Microsoft Word Free Math Add-In - Gonzaga Universityweb02.gonzaga.edu/faculty/nord/wordusersmanual/prec… · Web viewis one word followed by a description of each term, the increment

Microsoft Word Free Math Add-In

Precalculus

Conic Sections

Example 1: Graph an ellipse.

x2

4+ y

2

16=1

Selecting Plot in 2D gives the graph. The graph is a little misleading. It is not a circle. The distance for a unit length is different on the x-axis versus the y-axis. To make the graph proportional, click on the fourth button in the Display row.

The new graph is:

©2009 Nord Page 1

Toggles graph to be in and out of proportion.


Note that there is new labeling on the axes. The Display row can furthermore be used to alter how the graph is displayed.

Example 2: Graph a hyperbola.

x2

16− y

2

4=1

Select Plot in 2D.

Alternately, you can use the command:

©2009 Nord Page 2

Insert a desired domain and/or range.

Display axes (yes/no)

Diplay units on the perimeter (yes,no)

Show grid lines (yes/no)


PlotEq ( x2

16− y

2

4=1 , {x ,−10,10 } , {y ,−10,10 })

The insertion of {x ,−10,10 } , { y ,−10,10 } will control the plot range. However, the original equation can be inserted and the plot range established by using the last button on the Display row.

Insert the desired minimum and maximum values here.

Example 3: Graph a circle with radius √3 and center of the origin. Shade the outside of the circle.

Highlight and right click and select “Plot Inequality”.

x2+ y2>3

©2009 Nord Page 3


Precalculus

Graphing Trigonometric Functions

Set the angles to be in radian or degree mode by using Math Preferences.

Use Insert New Equation for an input. With an expression, the option of Plot in 2D will appear after a right click on the input.

©2009 Nord Page 4


Consider the example:

3sin(2 x)

The input will appear in blue as shown:

Right click and yield the graph:

Consider other examples.

©2009 Nord Page 5


Example 1: Graph ¿cos ( 1x) .

The option Plot in 2D also appears if the input is in the form of an equation. The graph is:

Example 2: Graph ¿sin xx .

The graph is:

sin x / x

Use the Fraction option as a possible way to enter. Notice that sin is recognized. To enter the argument after sin, press the space bar.

©2009 Nord Page 6


For our example, the graph is undefined at x=0. However, the graph appears continuous.

©2009 Nord Page 7


Precalculus

Solving Trigonometric Functions

To input a popular symbol, use the down arrow key.

The Basic Math feature will appear, and π will be an option.

Alternatively, in the Insert New Equation line type, “\pi” followed with the space bar. The input will automatically change to, “π”.

Consider some trigonometric equations and their solutions.

Example 1:

sin 2 x=0

Select Solve for x to yield:

x=π N1

2

©2009 Nord Page 8

This will bring up the Basic Math feature.


Example 2:

x=cos x

The answer in radians is:

x≈ 0.7390851332152

Example 3:

x= tan x

The output is:

x≈ 0

The option, Plot Both Sides in 2D, appears along with Solve for x. The graph is:

The answer is zero.

Example 4: Solve for the indicated variable. Solve for b.

b=tan (π)

The answer is:

b=0

Example 5: Solve for the indicated variable. Solve for c.

©2009 Nord Page 9


c=arctan( 12)

The Solve for c option brings up:

c=tan−1( 12 )

This does not help. Erase the “c =”. The option, Calculate, will appear.

tan−1( 12 )

Select Calculate to give the following answer in radian mode:

0.4636476090008

Select Calculate to give the following answer when the Math Preferences are set to Degrees:

26.565051177078

©2009 Nord Page 10


Precalculus

Complex Numbers

Example 1: Find the modulus of a complex number.−3+2 i.

The command is abs. Right-click and select Calculate.

|(−3+2i )|

The modulus of the complex number is:

√13

Example 2: Find the quotient of two complex numbers.

