EG31 MATHCAD NOTES

SESSION
2

Symbolic
Calculations

Mathcad can perform a number of operations which
are found in the Symbolics
pull-down menu. The results of these operations are
generally another expression. Here are examples of
some of these operations:

Evaluate

To evaluate a derivative symbolically

1. Select the differential operator
by selecting the
d/dx
operator from the Math palette or typing a ?

2. Type the differential expression into the placeholders

3. Select it by pressing the space bar or the
arrow keys until the editing lines are across the complete
bottom and right sides or drawing the mouse diagonally
over the expression.

4. Choose E
valuate
=> Symbolically from the Symbolics menu.

Mathcad returns the answer:

Mathcad places the evaluated expression
into the worksheet. The location of the evaluated
expression can be changed by selecting its math region
and dragging it to the desire location. The location
can also be pre-selected based on the Evaluation Style
selected. This will not be covered in these notes.

To evaluate a definite integral symbolically

1. Select the definite integral operator from
the Math palette and fill in the placeholders with
numerical limits and the expression. Hint: either
type the sin(x)
or select the sin
function from the Arithmetic palette.

2. Select the complete expression and choose
Evaluate
=> Symbolically
from the Symbolics
menu.

Mathcad returns the answer:

An alternate method for the solution to a
definite integral is to type the " =
" sign after the expression and press the Enter
key to complete the operation:

To evaluate an integral with symbolic limits, symbolically

1. Select the definite integral operator from
the Math palette and fill in the placeholders with
variables for the limits of integration and the expression
to be integrated

2. Select the complete expression and choose
Evaluate
=> Symbolically
from the Symbolics
menu.

Mathcad returns the answer:

To evaluate an indefinite integral symbolically

1. Select the indefinite integral operator from
the Math palette or press and hold the Ctrl key and type " i
". Fill in the placeholders with the expressionto
be integrated.

2. Select the complete expression and choose
Evaluate
=> Symbolically
from the Symbolics
menu.

Mathcad returns the answer:

Simplify

By using the keyword simplify, Mathcad can carry
out the simplification of algebraic and trigonometric
expressions.

1. Enter the expression to be simplified. Hint: use
captial X rather than lower case to avoid having Mathcad
plug in any previously used definintion of x.

2. Press and hold
the Ctrl
and Shift
keys and type " .
" key. Mathcad displays a placeholder to the
left of the arrow. Type " simplify "
into the placeholder and either click outside the math
region or press the Enter
key.

Mathcad returns the simplified expression to the right
of the arrow

A more direct method is to select the
expression to be simplified and then choose S
implify
from the S
ymbolics
menu.

To simplify a trigonometric expression by having Mathcad
apply the following two identities:

sin
2(x) + cos2
(x) = 1 and cosh2(x)
-sinh2
(x) = 1

1. Enter the trigonometric expression to be simplified.

2. Press and hold
the Ctrl
and Shift keys and type " .
" key. Mathcad displays a placeholder to the
left of the arrow. Type " simplify, trig "
into the placeholder and either click outside the math
region or press the Enter
key.

Expand

To expand all powers and products of sums in an expression,
the keyword
expand is used.

1. Enter the expression to be expanded

2. Press and hold
the Ctrl
and Shift
keys and press the "
. " key. Mathcad displays a placeholder to the
left of the arrow. Type expand
into the placeholder and either click outside the math
region or press the Enter
key.

Mathcad returns the expanded expression to the right
of the arrow

A more direct method is to select the
expression to be simplified and then choose Expand from the Symbolics menu.

Factor

To factor an expression using Mathcad, the keyword
factor
is used.

1. Enter the expression to be factored

2. Press and hold
the Ctrl
and Shift
keys and press the " .
" key. Mathcad displays a placeholder to the
left of the arrow. Type factor,
then the "
square root sign
" from the Arithmetic palette and then 2
into the placeholder and then either click outside
the math region or press the Enter
key.

Mathcad returns the expression factored with respect
to the square root of two to the right of the arrow

C
ollect

To simplify an expression by collecting terms containing
like powers of a variable, the keyword collect is used.

1. Enter the expression.

2. Press and hold
the Ctrl
and Shift
keys and press the " .
" key. Mathcad displays a placeholder to the
left of the arrow. Type collect,X
into the placeholder and theneither click outside
the math region or press the Enter
key.

