Let's do some simple operations.  Place your cursor in each of the equations below and click shift-enter to see the result.  Check the result in your head or with a calculator.

(3^2)

4*3+2

(9^(1/2))

(27^(1/3))

I made the equations above using the typesetting characters from the Basic Math Assistant Palette.  Now you try it.  create equations to find:
 1.  the forth root of 256,   
 2.  5!, 
 3.  (e^2)

Mathematica also has many built in functions like these.  You can find them under Help -> Function Navigator
Notice that built in Mathematica functions start with an upper case letter.  Square, not curved brackets are used.  Put your cursor in each of the input boxes below.  Press shift-enter and check that the answer agress with what you expect.

ArcSin[1]

Sqrt[9]

Now you do it.  Use the built in Mathematica functions to calculate:
1.  natural log of 2.718
2. base 10 log of 100
3. cosine of π/4
Check your answers with a calculator

We can also define variables for use in a calculation.

c=3;b=2;

c*b

Change b to 4 in the above equation.  Then put your curser in the equation c*B and press shift-enter.  Did you get 12?  If not figure out why.  Ask for help.  This is a very important point to understand.  You must always be thinking about this point!

The line of code below defines a function named myfnc that calculates y = (x^2)   It acts the same way as a built in finction like Cos[x] except that we have defined our own custom function.  Place your cursor in the box below and press Shift-Enter to make Mathematica process the definition.  No result is computed.  The definition is simply remembered by Mathematica.  I always begin the name of my own custom functions with lower case letters so they dont conflict with Mathematica functions that all start with upper csase letters.

myfnc[x_]:=((x )^2)

Try using the function, myfnc we just created to calculate the quantity (3^2)  by typing the quantity 3 inside the square brackets in the code below then pressing Shift-Enter.  You should get the value 9.

myfnc[]

Let's create a graph to plot the function y = (x^2) from x = -16 to x = +16.  The code below creates a range of x values from -16 to +16.  For each x value chosen by Mathematica within the range you specified, a  y value is calculated using the function, myfnc we created above. A plot of these x and y values is then created.  Place your cursor in the code below and click Shift-Enter to see the plot.

Plot[myfnc[x],{x,-16,16},AxesLabel->{"x","y"}]

Scroll back up  and change the definition of the function myfnc so that it calculates sin(x) instead of (x^2).  Your new function should look like this: myfnc[x_]:=Sin[x]  Take care to note that You must use square brackets and the Sin function must begin with a capital letter S.  Mathematica is very picky!!  Now go back to the plot code and press Shift-enter again.  Since you have changed the definition of myfunc the plot should change to y = sin(x).  Did it work for you?  If the plot does not change you have made a serious error.  try to figure out what you forgot to do!  get help if necessary.

Plot sin(x) and sin(x+ϕ), where ϕ is a phase factor with a value of π/2.  Make your plot over the range 0 to 10π.  To find out how to plot 2 functions together on the same graph Click Help on the Mathematica tool bar and search for Plot.

Suppose now we wish to make the same plot but restrict the y axis to plot only between 0 and +2.  Copy and paste your plot statement from above but this time add the option 
PlotRange -> {{0, 10 π}, {0, 2}}
to tell Mathematica what x and y range to plot.  The result below is what your plot should look like.

Use the graphics toolbar and drawing tools.  Create an arrow to identify the peaks in the above plot.  Label the peaks with the names of your group members.  Draw a verticle dashed line through the maximum of the peak.  The graphics inspector (right click in plot or use the graphics toolbar) can make the peak a dashed line.

One of the key things to look for are the colors that Mathematica uses.  Look at the code in the examples below.  The blue colored symbols are undefined.  Mathematica does not know what the x and y values are.  How can that be you might ask.  We just told Mathematica what the values were buy writing the code didnt we?  Well no because we have not yet pressed shift-enter.  Put your cursor in the the input box myFnc2[3].  Press shift-enter and you will not get an answer because myFnc2, x, and y, are blue.  They are undefined.  Now put your curser in the input box where x,y, and myFnc2 are defined.  Press shift-enter and you will see the colors change to black.  Now Mathematica knows the values of x and y.  Now execute myFnc2[3] again and this time you should get the answer 12.

Two variables in a function

x=2;
y=3;
myFnc2[x_]:=(x^2)+y

myFnc2[3]

Explain why the answer to myFnc2[3] is 12 rather than 7 when the definition x=2 and y=3 is used.

Use the solve function to find the 4 roots of the equation (1.5-2x)^2/((3x)^3(x))=2.8× (10^(-5))  Note that when you use solve on this equation it is good practice to make sure that x has not been previously defined otherwise solve will not work.  To ensure that x is undefined use Clear[x] before using Solve.

Use the built in function D[] to find the partial second derivative of (e^(-g x)) with respect to x.

Look at the examples below that contain code with mistakes.  Make a new input box by right clicking and use insert new cell then text.  Color this text box red (right click then background color).  In the text box you just made make 5 lines numbered 1 through 5.  On each line identify the mistakes in the corresponding 5 lines of code below.

Cos(π/6)

(e^2.718)

sin[π/2]

myfnc3[y]:=Cos[y]+Sin[y]

Solve[x^2+x-3=0,x]