2−4 i3+5 i

Enter as a fraction using:

The letter i is recognized equal to √−1. Right-click and select Calculate. The answer is the complex number:

−717

−1117i

Example 3: Raise a complex number to an exponent that is a natural number.

(√2+√2i )6

©2009 Nord Page 11


Right-click and select Calculate to give:

−64 i

Example 4: Simplify using two different approaches (Zill and Cullen, 2006 , p. 802).

(1−√3 i )3

Right-click and select Calculate to give:

−8

Apply DeMoivre’s Theorem, (r cosθ+r i sin θ )n ¿(cosnθ+i sin nθ)rn , for another approach. Let r=2 and θ=−π /3 and obtain the same answer.

(2 cos −π3

+2 isin −π3 )

3

−8

Example 5: Given a complex number, convert it to the polar form, z=r eθi.

Use the command topolar. Input the complex number, a + bi, following the command. Below is an example input and output.

topolar (0+1 i)

eπ i2

The command toRectchanges a number from polar form to rectangular form. The Calculateoption from the pull-down menu will execute without the needed command toRect. This is an example input and output.

√13etan−1 (2

3 )i

3+2 i

Reference

Zill, D. and Cullen, M. Advanced Engineering Mathematics, third edition, Jones and Bartlett, Sudbury, Massachusetts, 2006).

©2009 Nord Page 12


Precalculus

Series

When working with a series, do not use the letter, i. It is understood as the imaginary number equal to

√−1. It appears from the ribbon that using the letter i as a counter is acceptable. It is not.

Example 1: Find the sum of the first ten positive integers.

Insert the series using a summation.

∑k=1

10

k

Right-click and select Simplify to yield:

55

Example 2: Find a formula to sum the first n positive integers.

©2009 Nord Page 13


∑k=1

n

k

The output is:

n2+n2

Example 3: Find the sum of the square of n integers.

∑k=1

n

k2

The output is:

n3

3+ n

2

2+ n

6

Use the command Factor out the previous output.

factor ( n3

3+ n

2

2+ n

6)

The Simplify command yields:

n (n+1 ) (2n+1 )6

Example 4: Find the sum of cube of n integers.

∑k=1

n

k3

The output is:

n4

4+ n

3

2+ n

2

4

Using the Factor command with this output yields:

n2 (n+1 )2

4

Example 5: Use the seriessum command.

©2009 Nord Page 14


The command seriessum is one word followed by a description of each term, the increment variable, the starting value for the variable, and the ending value for the variable. An example is:

seriessum(n3 , n ,1,5)

The Simplify command gives:

225

©2009 Nord Page 15


Precalculus

Overview with Examples

Example 1: Solve an equation for x. x3−x2−4=0

The Solve for x command yields:

x= 3√ 4√219

+ 5527

+ 3√−4√219

+ 5527

+ 13

To give a numerical solution, use the nsolve command.

nsolve (x3−x2−4=0)

Solve for x or Simplify yields:

x≈2

Example 2: Solve an equation for x that will yield an answer containing a non-real number.

x2+4=0

Under Math Preferences, select Complex Numbers and click OK.

x2+4=0

The output is:

©2009 Nord Page 16


x=2 ix=−2 i

The output if the number field selected is Real Numbers is:

Example 3: Solve an equation with a degree 3 polynomial.

x3−6 x2−12 x−6=0

The Solve for x command yields:

x=3√√17+23+ 3√23−√17+2

To give a numerical solution use the nsolve command.

Nsolve (x3−6 x2−12 x−6=0)

The output is:

x≈7.667178637832

The nsolve command can be used with more than one equation. The user may opt to specify a variable with specific search window. The input is nsolve({eq1, eq2, …}),{var1,varmin, varmax},{var2, var2min, var2max},..}). An interval is elective. If a variable is specified with one number, then the search will occur around this value.

Below are some examples with the output provided.

Example 4: Specify an interval or target value for a solution for x.

