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Microsoft Word Free Math Add-In Precalculus Conic Sections Example 1: Graph an ellipse. π₯2 π¦2 + =1 4 16 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. Toggles graph to be in and out of proportion. The new graph is: ©2009 Nord Page 1 Microsoft Word Free Math Add-In Note that there is new labeling on the axes. The Display row can furthermore be used to alter how the graph is displayed. Show grid Diplay lines units on Display axes (yes/no) the Insert a desired (yes/no) perimeter domain and/or (yes,no) range. Example 2: Graph a hyperbola. π₯2 π¦2 β =1 16 4 Select Plot in 2D. Alternately, you can use the command: π₯2 π¦2 ππππ‘πΈπ( β = 1, {π₯, β10,10}, {π¦, β10,10}) 16 4 ©2009 Nord Page 2 Microsoft Word Free Math Add-In The insertion of {π₯, β10,10}, {π¦, β10,10} 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 β3 and center of the origin. Shade the outside of the circle. Highlight and right click and select βPlot Inequalityβ. π₯ 2 + π¦2 > 3 ©2009 Nord Page 3 Microsoft Word Free Math Add-In 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. Consider the example: 3 sin(2 π₯) The input will appear in blue as shown: ©2009 Nord Page 4 Microsoft Word Free Math Add-In Right click and yield the graph: Consider other examples. 1 Example 1: Graph = cos(π₯) . The option Plot in 2D also appears if the input is in the form of an equation. The graph is: Example 2: Graph = ©2009 Nord sin π₯ π₯ . Page 5 Microsoft Word Free Math Add-In The graph is: sin π₯/π₯ 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. For our example, the graph is undefined at x=0. However, the graph appears continuous. ©2009 Nord Page 6 Microsoft Word Free Math Add-In Precalculus Solving Trigonometric Functions To input a popular symbol, use the down arrow key. This will bring up the Basic Math feature. The Basic Math feature will appear, and π will be an option. Alternatively, in the Insert New Equation line type, β\piβ followed with the space bar. The input will automatically change to, βπβ. Consider some trigonometric equations and their solutions. Example 1: sin 2π₯ = 0 Select Solve for x to yield: π₯= ©2009 Nord π β1 2 Page 7 Microsoft Word Free Math Add-In Example 2: π₯ = cos π₯ The answer in radians is: π₯ β 0.7390851332152 Example 3: π₯ = tan π₯ The output is: π₯β0 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. π = tan(π) The answer is: π=0 Example 5: Solve for the indicated variable. Solve for c. ©2009 Nord Page 8 Microsoft Word Free Math Add-In 1 c = arctan( ) 2 The Solve for c option brings up: 1 π = tanβ1 ( ) 2 This does not help. Erase the βc =β. The option, Calculate, will appear. 1 tanβ1 ( ) 2 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 9 Microsoft Word Free Math Add-In Precalculus Complex Numbers Example 1: Find the modulus of a complex number. β3 + 2π. The command is abs. Right-click and select Calculate. πππ (β3 + 2π) The modulus of the complex number is: β13 Example 2: Find the quotient of two complex numbers. 2 β 4π 3 + 5π Enter as a fraction using: The letter π is recognized equal to ββ1. Right-click and select Calculate. The answer is the complex number: β 7 11 β π 17 17 Example 3: Raise a complex number to an exponent that is a natural number. (β2 + β2π) ©2009 Nord 6 Page 10 Microsoft Word Free Math Add-In Right-click and select Calculate to give: β64π Example 4: Simplify using two different approaches (Zill and Cullen, 2006 , p. 802). 3 (1 β β3π) Right-click and select Calculate to give: β8 Apply DeMoivreβs Theorem, (π cos π + π π sin π)π = (cos ππ + π sin π π)π π , for another approach. Let r=2 and π = βπ/3 and obtain the same answer. (2 cos β π π 3 + 2π sin β ) 3 3 β8 Example 5: Given a complex number, convert it to the polar form, π§ = ππ ππ . Use the command topolar. Input the complex number, a + bi, following the command. Below is an example input and output. π‘ππππππ (0 + 1π) π ππ 2 The command toRect changes a number from polar form to rectangular form. The Calculate option from the pull-down menu will execute without the needed command toRect. This is an example input and output. β13 π 2 tanβ1( ) π 3 3 + 2π Reference Zill, D. and Cullen, M. Advanced Engineering Mathematics, third edition, Jones and Bartlett, Sudbury, Massachusetts, 2006). ©2009 Nord Page 11 Microsoft Word Free Math Add-In Precalculus Series When working with a series, do not use the letter, i. It is understood as the imaginary number equal to ββ1. 