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# Review of Calculus Without Tears by William Flannery

Calculus Without Tears (CTW) is a collection of worksheets in 3 volumes that teaches the basic concepts of calculus very step-by-step, without a need of much algebra. The collections are intended to be self-teaching workbooks that students can study even before high school.

So what is calculus? It is the **mathematics of change**, extensively used in physics, biology, astronomy, economics, electronics, and other sciences... anywhere where scientists deal with CHANGE.

Calculus originally started as the mathematics of MOTION (Isaac Newton in 1600s). In CTW, calculus has to do with 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 the simplest of motions, which is a runner running at a 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 next volume, *Nature's Favorite Functions*, teaches how to find 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:

- The first chapter uses tables and simple linear functions to describe a
**runner running at a 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 the distributive property if the student hasn't learned it yet. - Chapter three introduces velocity. First it contains lots of exercises about the simple relationship
**velocity = distance / time**. Then comes the important notion of slope and how the slope of a graph equals the velocity of the motion.

Again, there are plenty of exercises dealing with plotting the position and velocity functions 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). - Chapter 4 deals with
**The Area Under a Curve**– usually known as the integral calculus. This volume starts out with a study of 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 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 a differential equation is the opposite operation of finding the derivative: 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.

Figure: Example problem from *Calculus Without Tears* first volume

I liked that the books had so many graphical exercises: plotting, shading, etc. Many times calculus students get to master the mechanical manipulations of symbols, such as finding the derivative, but don't grasp the basic IDEAS. CTW turns this backwards: it deals with the basic ideas and concepts first and 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. Several worksheets from this chapter could easily fit into a physics or science class.**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**proves that the derivative of*t*^{2}*t*^{2}is 2*t*.

This is done by studying a runner whose position*p*at time*t*is given by*p*(*t*) =*t*^{2}. First there are calculations for the 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 and the length of the time interval), the average velocity is 2*a*+*h*.

Then, there are several numerical examples (as exercises) that show how the average velocity approaches a certain number when*h*gets smaller and smaller – and a 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 t^{2}.

I feel that taking a whole chapter to prove this point may cause the student to lose the "forest from the trees." 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 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 (the 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 have to do with how to use the learned rules, graphing, and with the actual proofs.

I would have liked to see more about the graphical meaning of the first and second derivatives. - The second chapter,
**Polynomial Approximation and Taylor's Theorem**, again treats the topics carefully and step-by-step, aiming to help the student understand the reasoning behind it all.

The bound on the error of the polynomial approximation is a very abstract concept. I would have liked to see graphical examples and exercises with a function, the approximation, and the error. - The chapter
**Fundamental Theorem of Calculus**is 11 lessons long so it should convey a thorough understanding of this important result. It relies on graphical exercises a lot, thus emphasizing conceptual understanding. The chapter deals with step functions, approximating any function with a step function, areas, and how this all ties in with the familiar formula*velocity × time = distance*. - The next chapter,
**Roots and Radicals**, starts out explaining integer, fractional, and negative exponents. It reviews the chain rule and then that is used to find the derivative of the root function*t*^{1/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. In Lesson 4, we first find the derivatives of 2^{t}and 3^{t}– and then 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 e^{t}to calculate its values (which is what the calculator probably uses, too). - Then we get to applications of differential equations whose solutions are exponential functions: the velocity of a box with initial push and friction, a falling object (gravity and drag), and the voltage of a capacitor in a simple circuit. Lastly, the chapter has lessons about logarithms, their derivatives, and Taylor approximations.
**Trig Functions**starts with a nice review of angles, the basics of triangles, similar triangles, trigonometric ratios in a right triangle, the unit circle, sine wave, and finding/proving some basic trigonometric identities with the help of figures. The exercises are thoughtful, often graphical, and promote a deep understanding of the material. You cannot get by with just memorizing formulas here.

As usual, 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, and finds some shaded areas using integrals. Lastly the chapter deals with the Taylor polynomial approximation to sine.- The last chapter,
**Second Order Systems**, deals with a spring mass assembly (a spring attached to a wall pulling on a mass that rests on a 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 a quite detailed (and technical) discussion about how to solve this differential equation. The material in this chapter is a little more advanced. I wish that how the equations and parameters relate to the actual physical model would have been explained better. These lessons would of course make a good background material for actual experiments with measurements. The material also briefly introduces an electrical circuit with an inductor and a capacitor.

## In a summary

CTW's purpose is to first teach the 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, and tables;
- A constant review of previous topics;
- A very clear layout, not distracting colors, etc.;
- One double-sided page worksheet per lesson;
- The 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 a required math course for most students, and studying it before algebra definitely puts it as an "extra". But, these worksheets can help immensely even older students who are already studying calculus from a textbook. 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. About $17 per volume (at Amazon). Website BerkeleyScience.com.

At Amazon:

Calculus Without Tears: Easy Lessons for Learning Calculus for Students From the 4th Grade Up

Calculus without Tears, Vol. 2: Newton's Apple

Calculus Without Tears - Vol. 3 - Nature's Favorite Functions

*Review by Maria Miller*