Input:

nsolve ({xsin ( x )=14 }, {x ,0 , π })

Output:

x≈ 0.5111022402679

Input:

©2009 Nord Page 17


nsolve ({xsin ( x )=14 }, {x ,−π ,0 })

Output:

x≈−3.0597966989996

The next example will use a target place to look for the solution. An interval is not needed.

Input:

nsolve ({xsin ( x )=14 }, {x ,6π })

Output:

x≈18.8628099023054

Example 5: Use the nsolve command and specify an interval for a variable. Use more than one equation.

Input:

nsolve ({x sin y=14, x− y=8 },{ {x }, { y ,0,2 π } })

Output:

{x∧≈ 8.0311338842625y∧≈0.0311338842625

Example 6: Multiply polynomials.

Input:

( x−1 )∗( x−2 )∗(x−3)

The Expand command yields:

x3−6 x2+11 x−6

Example 7: Find the integer roots of a polynomial.

Input:

x3−6 x2+11 x−6=0

The Solve for x command finds the answer:

©2009 Nord Page 18


x=3x=1x=2

Example 8: Solve for an indicated variable.

Input:

x2+6 y2=4

The screen will show:

The option, Solve for y, gives:

y=√−x2

6+ 2

3

y=−√−x2

6+ 2

3

The option, Solve for x, gives:

x=√−6 y2+4x=−√−6 y2+4

Example 9: Simplify an expression.

Input:

©2009 Nord Page 19


x3 y5

x2 y7

The option, Simplify, will yield:

xy2

Example 10: Solve a trigonometric equation. Find all solutions.

Input:

cos (2 x )=0

©2009 Nord Page 20


The output where N1is an integer is:

x=π N1

2+ π

4

Example 11: Evaluate a limit where the answer is e.

Input:

limn→50 (1+ 1

n )n

The answer is:

e

To get an approximation, select Calculate.

Output:

2.7182818284591

Example 12: Evaluate a limit.

Input:

limx→∞sinx/ x

Output:

0

©2009 Nord Page 21


Example 13: Simplify an expression.

Input:

x2∗y3+3 y−x+82 y−12.6 y

Output:

x2 y3+ 362 y5

−x

Example 14: Create an example that will require the paid version of Microsoft Math.

Input:

max0≤ x ≤5

( x2−3x )

Use the function option.

Although it is possible to input the problem, the prompt will be:

Example 15: State the sequence of partial sums, S1, S2, S3, S4, and S5 for each of the two series below:

©2009 Nord Page 22


a) ∑n=1

∞ 1n!

b) ∑n=1

∞ 2n

√n !

Note that the series uses n as the variable, and not i. If i were used, it would be understood as the irrational constant number, √−1 , and the line would not execute. For the first example, input:

∑n=1

∞ 1n !

Substitute 5 to get:

∑n=1

5 1n !

Select Simplify to yield:

10360

The partial sums for our problem are:

1

32

53

4124

10360

Consider the original problem:

∑n=1

∞ 1n !

©2009 Nord Page 23


Select Simplify to get:

e−1

Select Calculate to have a decimal approximation:

1.7182818284591

Right-click and select Calculate for each output for a partial sum. The answers are:

1

1.5

1.6666666666667

1.7083333333333

1.7166666666667

Consider the input for the second example:

∑n=1

∞ 2n

√n!

Substitute 5 to get:

∑n=1

5 2n

√n!

Right-click and select Simplify. The output is:

8√63

+ 8√3015

+2√2+2

Select Calculate to receive:

©2009 Nord Page 24


14.2815867455289

Similarly, substitute 1, 2, 3, and 4, also. The answer for the partial sums is:

2

2√2+2

4 √63

+2√2+2

8√63

+2√2+2

8√63

+ 8√3015

+2√2+2

Select Calculate for each output and the answer as a decimal for the partial sums is:

2

4.8284271247462

8.0944134484571

11.360399772168

14.2815867455289

Consider the original problem:

∑n=1

∞ 2n

√n!

