Read about this material here: Symbolic math in Julia.
","metadata":{}}, {"cell_type":"markdown","source":"Begin by loading our package for plotting and a new package that allows symbolic math:
","metadata":{}}, {"outputs":[],"cell_type":"code","source":["using Plots\nusing SymPy"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"There are many computer algebra systems (CAS) that do symbolic math. Most students are familiar with Wolfram's alpha site, as many homework problems can be easily done there. That site uses Wolfram's Mathematica program, though the interface relaxes greatly the syntax of that program. Other CASs include Maple and Sage. Both Mathematica and Maple are available for free to CUNY students and Sage is free to everyone, as it is open source.
In addition to these, there is SymPy, an add-on to the popular Python programming language. Julia's SymPy package interfaces with Python's SymPy package in a fairly easy to use manner.
The first thing one does is create some symbolic variables:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["(x, y, z, h)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["x, y, z, h = symbols(\"x, y, z, h\")"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"(This can also be done with just @vars x y z h
That command creates symbolic variables that (more or less) magically interact with Julia functions. So for example, we can create a symbolic expression as follows:
Note a subtle but big difference: we did not define a function, rather p is an expression. (A function would be p(x) = ...). Symbolic expressions print slightly differently than functions, so this can be a clue. Also note that we could have defined a function, then evaluated it on the symbolic value of $x$:
What is so great about symbolic expressions? Well, they can be manipulated symbolically!
","metadata":{}}, {"cell_type":"markdown","source":"For example to factor the polynomial p we have the factor function:
Or if we used a function:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$16 \\left(x - 4\\right) \\left(x - 2\\right)$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["factor(q(x))"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"(factor applied to a symbolic expression tries to factor as a polynomial, say; whereas factor applied to an integer tries to find prime factors. The factor function is generic, so can have different implementations depending on the type of its argument.)
To solve for p=0 we have solve:
The two answers are $2$ and $4$, as one could read from the factorization of p. The solve function tries to solve when the expression is equal to $0$. To solve something of the type $g(x) = h(x)$, use $g(x) - h(x) = 0$, as is done with fzero, our numeric solver.
Symbolic expressions are not functions, but they can often be called like functions. In the simplest case – with just one free symbol, the notation is similar:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$2$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ex = x^2 - 2x + 2\nex(2)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"For expressions with one or more variable, then an indication must be made as to which variable gets which value. There are a few ways to do this, but using => might be the most direct:
The use of () to substitute a value is only valid for version 0.4 or greater. This is just a convenient interface for subs to substitute a value into a expression. The syntax for that is: subs(ex, (x,xvalue), (y, yvalue), ...). So, we could have done:
Algebra functions include expand to expand a polynomial expression, simplify to simplify algebraically an expression, together to combine expressions, and factor to factor expressions.
Factor the polynomial $p(x) = -2x^4 - x^3 + 3x^2$
\nFind the zeros and any vertical asymptotes of the rational function
\n(Use factor to get the factors, then read these values off.)
Factor the polynomial $p(x) = 4x^4 + 2x^3 - 2x^2 -3x - 1$. How many roots are there? How many real roots are there?
\nSolve the equation $x^4 - 8x^2 + 8 = 0$. How many real roots are there?
\nSimplify
\nFrom your simplified answer, what would be the value if $h=0$?
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"The limit function from SymPy implements Gruntz's algorithm to find symbolic derivatives. It does not have the issues with floating point that a numeric approach does.
The basic form is limit(expr, x, c), where x is the symbolic variable and c is where the limit is being taken. Optionally one can include dir="+" or dir="-" to find limits from the right or left. For example, the right limit at $c=0$ of $x^x$ is given by:
Symbolically find
\nSymbolically find
\nSymbolically find
\nYou can define $m$ and $n$ as symbols and your answer will include them:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["(m, n)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["m,n = symbols(\"m, n\")"],"metadata":{},"execution_count":1}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Symbolically find
\nLimits at infinity just need c=oo (oh-oh, not zero-zero).
What is the value of this expression and does it make sense?
\nSymPy provides the diff function for finding derivatives: diff(ex, x) will find the derivative in x, whereas diff(ex, x, 2) will find the second derivative. For example, here we see the chain rule in action:
(Just to be clear, the D function from the Roots package was D(f) which creates a function. Here we use f(x) which creates a symbolic expression. It is an important distinction, diff(f,x) will not work, as desired.)
What is the derivative of $f(x) = \\sin(x) / (\\tan^{-1}(x) + \\tan^2(x))$?
\nWhat is the derivative of
\nFind the second derivative of $f(x) = \\tan^{-1}(x)$.
\nFind the 10th derivative of $f(x) = xe^{-x}$.
\nDoes this limit give the first derivative? Check that it does or doesn't for $f(x) = \\sin(x)$.
\nDoes this limit give the second derivative? Check that it does or doesn't for $f(x) = \\sin(x)$.
\nFind the critical points of $f(x) = 3x^4−32x^3+114x^2−144x + 2$ by solving for when the derivative is $0$.
\nFind the inflection points of $f(x) = e^{-x^2}$
\nFor the function $\\sin(x)$ over the interval $[0, \\pi/2]$ find a point $c$ such that the slope of the tangent line at $c$ is equal to the slope of the secant line from $0$ to $\\pi/2$.
\nThe plot function is overloaded to also plot symbolic expressions of a single variable. The use is not so different than how it is used to plot a function:
The difference is we plot the expressions sin(x) and cos(x) rather than function objects sin and cos.
Graphically solve for when $f(x) = e^x$ and $g(x) = 10 + 5x$ intersect for $x > 0$.
\nFor the function $f(x) = e^x$. Over the interval $[0,3]$, plot both $f(x)$ and the expression
\nWhere $n!=n \\cdot (n-1) \\cdots 2 \\cdot 1$, and can be evaluated with factorial(2). Do the two mostly agree on this interval?
(One tedious aspect of finding $f'(a)$ is that it is done in two steps like replace(diff(f, x), x, a) (find the derivative in x, the replace x with a). The following function can shorten the above so that D(f,k)(a) works on symbolic expression.
The integrate function can be used for integration – when an antiderivative exists. Integration comes in two flavors
indefinite integrals (or basically antiderivatives)
\ndefinite integrals.
\nThe two are related, as the definite integral $\\int_a^b f(x) dx = F(b) - F(a)$ where $F(x)$ is any antiderivative (they all differ by atmost a constant).
","metadata":{}}, {"cell_type":"markdown","source":"In SymPy, we simply pass in a symbolic expression and optionally limits of integration:
or
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$- \\frac{11}{e^{10}} + 1$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["integrate(x*exp(-x), (x, 0, 10))\t# definite"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"The limits of integration are lumped together as a tuple of the form (symbol, a, b), this syntax allows for multiple integration.
Find an antiderivative of $\\sin(x) \\cdot \\cos(2x)$.
\nFind the area under the curve of $\\sin(x)$ bewteen $0$ and $\\pi$.
\nIf you use a symbolic variable for the limits of integration the answer can be expressed that way. Find
\nThen differentiate it to confirm that the fundamental theorem of calculus holds.
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"The mean value theorem for integrals states that there exists a $c$ in $[a,b]$ for which
\nFor $f(x) = 10e^{-5t}$ find a $c$ in the interval $[0,1]$.
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