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# Review of Calculus Without Tears by William Flannery

Calculus Without Tears (or CTW) is a collection of worksheets (in 3 volumes) that teach basic concepts of calculus very step-by-step, without need of much algebra. They are intended to be self-teaching workbooks that even students before high school can study. The price is \$15 per workbook if bought at the author's website - very reasonable.

So what is calculus? It is the mathematics of change, extensively used in physics, biology, astronomy, economics or other sciences... anywhere where people study CHANGE.

Calculus originally started as mathematics of MOTION (Isaac Newton in 1600s). In CTW, calculus is studying motion in these terms: position (where you're at), velocity (how quickly you're going and where), and acceleration (how quickly your speed is changing).

Calculus Without Tears starts with studying the simplest of motions, which is a runner running with constant speed (or sometimes standing still!). The entire volume 1 of CTW uses this simple motion while introducing the basic concepts of calculus as they apply to it.

Volume 2 is named Newton's Apple and concentrates on the motion of a falling apple (motion with constant acceleration such as under gravity of Earth, moon, or Mars).

The last volume, Nature's Favorite Functions, goes about finding the derivatives of polynomials, trigonometric functions, roots, exponential and logarithmic functions.

## Overview of the 1st Volume - Constant Velocity Motion

The first volume has five chapters:

• First chapter uses tables and simple linear functions to describe a runner running at constant velocity. It also explains and has exercises for negative numbers and how to use expressions.

• Chapter two teaches coordinate graphing and how to graph the position of the runner. It also teaches distributive property if the student hasn't yet learned it.

• Chapter three introduces velocity. First it has lots of exercises about the simple relationship velocity = distance / time. Then comes the important notion of slope and how slope of the graph equals the velocity of the motion.

Again, there are plenty of exercises dealing with plotting the position function and velocity function on the coordinate graph. Everything proceeds in very small steps - the student won't be left behind. Also, the word derivative is introduced, as are acceleration and the notation f' (f prime).

• Figure: Example problem from Calculus Without Tears first volume

• Chapter 4 studies The Area Under a Curve - usually known as integral calculus. CTW starts by studying the shaded areas between linear functions and the x-axis.

The fundamental idea in the chapter is that the area under the velocity curve gives the distance traveled in that time. Again, CTW presents the basic idea with the simplest of situations without any complex-looking equations. CTW devotes many pages to this topic and to learning the term 'anti-derivative'.

• The last chapter deals with differential equations. Again, it is very simplified. Basically differential equations are the opposite operation of finding the derivative: this time you know the derivative but need to find the function. This chapter just deals with the simple situation of constant-velocity motion, its position, velocity, and acceleration functions.

I liked the fact the books had so much graphical exercises - plotting, shading, etc. Many times calculus students get to master the mechanical symbol manipulations such as finding the derivative, but don't grasp the basic IDEAS. CTW turns this backwards: it deals with the basic ideas and concepts foremost and puts much less emphasis on finding derivatives or integrals of various functions (though the different functions are not forgotten - they are the subject of volume 3).

## Volume 2: Newton's Apple

In this volume, everything wraps around the falling apple situation: motion where acceleration is constant, velocity is a linear function, and position is a quadratic function. Again there are five chapters:

• Physics of a Falling Object - studying the law of gravity and motion with constant acceleration. Many worksheets from this chapter could easily fit into physics/science classes.

• Graphing Solutions to Differential Equations is about plotting a piecewise linear approximation solution to the DE. The differential equations used are very simple.

This method for solving DE's is very important because it is easy, it helps the student understand differential equations and become comfortable with them, AND because finding a piecewise linear approximation is the way engineers solve most DEs! This is because often it is impossible to find an analytical solution to them.
• The chapter Derivative of t2 proves that the derivative of t2 is 2t.
This is done by studying a runner whose position p at time t is given by p(t) = t2. First there are calculations about average velocity in a time interval from a to a+h. It is noted that no matter what a or h are (the beginning point of the time interval and the length of time interval), the average velocity is 2a+h.

Then, there are several numerical examples (as exercises) about the average velocity approaching a certain number when h gets smaller and smaller - and discussion about "instant velocity". This is then tied to the slope of p at the given time.

This is exactly the mathematical defition of derivative - used here for the special case of t2.

I feel that taking a whole chapter to prove this point may cause the student to lose "forest from the trees" - and the teacher definitely needs to be aware of what the book is coming to in this chapter. At this point it might be easier to go through the lessons quicker than one a day.

Then again, it is NOT easy for most (average) students to grasp the mathematics behind "instant velocity" as it is usually presented in math books (as a limit), or the how this is the same as the slope. CTW does a fine job of 'dissecting' this proof into very small steps.

• Chapter 4 is Integrating Linear Functions - this is done by shading and finding the area and then using the antiderivative (fundamental theorem of integral calculus).

