Estimate Pi Using Monte Carlo Sampling

A classic use of the Monte Carlo random sampling technique is to estimate Pi. It's very simple.

1. We construct a square by randomly selecting point in the interval (-1, 1).

2. We select a subset of points that satisfy the predicate that places them on or within the unit circle:

x^2 + y^2 < =1.

3. The area of the square extending from -1 to 1 is 2^2 = 4. The area of the unit circle is 1 (Pi*r^2 = Pi*1^2). Therefore we multiply the ratio of the number of points lying within the unit circle divided by the number of points within the square by 4.

Here is an implementation. Below that is an implementation I found on the web that shows how newbies to Mathematica may transfer inefficient coding techniques from other languages to Mathematica.

Clear@circleRegionMonteCarlo;circleRegionMonteCarlo[numberSamplePoints_Integer]:=Module[{samplePoints=RandomReal[{-1,1},{numberSamplePoints,2}],
areaOfCirclePoints},

(* Select points within the unit circle using the parametric equation for a circle *)
areaOfCirclePoints=Select[samplePoints,First@#^2+Last@#^2<=1&];

Print@StringForm["Approximation to Pi using `` sample points is ``.",numberSamplePoints,4*Length@areaOfCirclePoints/numberSamplePoints//N];

If[numberSamplePoints<50001,ListPlot[areaOfCirclePoints,AspectRatio->Automatic]]
]

circleRegionMonteCarlo[50000]

Approximation to Pi using 50000 sample points is 3.14104`.

This represents an obsolete coding technique: 1) initializing the Pin (points inside the circle) and Pout (points outside the circle) arrays, 2) the For loop calling named random variable functions, 3) testing the distance predicate in the loop instead of all at once as is typical in functional programming, 4) having to take the Length of Pin and Pout in the loop (obviously you don't need to test the Length every time anyway), and 5) using Return instead of just leaving off the semi-colon after approx, are all obsolete techniques.

MonteCarloPi[n_]:=Module[{d,i},
n=n0;
Pin=Pout={};
For[i=1,i<=n,i++,
x=RandomReal[];
y=RandomReal[];
d=x^2+y^2;
If[d<=1, Pin=Append[Pin,{x,y}],Pout=Append[Pout,{x,y}]  ];
n=Length@Pin;
m=Length@Pout;
\[Rho]=m/n;
approx=4.0;]
Return@approx

]

String Replacement Methods: Overview

Here are String replacement methods that I have used in code from one-liners up to programs producing hundreds of thousands of text and html files. In general, use the simplest method or one that you understand clearly. Use StringForm to output messages from your functions and programs. For longer functions or programs, StringTemplate is the new best practice.

There is a function I don't discuss, StringInsert, which inserts a substring at a given StringPosition in a control String. I don't advocate its use since it's very brittle in that if you add or delete even one character before the StringPosition then the insertion point will be wrong.

StringForm

Literal Replacement, Markers, and Delimiters

StringTemplate

String Replacement Methods: Literal Replacement, Markers, and Delimiters

A String Replacement Overview is here.

Note that the next three methods all use StringReplace. This is in keeping with my principle that the fastest way to learn Mathematica is to become a power user of its 70 or so core functions. In String processing, for instance, StringInsert is not a function you need to know. Instead learn to use the more powerful and robust function, StringReplace.

Literal Replacement

Literal replacement works by using StringReplace to find a literal substring within a String and substitute another substring for it. Literal replacement is very simple and easy to use.

string1="The quick brown fox jumped over the lazy white dog.";

StringReplace[string1,{"fox"->"mink","dog"->"pecadillo"}]

The quick brown mink jumped over the lazy white pecadillo.

Markers

Using markers to indicate the replacement position can improve code legibility. Use StringReplace to replace just the marked text.

string2="The quick brown <animal1> jumped over the lazy white <animal2>.";

StringReplace[string2,{"<animal1>"->"mink","<animal2>"->"pecadillo"}]

The quick brown mink jumped over the lazy white pecadillo.