Right-click and select Simplify to get:

∑n=1

∞ 2n

√n!

The answer is not given. However, a substitution of a value for n

©2009 Nord Page 25


∑n=1

60 2n

√n!

such as n=60 gives the answer (with a little pause for the calculation) for the partial sum to be:

8√63

+ 8√3015

+ 16√515

+ 16√7063

+ 32√35105

+ 64√7315

+ 128√773465

+ 128√23110395

+ 256√858135135

+ 256√1430225225

+ 256√3003135135

+ 512√2431011486475

+1024 √1215534459425

+ 2048√46189654729075

+ 2048√230945654729075

+ 4096√17635813749310575

+ 4096 √96996913749310575

+ 8192√676039225881530875

+ 8192√4056234316234143225

+ 16384√1040061581170716125

+ 32768√4457414230536445125

+ 32768√31201814230536445125

+ 65536√1292646412685556908625

+ 65536√6678671063966261320836875

+ 131072√10772052063427784543125

+ 131072√22039614302110886623587616875

+ 262144√3339335563966261320836875

+ 262144 √648223952110886623587616875

+ 1048576√3534526381640158906527578311875

+ 1048576√33578000611640158906527578311875

+ 2097152√9075135344328619095339954375

+ 2097152√153162809821322065784858518054375

+ 2097152√3829070245106610328924292590271875

+ 4194304√13456446861013113070457687988603440625

+ 4194304√1569918800454371023485895996201146875

+ 8388608√526024740930563862029680583509947946875

+ 8388608√5786272150230563862029680583509947946875

+ 16777216√1052049481861691586089041750529843840625

+ 33554432√22870640911691586089041750529843840625

+ 33554432√3583067075961836869254970658257624840625

+ 67108864√716613415182782659116473679621593117828125

+ 67108864√10749201227779504546184962274902660509375

+ 134217728√281132955186141915614940157660701249009234375

+ 134217728√3654728417418141915614940157660701249009234375

+ 268435456√149000466248587521527591828356017166197489421875

+ 536870912√248334110414322564582775485068051498592468265625

+ 536870912√19121726501901108687364368561751199826958100282265625

+ 1073741824√1365837607278651241052052651678742832422585754609375

+ 1073741824√108993841060836270495179769008019818390136611716089140625

+ 2147483648√1879204156221315495179769008019818390136611716089140625

+ 4294967296√739153634780383929215606371473169285018060091249259296875

+ 4294967296√11087304521705758529215606371473169285018060091249259296875

+2√2+2

Select Calculate to get:

21.8586197886637

Example 16: Use the nsolve command to solve a system of equations involving trigonometric functions.

The syntax for the nsolve command is: nsolve({eq1, eq2, ..,eqn}). Consider the following equations:

2.9=rcosθ2=rsinθ

There is an alternate way to solve this system. Insert each equation using a separate Insert New Equation prompt. Highlight both equations simultaneously using the left mouse to drag (and press) over both equations. Right-click and select Solve for θ , r. The answer is:

{r∧≈3.5227829907617θ∧≈0.6037493333974

The executable line using nsolve is:

nsolve ({2.9=r cosθ ,2=r sinθ })

Select Simplify to yield:

{r∧≈3.5227829907617θ∧≈0.6037493333974

Example 17: Solve the following system of polar equations by giving an approximate solution for x , y , r ,∧θ.

r=3+sin θ

©2009 Nord Page 26


r=2/sin θ

Insert each polar equation individually using Insert New Equation. Left click and drag over both equations to highlight both equations simultaneously. Right-click and select Plot in 2D. Use the Trace feature to find the approximate solution for x , y , r ,∧θ.

©2009 Nord Page 27

Documents

Microsoft Word Free Math Add-In - Gonzaga Universityweb02.gonzaga.edu/faculty/nord/wordusersmanual/prec… · Web viewis one word followed by a description of each term, the increment