Review for test 1 in MTH 229

[Get this a an ipynb file.]

Test 1 will cover the first 4 projects. This review workbook serves two purposes:

• to go over the major concepts we've learned

• to give you a workbook to start your exam from. The exam itself will be distributed on paper and the answers written on paper.

To begin, you should "run all" so that these packages are loaded into your Julia session.

using MTH229

Calculator skills:

The basic ideas of a calculator easily correspond to ideas of julia:

• basic computations involving +, -, *, / and ^ utilize the same order of operations (PEMDAS) that they do on most calculators, save for expressions like 5x where that multiplication happens before the "P".

• The buttons on a calculator for functions are replaced by names. So the "sin" button becomes sin(). Buttons like "sin$^{-1}$" are not spelled that way as functions (asin in this case). In fact the biggest gotcha is translating the shorthand of math related to powers to valid commands in julia.

• The = in julia is assignment. In math it usually implies solving an equation. This is completely different. The left hand side of the = sign in julia is just a name or function name with arguments, not an expression.

Functions

Basic functions in julia and in math look almost identical: $f(x) = \sin(x) + \cos(2x)$ becomes

f(x) = sin(x) + cos(2x)

f (generic function with 1 method)

Julia has some alternative forms though, for example when parameters are involved these can be added as additional arguments to the function with default values.

Functions in math that involve cases are often easily handled with the ternary operator: predicate ? iftrue : iffalse. An example was a model for cost when the first 100 cost 5 dollars per unit and afterwards 4 dollars per unit:

cost(x) = (x <= 100) ? 5*x : 5*100 + 4*(x - 100)

cost (generic function with 1 method)

Here are some examples:

• A function describing a secant line between 0 and $\pi/2$ for the sine function is given using the point-slope form of a line $(y=y0 + m \cdot (x-x0))$:

f(x) = sin(0) + (sin(pi/2) - sin(0)) / (pi/2 - 0) * (x - 0)

f (generic function with 1 method)

• A function modeling exponential growth with $r=5$% per year and initial amount of 1,000 would be given by the following. It's value at 2 is given:

f(x) = 1000 * exp(0.05 * x)
f(2)

1105.1709180756477

Plotting

Plotting is done in julia through external packages. Loading the MTH229 package loads the Plots package which makes it easy to plot functions by name: plot(f, a, b). We can also plot two or more functions over $[a,b]$ – just place the functions into a [] container:

f(x) = x^2 - 2x + 3
plot(f, -3, 4)

f(x) = cos(x)
g(x) = 1 + f(2*x)
plot([f,g], 0, 2pi)

Making a plot is easy, where to plot is the possibly difficult question. This is because there are many different things we do with plots: looking at the "large" to find horizontal or slant asymptotes, or vertical asymptotes and looking small to see, for example, zeros of a function. Don't be afraid to change the values of a and b to make new plots over different domains.

Zeros

Finding zeros of a function is aided by some functions in the Roots package:

• If f is a polynomial function, then fzeros(f) will return all the real zeros of f

f(x) = (x-1) * (x-2) * (x^2 + 2x + 2)
fzeros(f)

2-element Array{Real,1}:
 1
 2

• In general, for a function f the command fzeros(f, a, b) will look for all zeros within [a,b]. This function may miss values.

• if f is not a polynomial,: fzero(f,a,b) is guaranteed to find a zero in the interval $[a,b]$ provided there is obviously one there – when $f$ is continuous and $f(a)$ and $f(b)$ have different signs. This is because the bisection method – which uses the intermediate value theorem – provides a guarantee.

f(x) = exp(-3x) - 1/2
plot(f, 0, 5)			# zero between 0 and 1
fzero(f, [0, 1])

0.23104906018664842

Okay, start your engines

Use these cells or add some more to answer the questions on the exam.