Delimiters

Use StringReplace to replace text between the delimiters. This is very useful when you want to replace a lot of text in a document, especially in a long document. However, the new function StringTemplate is a superior method overall.

sitemapTemplate="<?xml version=\"1.0\" encoding=\"UTF-8\"?>
<urlset xmlns=\"http://www.sitemaps.org/schemas/sitemap/0.9\">
<!-- put list of urls here with a line feed after each one -->
</urlset>";

urls="<url><loc>http://www.blah.net/page1.html</loc></url>
<url><loc>http://www.blah.net/page2.html</loc></url>";

Note that you use StringExpression (shorthand "~~") to concatenate quoted Strings with Blanks in the String to be found by StringReplace, but you must use StringJoin (shorthand "<>") if you concatenate different Strings in the replacement String.

sitemapTemplateWithURLs=StringReplace[sitemapTemplate,"<!-- put list"~~urlsList__~~"each one -->"->urls]

<?xml version="1.0" encoding="UTF-8"?>
<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
<url><loc>http://www.blah.net/page1.html</loc></url>
<url><loc>http://www.blah.net/page2.html</loc></url>
</urlset>

String Replacement Methods: StringForm

A String Replacement Overview is here.

StringForm

StringForm is a simple, elegant String template function. Use it in your functions where Print isn't enough since you need to fill in some variables, such as calculations on the fly. In a function use the following form with Print to output it since lines end with an output-suppressing semi-colon.

Print@StringForm["control string", variables];

Here the double backtick marks tell Mathematica where to fill in the blanks with arguments you give in the order in which they are inserted into the String. An argument can be a String or an Expression of unlimited complexity, which will be evaluated before insertion. If you don't want the inserted Expression to be evaluated, though, use HoldForm (example below).

StringForm["Use `` for relatively short and simple String templates, such as output messages in your functions. For example, the cube root of `` is ``.","StringForm",27,27^(1/3)]

Use StringForm for relatively short and simple String templates, such as output messages in your functions. For example, the cube root of 27 is 3.

If you're going to use an argument twice, switch the order, or use a number of arguments and you want to prevent mistakes, use numbered, rather than ordered, backticks. You often want a line break, for which the \n escape character is used within the quotation marks that are Mathematica's String delimiters.

StringForm["Flying or gliding mammals include `1`, `2`,\n`3`, `4`, `5`, `6`, and `7`.\nThe most common species are in the `3` family.","flying possums","greater glider","bats","flying squirrels","flying lemurs","flying monkeys","cats"]

Flying or gliding mammals include flying possums, greater glider,
bats, flying squirrels, flying lemurs, flying monkeys, and cats.
The most common species are in the bats family.

To prevent the inserted Expression from being evaluated, use HoldForm:

StringForm["For example, the sixth term of the Fibonacci series is the sum of the preceding two terms: ``.",HoldForm[1+1+2+3+5=8]]


For example, the sixth term of the Fibonacci series is the sum of the preceding two terms: 1+1+2+3+5=8.

Memory Management Tools

While Mathematica is designed to manage memory for you, under certain circumstances it can get bogged down, mainly because it keeps a record of all your inputs and outputs with In and Out. So if you're using functions that output a lot of computation, or working with large files, you may notice Mathematica slowing down.

There are a number of ways that you can manage memory in Mathematica. Here is a summary (see also How to Find Memory Used in Computations).

Command	Effect
?Global`*	Shows all Symbols in a non-accessible table
Names@”Global`*”	Returns a List of all Symbols that you can access
Clear@symbol	Clears the value of symbol but leaves its name in memory
Clear@”Global`*”	Clears the values of all Symbols but leaves their names in memory
Remove@symbol	Removes the name symbol and its value from memory
Remove@”Global`*”	Removes all Symbols and their values from memory

If you're going to go as far as removing all Global Symbols, consider starting a new session by entering Quit[] in your Notebook or Quit Kernel → Local under the Evaluation menu.

Beginners hesitate to Quit the kernel, but there's little downside. Even if you haven't saved your Notebooks, the kernel is a separate entity and you can save them.

To automate resuming after quitting the kernel or in general, use Initialization Cells. You can set Initialization in the menu under Cell → Cell Properties or by right-clicking on the cell and selecting Initialization Cell. A little downward tick mark appears in the upper right corner of the cell.

Then when you re-start the kernel by selecting any cell, selecting Evaluation → Evaluate Initialization Cells, or re-open the Notebook, all the Initialization cells are automatically re-Evaluated. In this way you lose very little time by quitting the kernel and re-starting.

