopital’s Rule.

Part 9: A formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 10: A second formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 11: Numerical computation of $e$ using Riemann sums and the Trapezoid Rule to approximate areas under $y = 1/x$ .

Part 12: Numerical computation of $e$ using $\displaystyle \left(1 + \frac{1}{n} \right)^{1/n}$ and also Taylor series.

An Evaluative Calculus Project: Applying Bloom’s Taxonomy to the Calculus Classroom

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight Gizem Karaali (2011) An Evaluative Calculus Project: Applying Bloom’s Taxonomy to the Calculus Classroom, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:8, 719-731, DOI: 10.1080/10511971003663971

Here’s the abstract:

In education theory, Bloom’s taxonomy is a well-known paradigm to describe domains of learning and levels of competency. In this article I propose a calculus capstone project that is meant to utilize the sixth and arguably the highest level in the cognitive domain, according to Bloom et al.: evaluation. Although one may assume that mathematics is a value-free discipline, and thus the mathematics classroom should be exempt from focusing on the evaluative aspect of higher-level cognitive processing, I surmise that we as mathematics instructors should consider incorporating such components into our courses. The article also includes a brief summary of my observations and a discussion of my experience during the Fall 2008 semester, when I used the project described here in my Calculus I course.

The full article can be found here: http://dx.doi.org/10.1080/10511971003663971

Calculators and Complex Numbers: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how the trigonometric form of complex numbers, DeMoivre’s Theorem, and extending the definitions of exponentiation and logarithm to complex numbers.

Part 1: Introduction: using a calculator to find surprising answers for $\ln(-5)$ and $\sqrt[3]{-8}$ . See the video below.

Part 2: The trigonometric form of complex numbers.

Part 3: Proving the theorem

$\left[ r_1 (\cos \theta_1 + i \sin \theta_1) \right] \cdot \left[ r_2 (\cos \theta_2 + i \sin \theta_2) \right] = r_1 r_2 (\cos [\theta_1+\theta_2] + i \sin [\theta_1+\theta_2])$

Part 4: Proving the theorem

$\displaystyle \frac{ r_1 (\cos \theta_1 + i \sin \theta_1) }{ r_2 (\cos \theta_2 + i \sin \theta_2) } = \displaystyle \frac{r_1}{r_2} (\cos [\theta_1-\theta_2] + i \sin [\theta_1-\theta_2])$

Part 5: Application: numerical example of De Moivre’s Theorem.

Part 6: Proof of De Moivre’s Theorem for nonnegative exponents.

Part 7: Proof of De Moivre’s Theorem for negative exponents.

Part 8: Finding the three cube roots of -27 without De Moivre’s Theorem.

Part 9: Finding the three cube roots of -27 with De Moivre’s Theorem.

Part 10: Pedagogical thoughts on De Moivre’s Theorem.

Part 11: Defining $z^q$ for rational numbers $q$ .

Part 12: The Laws of Exponents for complex bases but rational exponents.

Part 13: Defining $e^z$ for complex numbers $z$

Part 14: Informal justification of the formula $e^z e^w = e^{z+w}$ .

Part 15: Simplification of $e^{i \theta}$ .

Part 16: Remembering DeMoivre’s Theorem using the notation $e^{i \theta}$ .

Part 17: Formal proof of the formula $e^z e^w = e^{z+w}$ .

Part 18: Practical computation of $e^z$ for complex $z$ .

Part 19: Solving equations of the form $e^z = w$ , where $z$ and $w$ may be complex.

Part 20: Defining $\log z$ for complex $z$ .

Part 21: The Laws of Logarithms for complex numbers.

Part 22: Defining $z^w$ for complex $z$ and $w$ .

Part 23: The Laws of Exponents for complex bases and exponents.

Part 24: The Laws of Exponents for complex bases and exponents.

Two ways of doing an integral: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on different ways of computing the integral $\displaystyle \int \frac{dx}{\sqrt{4x-x^2}}$ .

Part 1: The two “different” answers.

Part 2: Explaining why the two “different” answers are really equivalent.

Day One of My Calculus Class: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on what I teach my students on the first day of calculus in order to start the transition from Precalculus and to get them engaged for what we’ll be doing throughout the semester.

Part 1: The two themes of calculus: Approximating curved things by straight things and passing to limits.

Part 2: Using the distance-rate-time formula to estimate how fast an accelerating object lands when dropped from a tall building.

Part 3: Passing to limits to precisely calculate the above velocity.

Part 4: Using rectangles to estimate the area under a parabola.

Part 5: Passing to limits to precisely calculate the area under a parabola.

Part 6: Final comments: these two questions apparently have nothing to do with each other, but are in fact highly interrelated. The connection between these two topics, the Fundamental Theorem of Calculus, is one of greatest discoveries in the history of mankind, which my students are now privileged to understand at the ripe old age of 18 or 19 years old.

Why Does 0.999… = 1? (Index)

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on different techniques that I’ll use to try to convince students that $0.999\dots = 1$ .

Part 1: Converting the decimal expansion to a fraction, with algebra.

Part 2: Rewriting both sides of the equation $1 = 3 \times \displaystyle \frac{1}{3}$ .

Part 3: Converting the decimal expansion to a fraction, using infinite series.

Part 4: A proof by contradiction: what number can possibly be between $0.999\dots$ and $1$ ?

Part 5: Same as Part 4, except by direct reasoning.

Category: Calculus

Integration by parts

Schoolhouse Rock and Calculus