Mathcad returns the expression with the like powers
of X collected to the right of the arrow

An alternate method for collecting terms of an
expression is to click on a variable in the expression
and choose Collect
from the Symbolics
menu.

Finding the solutions to equations:

To solve a single equation in a single unknown,the
root
function can be used. This Mathcad function uses an
expression and one of the variables from the expression.
It varies the variable until the expression is equal
to zero. It then returns the value that makes the
expression equal zero. The root function has the form:

root ( f(z) , z ) where
f(z) is the function, and the value of z which makes
the function equal to zero is returned as the solution.
The function can be a function which is defined elsewhere
in the worksheet or an expression.

To find the solution to the equation
x
3 - e x

1. First define a guess for the value of the variable.
This guess value must be placed before the root function
in the Mathcad worksheet. Define the guess value of
x
as 3:

2. Arrange the function in a form that is equal
to zero. Select a variable name for the scalar which
will be used to define the root of the equation. Using
the variable a
for this:

3. To have Mathcad show the root of the equation
simply type a=

Mathcad returns
the answer:

4. If the equation has more than one root, the
guess value will determine the root which is closest
to the guess value. An alternate way to find multiple
roots is to use the polyroot
function.

The polyroot function does not require a guess
value. However, it returns all of the roots of an
equation at once, whether they are real or complex.
In order to use the polyroot function, the equation
must be expressed as a vector. This will not be covered
in this material.

Systems
of equations:

Mathcad allows you to solve a system of up to fifty
simultaneous equations in fifty unknowns.

There are four steps to solving a system of simultaneous
equations. These are:

1. Provide an initial guess for all of the unknowns
that are desired. The initial guesses give Mathcad
a place to start searching for solutions.

2. Type the word
Given
. This tells Mathcad that what follows is a system
of equations. It can be typed in any combination of
upper and lower case letters and in any font. Be
sure that it is not typed in a text region or paragraph

3. Type the equations and inequalities in any
order below the word Given
. Make sure to use the symbol "=" to separate
the left and right sides of each equation. To get
the correct = sign, press and hold the Ctrl key and then press the =
key. The left and right sides of an inequality can
be seperated with any of the symbols <, >, <, or >.

4. Type an equation that
invloves the Find
function. It can be in any combination of upper and
lower case letters in any font, size or style.

The answer is expressed
as a vector.

If Find
has one argument, it returns the value of that variable
that solves the equation between it and the Given.

If Find
has more than one argument, it returns a vector of
answers.

Mathcad is very specific about the types of expressions
that can appear between the Given and the Find
. The table below lists all the expressions that can
be placed in a solve block between these two functions.
These expressions are often called constraints. In
the table below, x and y represent real-valued
scalar expressions, z and w represent arbitrary
scalar expression.

Condition Keystroke
Description

z=w [Ctrl]=
Constrained to be equal

x>y >
Greater than

x<y <
Less than

x>
y [Ctrl]0 Greater than
or equal to

x<
y [Ctrl]9 Less than
or equal to

Mathcad does not allow contraints with the equals
sign with a slash through it to indicate not equal
to, range variable or expressions involving range variables
of any kind, and inequalities of the form a<b<c

Solve blocks cannot be nested inside each other.
Therefore, each solve block can have only one Given
and one Find. a function like f( x ) :=Find( x ) cna
be defined at the end of one solve block and the same
function used in another solve block.

Never use assignment statements such as initial
guesses ( x
:= 1
) inside a solve block. Mathcad marks them with an
appropriate error message.

EG31 MATHCAD EXERCISES

SESSION
2

Exercise 1: Taking a Derivative

a. Find the first derivative of the expression

b. Find the second derivative of the same expression.

c. Find the third derivative of the same expression.

Exercise
2: Integrating

a. Find the indefinite integral of the following
expression

b. Find the definite integral for the same expression
between 0
and 5

c. Find the definite integral for the same expression
between the symbolic limits of a
to c

Exercise
3: Expanding

Expand
the following expressions

a.

b.

Exercise
4: Factoring

Factor
the following expression

Exercise
5: Collecting

In the following expression:

a.
First collect x

b.
Then collect y

c.
Collect x
and y
at the same time

This is not functioning
correctly

Exercise 6: Solving and equation

Solve the following equation for its roots