Memory-Management Commands to Use Occasionally

Memory currently used by the kernel:

In[157]:= MemoryInUse[]

Out[157]= 135450976

Memory currently used by the front end (all of your open Notebooks):

In[158]:= MemoryInUse@$FrontEnd

Out[158]= 543264768

The maximum memory used by the kernel during your current Mathematica session:

In[159]:= MaxMemoryUsed[]

Out[159]= 137155304

Clear a cell that consumed lots of memory in your session:

Unprotect[Out]; Out[537] =.;
Protect@Out;

Ways to Create Arrays

Programmers often have to create an array. Of course, in Mathematica, you don't have to allocate memory for an array or dimension it or any of that outdated nonsense. Here are the six standard functions and common methods used to create arrays in Mathematica. Use the easiest-to-implement and most readable method for your application.

First there is the versatile function Array that applies a function to the array indices of any desired array dimension.

Array[g,{2,3}]

{{g[1,1],g[1,2],g[1,3]},{g[2,1],g[2,2],g[2,3]}}

To simply create a nested List of Integers you can use List as the function to apply to the array. "Give me 5 rows of 3 elements each:"

dataset=Array[List,{5,3}]

{{{1,1},{1,2},{1,3}},{{2,1},{2,2},{2,3}},{{3,1},{3,2},{3,3}},{{4,1},{4,2},{4,3}},{{5,1},{5,2},{5,3}}}

It is quite easy to create a similar array with one of the most versatile and commonly-used functions in functional programming, Table. It pays handsomely to become a power user of Table.

Table[{i,j},{i,5},{j,3}]

{{{1,1},{1,2},{1,3}},{{2,1},{2,2},{2,3}},{{3,1},{3,2},{3,3}},{{4,1},{4,2},{4,3}},{{5,1},{5,2},{5,3}}}

Table is highly versatile and can certainly be used to do many things. In some cases there is a built-in function that is more specialized. For instance, Table can be used to create a constant array. "Give me 25 7s:"

Table[7,{25}]

{7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}

ConstantArray is a specialized function that does the same thing, but except for a little bit of clarity in reading the code doesn't do more than Table.

ConstantArray[7,25]

{7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}

Table can generate random numbers using any of the random function family:

Table[RandomInteger[],{10}]

{1,1,1,0,0,0,0,1,1,1}

But for simple as well as higher-dimension arrays, the several half-dozen built-in random functions are superior and more readable than doing the same thing in Table.

RandomInteger[{1,5},{2,3}]

{{2,4,3},{2,1,3}}

Likewise, while Table can easily create a range of numbers, Range is superior.

Table[i,{i,7}]

{1,2,3,4,5,6,7}

Range@7

{1,2,3,4,5,6,7}

From 14 to 28 in steps of 2:

Range[14,28,2]

{14,16,18,20,22,24,26,28}

Those applications yield vectors, but here is a lesser-known usage of Range with a List as argument that generates an array. For each number in the List, Range yields a List of integers from 1 up to that number.

Range[{2,5,3}]

{{1,2},{1,2,3,4,5},{1,2,3}}

To more carefully control aspects of the array such as start, end, and step size, you may need to manually assemble the array with Table or Range.

{Range[14,28,2],Range[10^6,10^5,-2*10^5]}

{{14,16,18,20,22,24,26,28},{1000000,800000,600000,400000,200000}}

{Table[i,{i,14,28,2}],Table[j,{j,10^6,10^5,-2*10^5}]}

{{14,16,18,20,22,24,26,28},{1000000,800000,600000,400000,200000}}

Finally, MapIndexed is used in general to generate a List where a function is applied to a List of arguments along with an index number, which is the second argument.

MapIndexed[f,{a,b,c,d}]

{f[a,{1}],f[b,{2}],f[c,{3}],f[d,{4}]}

Using First strips off the enclosing List of the index.

MapIndexed[{#^2,First@#2}&,Range@5]

{{1,1},{4,2},{9,3},{16,4},{25,5}}

Getting Help: Find a Function Using Wildcards

There are over 4000 built-in functions in Mathematica, not including those in add-on Packages. 

