As CME, Chapter 7 notes, suppose:

```
```

That is to say, *y* is some function of *x*. If that is so, then:

```
```

is sometimes denoted:

```
```

with the apostrophe indicating differentiation.

Because of Sway's liberal views on naming variables, we can use the same notation. Consider defining a line:

var f = line(3,-5); //equivalent to y = 3x - 5

where the *line* constructor is just a wrapper for a call to the *plus* constructor:

function line(m,b) { plus(term(m,1),term(b,0)); }

Then, we can differentiate our line thusly:

var f' = f . diff();

Since *f* represents a line and the differentiation of a line yields the slope of a line and the slope of the line is always a constant *m*, we would expect that evaluating at different points would always yield the slope:

sway> f' . toString(); STRING: 3x^0 + 0x^-1 sway> f' . value(0); EVALUATION ERROR: :mathError stdin,line 9: exponentiation: cannot divide by zero

What happened? The problem is the second term of *f'*. We end up evaluating that term with *x* = 0. What is `x ^ -1` when *x* is zero? It's . That's what caused the divide-by-zero error. What should we do about it? Whenever a term's exponent goes to zero, we should force the exponent to stay at zero, when differentiating.

Here's a new version of *diff* for *term*:

function diff() { if (n == 0) { term(0,0); } else { term(a * n,n - 1); } }

Does it work? After remaking, *f* and *f'*, we have

sway> f' . toString(); STRING: 3x^0 + 0x^0 sway> f' . value(0); REAL_NUMBER: 3.000000000

sway> f' . value(5); REAL_NUMBER: 3.000000000

we see that it does.

If we repeat the differentiation process again, we find out how fast the slope is changing at any given point. For a line, the slope never changes so we would expect the derivative of the derivative (also known as the *second* derivative) always to yield zero.

var f'' = f' . diff(); sway> f'' . toString(); STRING: 0x^0 + 0x^0 sway> f'' . value(0); INTEGER: 0 sway> f'' . value(5); INTEGER: 0

Confirmed.

## Contents

## More on visualizationEdit

We really need to do something about our visualization. It's printing out terms that we really don't need to see. Let's simplify the output by making *term'*s *toString* method more complicated. Please implement the following rules for term's *toString* method:

- if the coefficient is zero,
*toString*should return "0" - if the exponent is zero,
*toString*should return "" + a - if both the coefficient and the exponent are 1,
*toString*should return "x" - if the exponent is one,
*toString*should return "" + a + "x" - if the coefficient is one,
*toString*should return "x^" + n - otherwise,
*toString*should return "" + a + "x^" + n

Earlier rules take precedence over later rules. After implementing these rules and remaking `f`, `f'`, and `f''`, we get the following visualizations:

sway> f . toString(); STRING: 3x + -5 sway> f' . toString(); STRING: 3 + 0 sway> f'' . toString(); STRING: 0 + 0

Somewhat better. Now we need to improve *plus* to throw away constant terms with a value of zero. We can do this in the main body of the *plus* constructor:

- if the first argument is equivalent to zero, have
*plus*return the second argument - if the second argument is equivalent to zero, have
*plus*return the first argument - otherwise, have
*plus*return`this`

Your logic should look like this:

function plus(p,q) { ... if (p is equivalent to zero) { q; } else if (q is equivalent to zero) { p; } else { this; } }

How do we determine if an argument to *plus* is equivalent to zero? Since we are representing zero as a term with a zero coefficient, we could use logic like this:

if (p is :term && p . coefficient == 0)

Now our hack of using a term to represent zero is starting to raise its ugly head. Our code is becoming non-obvious to someone who might be reading our code (for maintenance purposes, perhaps) without benefit of reading this text. Zero is a constant, in that it doesn't ever change; zero is always zero. A term, on the other hand, often changes, depending on the value of the independent variable. This leads to a cognitive dissonance that increases the difficulty of understanding our code.

Let's solve this problem by building a 'constant' constructor. This constructor will have all the methods of *term*, but will better represent a value that never changes. Here is an attempt:

function constant(value) { function toString() { "" + value; } function diff() { ... } function y(x) { ... } this; }

The implementations of *diff* and *y* are left as an exercise and should implement the following logic:

- the differential of a constant is a constant zero
- the y value of a constant is always the value of the constant, regardless of the value of the independent variable

Now, we can test for a zero object in plus like this:

if (p is :constant && p . value == 0)

See how this code is much more understandable. We now update our diff method in term to take advantage of our new *constant* constructor.

function term(coefficient,exponent) { ... function diff() { if (n == 0) { constant(0); } else { term(a * n,n - 1); } } ... this; }

With these changes and after remaking `f`, `f'`, and `f''`with the new versions of *plus*, *term* and *constant*, we get the following visualizations:

sway> f . toString(); STRING: 3x + -5 sway> f' . toString(); STRING: 3 sway> f'' . toString(); STRING: 0

Much better.

## Successive Differentiations of Complex PolynomialsEdit

It's rather uneventful to repeatedly differentiate a line. Higher-order polynomials^{[1]} are a little more interesting. Rather than build up a high order polynomial of many terms piece by piece, as in:

var a = term(1,0); var b = plus(term(2,1),a); var c = plus(term(3,2),b); var y = plus(term(4,3),c);

We can build it up *en masse*:

var y = plus(term(4,3),plus(term(3,2),plus(term(2,1),term(1,0))));

or since this whole scale building can be hard to read, build *y* using infix operator syntax:

var y = term(4,3) plus term(3,2) plus term(2,1) plus term(1,0);

No matter how you make *y* (it's all a matter of preference), we can differentiate it successively:

var y' = y . diff(); var y'' = y' . diff(); var y''' = y'' . diff(); sway> y . toString(); STRING: 4x^3 + 3^x2 + 2x + 1 sway> y' . toString(); STRING: 12x^2 + 6^x + 2 sway> y'' . toString(); STRING: 24x + 6 sway> y''' . toString(); STRING: 24

To check numerically, let's evaluate y'' at x = 1 and 2.

sway> y'' . value(1); INTEGER: 30 sway> y'' . value(2); INTEGER: 54

The object y''' should represent the constant 24:

sway> y''' . value(1); INTEGER: 24 sway> y''' . value(2); INTEGER: 24

Everything looks good.

## QuestionsEdit

**1**. Justify mathematically each of the simplifying visualization rules for a term.

**2**. Add simplification to the *minus* constructor. The simplification is similar to *plus* but is a tiny bit trickier.

**3**. Add simplification to the *times* constructor. You should return constant zero if either argument is zero and if one of the arguments is constant one, return the other. Otherwise, return *this*.

**4**. Add simplification to the *div* constructor. You should return constant zero if the numerator (the first argument) is constant zero. You should return the numerator if the denominator (the second argument) is constant one. Otherwise, you should return *this*.

**5**. Represent and plot in sway using *gnuplot*. What is ? Plot it versus *x*. What does it represent? What is ? Plot it versus *x*. What does it represent? What is ? Plot it versus *x*.

**6**. Use both sway or pencil and paper to do 1-3 on p. 82.

## FootnotesEdit

- ↑ The higher the order of a polynomial, the larger the largest exponent in the terms making up the polynomial.