• Chapter 5 - Differential Equations is about finding a function when its derivative is a linear function. It also has a few lessons on electrical circuits in the end.

## Volume 3: Nature's Favorite Functions

This volume goes beyond the earlier volumes, into more advanced (and interesting) topics: differentiation rules, polynomial functions, Taylor approximations, roots, radicals, exponential functions, and trigonometric functions. The last chapter contains detailed examples of solving certain differential equations, using the techniques learned.

• The first chapter, Differentiating Polynomials, also teaches the product rule, the chain rule, and the quotient rule of differentiation. These rules are all proven well, typically by using linear approximations to the two functions and the definition of derivative. The product rule is used to find and prove the derivative functions for all polynomials.

The exercises in these lessons generally deal with using the learned rules, graphing, and with the actual proofs.

I would have wished for more stuff about the first and second derivative's graphical meaning.

• The second chapter is Polynomial Approximation and Taylor's Theorem, again with a careful step-by-step treatment aiming to let the student understand the reasoning behind it all

The bound on the error of the polynomial approximation is a very abstract concept and it might be easy to "lose forest from the trees". So I would have wished for graphical examples and problems of a function, the approximation, and the error.

• The chapter Fundamental Theorem of Calculus is 11 lessons long - so should convey a thorough understanding of this important result. It relies a lot on graphical exercises, thus emphasizing the understanding of concepts. The chapter deals with step functions, approximating any function with a step function, areas, and how it all ties in with the familiar formula velocity * time = distance.

• The next chapter, Roots and Radicals, starts out explaining integer, fractional, and negative exponents. We get a chain rule review, and then that is used to find the derivative of a root function t1/n.

• Finally we get to the Exponentials and Logs (my favorites!). The chapter starts out with lessons on interest, population growth/decline, and carbon 14 dating. Lesson 4 first finds the derivatives of 2t and 3t - and then finds a number between 2 and 3 for which the function is equal to its derivative - e. The lessons show how to use Taylor's approximation to et to calculate its values (which is what the calculator probably uses, too).

• Then we get to applications that have a differential equation to solve with exponential function as a solution: a box's velocity with initial push and friction, ball falling (gravity and drag), and the capacitor voltage in a simple circuit. Lastly, the chapter has lessons about logarithms, their derivatives, and Taylor approximations.

• Trig Functions starts with a nice geometry review of angles, basics of triangles, similar triangles, trigonometric ratios in a right triangle, the unit circle, sine wave, and finding/proving some basic trigonometric identities with help of figures. The exercises are thoughtful, often graphical, and promote deep understanding of the material. You cannot get by with just memorizing formulas here.

As always, the derivatives of sine and cosine are proved, not just announced. The student also practices using the chain rule, product rule, and quotient rule with sine and cosine, finds some shaded areas using integrals, and lastly learns about the Taylor polynomial approximation to sine.

• The last chapter, Second Order Systems, deals with a spring mass assembly (a spring attached to wall pulling on mass that rests on table). It turns out that Newton's Second Law of Motion, when applied to this simple situation, yields a second order differential equation.

What follows is quite detailed (and technical) discussions about how to solve this differential equation. The material in this chapter is a little more advanced. I would have hoped for a little better explanations as to how the equations and parameters relate to the actual physical model. These lessons would of course make a good background material for actual experiments with measurements. Also briefly studied is an electrical circuit with an inductor and a capacitor.

## In a summary

CTW's purpose is to first teach basic concepts of calculus - differentiation, integration, and differential equations - without much algebra or complicated equations. In volume one, only linear and constant functions are used (for position). In volume 2, this is expanded to quadratic functions. Then, volume 3 expands the horizons as more complicated functions and situations are studied.

### WHAT I LIKED:

• The basic idea of emphasizing the concepts over symbolic manipulations
• Lots of problems about graphing, plotting, tables
• constant review of previous topics
• very clear layout, not distracting colors etc.
• one double-sided page worksheet per lesson
• concepts progress step-by-step - and the steps are small.

### WHAT I DIDN'T LIKE:

• I personally would have not used * for multiplication, or / for division all through the books.
• Sometimes the lessons go so slowly that a student without a teacher might not see the point the book is coming to.

### WHO CAN I RECOMMEND THIS TO?

Calculus is not required math for most students, and studying it before algebra definitely puts it as an "extra". But, these worksheets can help immensely even older students who already are studying it with some other book. And, they are a good resource for teachers. It is best if the teacher (parent) already knows the subject material, especially in volume 3.

• Anyone interested in learning calculus - middle school or high school.
• Physics teachers will find lots of worksheets to print.
• Calculus teachers.

Calculus Without Tears, by William Flannery. \$19.95 per volume (at Amazon.com). Website BerkeleyScience.com.

Review by Maria Miller, MSc, author of HomeschoolMath.net