In[340]:= Names["System`*"]//Length

Out[340]= 4131

Therefore it makes sense to see if there is a built-in function before we write one. Likewise, the large number of functions means that we often have a vague memory of a function we've used but can't put our finger on its name when we need it again. If I have some idea of the name of a function I've used, or want to see if there is an existing function and guess at part of its name, I use a wildcard in my current Notebook with the short form of Information (?). To find a function or survey related functions, this is faster than using the Doc Center.

Here I want to know the filepath of my current Notebook, which I cannot find in any Mathematica menu item. Maybe there's a Notebook function to tell me. I scan down the results until I see NotebookFileName, which is the one I want. I also quickly click on a half-dozen or so functions to refresh my memory of them or explore new ones and read the short info displayed below the results.

Using Information this way is always fruitful for me since I efficiently learn about functions I didn't remember or didn't know. But if it fails to flush out the function I want, I go to the Doc Center and search there.

How to Lose List Brackets in Mathematica

First, Apply, Sequence, ReplaceAll

There come times in every Mathematica user's life when the all-pervasive List brackets get in the way. Sometimes it's easy to redesign the function to accept the arguments in a List. Here are other solutions.

First

For a single element enclosed by brackets, you can use First to extract the element. In this example, since all Mathematica arithmetic functions are Listable (they will automatically Map over a List) we specify that the argument must be an Integer so the function will fail.

enclosed1={2};
Clear@function1;function1@x_Integer:=x^3;
function1@enclosed1

function1[{2}]

function1@First@enclosed1

8

Apply

First, see if you can use Apply @@ to apply a function to the contents of your List. Apply will replace the List head with your desired function and Evaluate it.

In[107]:=  Range@15
Out[107]=  {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}

In[108]:=  Plus@@%
Out[108]=  120

Here's a single filename that is unnecessarily surrounded by List brackets. It's a String, so we just Apply ToString to it. It then stands alone – no List brackets – but its Head is String.

FileNames[]

{TableFiredFibers-Control-CathodeOnly-0p644.txt}

ToString@@%141

TableFiredFibers-Control-CathodeOnly-0p644.txt

Head@%
String

FullForm@%%
"TableFiredFibers-Control-CathodeOnly-0p644.txt"

Sequence

Mathematica was designed as a list processing language in which the list is the fundamental data structure. Allan Newell's IPL (Information Processing Language) was the original prototype, after which John McCarthy coupled the list as a universal data structure with the lambda calculus and logic functions into the major programming language, LISP.

We can Apply Sequence to a List to replace List as the container (the Head). Here are two examples, first to a single List, then to a List of Lists.

In[115]:= Sequence@@{0,4}

Out[115]= Sequence[0,4]

In[119]:= Sequence@@{{0,1,1},{0,2},{0,1,3},{0,4}}

Out[119]= Sequence[{0,1,1},{0,2},{0,1,3},{0,4}]

FileNameJoin is the safe way to assemble filepaths in Mathematica since it knows your operating system syntax. However FileNameJoin only accepts a single-level List as its argument. To feed it a nested List, use Sequence instead of List as the Head of the inner List.

In[350]:= twoPathElements={"webPages","index.html"};

In[351]:= FileNameJoin@{$HomeDirectory,twoPathElements}

Out[351]= FileNameJoin[{C:\Users\kwcarlso,{webPages,index.html}}]

Since it didn't see a List of Strings (filepath names) as the correct argument types, FileNameJoin returned the input. Using Sequence solves this.

In[352]:= twoPathElements=Sequence["webPages","index.html"];

In[353]:= FileNameJoin@{$HomeDirectory,twoPathElements}

Out[353]= C:\Users\kwcarlso\webPages\index.html

The context of one function I use to illustrate Sequence is worth mentioning (but skip this to get to more usages of Sequence). I have digitally sampled a curve from the literature (a paper by Holsheimer from 1999) in the graphing program Origin. In other words, you can import an image, in this case a graph for which Holsheimer doesn't give either the underlying data or an equation, and repeatedly click on it to harvest {x,y} values. We captured 163 sample points on the solid curve below, which is a vast improvement on the antiquated method of using a straightedge to pick up x and y values off the axes.

With that many sample points, finding the x value closest to the one desired is a reasonable alternative to fitting a curve and applying the fit equation to values you wish to interpolate or extrapolate.

So this is a function to find the nearest x value (a neuron axon diameter, fiberDiameter) to one for which we want to know the threshold as determined by Holsheimer (which is the y-value retrieved). The problem is Nearest returns values in List brackets, which don't match the un-bracketed x values in the List of data. Applying Sequence to the result of Nearest feeds the unbracketed value to Cases, which then performs the match.

Clear@diameterThreshold; diameterThreshold[fiberDiameter_, fiberThresholdSourceData_:holsheimerData1999] := Cases[fiberThresholdSourceData, {x_, y_} /; x == Sequence@@Nearest[fiberThresholdSourceData[[All, 1]], fiberDiameter] ]

So in sum this function says: Clear the function name (which I do habitually at the start of a function definition to prevent accidentally testing a previous version as I revise it). Then define it as the Cases in the source data file (fiberThresholdSourceData--default to the holsheimerData1999 file unless I specify another file) such that ( /; ) the x-value is Nearest to the fiberDiameter parameter I give it.

ReplaceAll

Here is a third way using rule patterns.

In[128]:= {{0,1,1},{0,2},{0,1,3},{0,4}}/. List[a__] ->a

Out[128]= Sequence[{0,1,1},{0,2},{0,1,3},{0,4}]

Mathematica Plot: Using Table to See Numerical Values

Plot with Table to See Numerical Values

We saw how to use Table to generate sample points for a ListPlot, and how to compare your own sample to Plot showing its mesh points. Here is the converse usage, using Plot to graph a function and then Table to see it numerically. We'll start with a simple sine curve.

To see numerical values, I click Control+L to regenerate the input, replace "Plot" with "Table", and specify a third argument for the iterator to give the sampling frequency. The Postfix function N tells Mathematica to give us decimal output instead of its default, exact fractional values.

In[314]:= Table[{x, Sin@x // N}, {x, 0, 2 Pi, Pi/4}] // TableForm

Let's look at a more complicated example with two decay curves that are identical except for their asymptotes, which are 0 for decay1 and 0.7 for decay 2 (shown in the Plot).

In[347]:= Clear@decay1; decay1@d_ := (q/d) + 0.7; q = .2;

In[348]:= Clear@decay2; decay2@d_ := (q/d); q = .2;

In[363]:= Plot[{decay1@d, decay2@d}, {d, .1, 4}, PlotRange -> {Automatic, {0, 3}}]

When you need to keep track of three or more columns, TableHeadings can be helpful. Here we say no row headings, but three column headings.

Table[{d, decay1@d, decay2@d}, {d, .1, 1.2, .1}] // 

 TableForm[#, TableHeadings -> {None, {"d", "decay1@d", "decay2@d"}}] &

Mathematica Plot: Options Overview

Plot Has over 80 Options to Control its Behavior


Here is a Plot of two curves where we use Plot options to add a Frame, tick marks to the left and right y-axes, but to the bottom x-axis only, and to Plot the full -y-range instead of leaving out extreme values.

Plot[Tooltip@{Sin@x, x Sin@x}, {x, 0, 20}, Frame -> True,
 FrameTicks -> {{Automatic, All}, {Automatic, None}}, PlotRange -> Full]

You can ask Mathematica for all the available Options for a function with Options@function. When you do, Mathematica tells you their default values as well.

Options@Plot // Partition[#, 3,3,1,{}] & // TableForm

The Partition syntax may look a little complicated, but is explained in Capturing the Remnant of a Partitioned List. It says: "Partition the list into sublists of length 3, take them in blocks of 3 (i.e. do not overlap them), start the first element of the first list at position 1 of the first sublist (at the beginning), and do not pad the last uneven list (i.e. pad it with the empty set)." While Plot has 57 Options, which divide evenly by 3, the partition syntax will work if Plot has a non-divisible-by-3 number of Options in the future.

I thank a diligent reader, Murta, for suggesting this correction to my original post.

Mathematica Plot: Plotting More than One Function

Using Plot to Plot More than One Function


It is often useful to Plot different function parameter values and compare them. Just give Plot a list of functions with the parameter values you wish to compare and it will Plot them together without your having to use Show. Let us use the function defined in Basic Plotting:


f[x_, t_] := x^t;


You can use Table to automate the parameter sweep, but pre-generate the sweep functions or (for reasons I don't understand) Plot won't automatically color-code them.

In[278]:= sweepFunctionTable = Table[f[x, t], {x, 2, 5}]

Out[278]= {2^t, 3^t, 4^t, 5^t}

In[279]:= Plot[sweepFunctionTable, {t, 0, 5}]

Plot with Tooltip to Identify the Plots

Sometimes you can lose track of which function is which when plotting several function together. Plot helps by color-coding them. But adding Tooltip labels is another technique. Let's compare the three functions Sin x, Cos x, and Sin x + Cos x. You will need to Plot this yourself in a Notebook to mouseover and see the tooltips that label each curve with the three function names ("Sin@x", "Cos@x", "Cos@x + Sin@x").

Plot[Tooltip@{Sin@x, Cos@x, Cos@x + Sin@x}, {x, -3 Pi, 3 Pi}]

Now let's specify that the tick marks should be labeled in radians:

Plot[Tooltip@{Sin@x, Cos@x, Cos@x + Sin@x}, {x, -3 Pi, 3 Pi},
 Ticks -> {{-3 Pi, -2 Pi, -Pi, 0, Pi, 2 Pi, 3 Pi}, Automatic}]

Mathematica Plot - Basic Plotting

Plot

For starters, be aware that Mathematica's data plotting functionality is extensive and there are a variety of Plot functions. Using Information "?" with the wildcard characters "*" shows all functions that include "Plot". I won't show them here, but you can execute the command if you wish to see them.

?*Plot*

Basic Plotting

Let's say you have a function and you want to understand its behavior. We'll do a simple one.

In[238]:= y == x^n // TraditionalForm

Out[238]//TraditionalForm= y==x^n

Let's start by generating a few sample points of the function and plotting them.  You might think in the age of computers we can dispense with plotting sample points and just let Plot do the job. But when Plot fails to give you output and you don't know why, one measure is to go ahead and plot a few sample points by hand and compare them to what Plot is doing. So let's generate the sample points the easy way with Table. We Clear the function name, define the function, give x a value of 2, and use Table to sample the domain from -5 to 5 in increments of 0.5.

In[249]:= Clear@f; f[x_, t_] := x^t;

In[250]:= samplePoints = Table[f[2, t], {t, -5, 5, .5}]

Out[250]= {0.03125, 0.0441942, 0.0625, 0.0883883, 0.125, 0.176777, 0.25, 0.353553, 0.5, \
0.707107, 1., 1.41421, 2., 2.82843, 4., 5.65685, 8., 11.3137, 16., 22.6274, \
32.}

Now we plot these values using ListPlot. Ignore for now the x-tick points, which are just the number of the data point in the List.

In[242]:= plot1 = ListPlot@samplePoints

To better visualize the function, it's often helpful to connect the sample points. We use ListLinePlot. Another way is to use ListPlot's option, Joined -> True). If you look closely, you can see that the sample points are joined by line segments; our plot is not a continuous curve. Again, ignore the abscissa values.

plot2 = ListLinePlot@samplePoints

If we wished, we could combine the plots with Show--one of the most commonly-used plot techniques.

Show[plot1, plot2]

And this is similar to what Plot does--sampling a function and plotting the connected sample points--but of course in a much more sophisticated way. For one thing, Plot samples more frequently where it thinks it might need to, such as where the value of the function changes more frequently, maybe with several tries, and automatically connects the sample points in a continuous curve. Using Plot the x-axis values are correctly labeled.

Plot[f[2, t], {t, -5, 5}]

But notice that Plot is not showing the full domain and range that we asked for. By default it tries to, in effect, show the most important part of a function by excluding what it thinks might be outliers. Two options allow you to override this behavior. PlotRange->Full tells Plot to not clip the y-axis. PlotRange->All tells Plot to include all domain points and their y-values. I usually use PlotRange->All.

Plot[f[2, t], {t, -5, 5}, PlotRange -> All]

If you wish, reveal the sample points with the Option, Mesh->All. When you don't understand what Plot is doing, revealing the Mesh is a good first step. And then as I said above, you can compare Plot's sample points to yours, which might illuminate what's going on. There are more Mesh options to allow you to control how Plot uses its sample points.

Plot[f[2, t], {t, -5, 5}, Mesh -> All, PlotRange -> All]

Next: Using Plot to Plot More than One Function