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. β10 π=1 π Right-click and select Simplify to yield: 55 Example 2: Find a formula to sum the first n positive integers. ©2009 Nord Page 12 Microsoft Word Free Math Add-In π βπ π=1 The output is: π2 + π 2 Example 3: Find the sum of the square of n integers. π β π2 π=1 The output is: π3 π2 π + + 3 2 6 Use the command Factor out the previous output. π3 π2 π ππππ‘ππ( + + ) 3 2 6 The Simplify command yields: π (π + 1) (2 π + 1) 6 Example 4: Find the sum of cube of n integers. π β π3 π=1 The output is: π4 π3 π2 + + 4 2 4 Using the Factor command with this output yields: π2 (π + 1)2 4 Example 5: Use the seriessum command. ©2009 Nord Page 13 Microsoft Word Free Math Add-In 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: π πππππ π π’π (π3 , π, 1,5) The Simplify command gives: 225 ©2009 Nord Page 14 Microsoft Word Free Math Add-In Precalculus Overview with Examples Example 1: Solve an equation for x. π₯3 β π₯2 β 4 = 0 The Solve for x command yields: 3 4 β21 55 3 4 β21 55 1 π₯=β + + ββ + + 9 27 9 27 3 To give a numerical solution, use the nsolve command. ππ πππ£π(π₯ 3 β π₯ 2 β 4 = 0) Solve for x or Simplify yields: π₯β2 Example 2: Solve an equation for x that will yield an answer containing a non-real number. π₯2 + 4 = 0 Under Math Preferences, select Complex Numbers and click OK. π₯2 + 4 = 0 The output is: π₯ = 2π π₯ = β2π ©2009 Nord Page 15 Microsoft Word Free Math Add-In The output if the number field selected is Real Numbers is: Example 3: Solve an equation with a degree 3 polynomial. π₯ 3 β 6π₯ 2 β 12π₯ β 6 = 0 The Solve for x command yields: 3 3 π₯ = ββ17 + 23 + β23 β β17 + 2 To give a numerical solution use the nsolve command. ππ πππ£π(π₯ 3 β 6π₯ 2 β 12π₯ β 6 = 0) The output is: π₯ β 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, β¦}),{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: 1 ππ πππ£π({π₯π ππ(π₯) = } , {π₯, 0, π}) 4 Output: π₯ β 0.5111022402679 Input: 1 ππ πππ£π({π₯π ππ(π₯) = } , {π₯, βπ, 0}) 4 Output: ©2009 Nord Page 16 Microsoft Word Free Math Add-In π₯ β β3.0597966989996 The next example will use a target place to look for the solution. An interval is not needed. Input: 1 ππ πππ£π({π₯π ππ(π₯) = } , {π₯, 6π}) 4 Output: π₯ β 18.8628099023054 Example 5: Use the nsolve command and specify an interval for a variable. Use more than one equation. Input: 1 ππ πππ£π({π₯ sin π¦ = , π₯ β π¦ = 8}, {{π₯}, {π¦, 0,2π}}) 4 Output: { π₯ β 8.0311338842625 π¦ β 0.0311338842625 Example 6: Multiply polynomials. Input: (π₯ β 1) β (π₯ β 2) β (π₯ β 3) The Expand command yields: π₯ 3 β 6 π₯ 2 + 11 π₯ β 6 Example 7: Find the integer roots of a polynomial. Input: π₯ 3 β 6 π₯ 2 + 11 π₯ β 6 = 0 The Solve for x command finds the answer: π₯=3 π₯=1 π₯=2 Example 8: Solve for an indicated variable. Input: ©2009 Nord Page 17 Microsoft Word Free Math Add-In π₯ 2 + 6π¦ 2 = 4 The screen will show: The option, Solve for y, gives: π¦ = ββ π₯2 2 + 6 3 π¦ = βββ π₯2 2 + 6 3 The option, Solve for x, gives: π₯ = ββ6 π¦ 2 + 4 π₯ = βββ6 π¦ 2 + 4 Example 9: Simplify an expression. Input: π₯3π¦5 π₯2π¦7 The option, Simplify, will yield: ©2009 Nord Page 18 Microsoft Word Free Math Add-In π₯ π¦2 Example 10: Solve a trigonometric equation. Find all solutions. Input: πΆππ (2π₯) = 0 The output where β1 is an integer is: π₯= ©2009 Nord π β1 π + 2 4 Page 19 Microsoft Word Free Math Add-In Example 11: Evaluate a limit where the answer is e. Input: 1 π lim (1 + ) πβ50 π The answer is: π To get an approximation, select Calculate. Output: 2.7182818284591 Example 12: Evaluate a limit. Input: lim π πππ₯/π₯ π₯ββ Output: 0 Example 13: Simplify an expression. Input: π₯ 2 β π¦ 3 + 3π¦ β π₯ + 82π¦ β 12.6π¦ Output: ©2009 Nord Page 20 Microsoft Word Free Math Add-In π₯2 π¦3 + 362 π¦ βπ₯ 5 Example 14: Create an example that will require the paid version of Microsoft Math. Input: max (π₯ 2 β 3π₯) 0β€π₯β€5 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: a) ββ π=1 b) ββ π=1 1 π! 2π βπ! 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, ββ1 , and the line would not execute. For the first example, input: ©2009 Nord Page 21 Microsoft Word Free Math Add-In β β π=1 1 π! Substitute 5 to get: 5 β π=1 1 π! Select Simplify to yield: 103 60 The partial sums for our problem are: 1 3 2 5 3 41 24 103 60 Consider the original problem: β β π=1 1 π! Select Simplify to get: πβ1 Select Calculate to have a decimal approximation: 1.7182818284591 ©2009 Nord Page 22 Microsoft Word Free Math Add-In 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: β β π=1 2π βπ! Substitute 5 to get: 5 β π=1 2π βπ! Right-click and select Simplify. The output is: 8 β6 8 β30 + + 2 β2 + 2 3 15 Select Calculate to receive: 14.2815867455289 Similarly, substitute 1, 2, 3, and 4, also. The answer for the partial sums is: 2 2 β2 + 2 ©2009 Nord Page 23 Microsoft Word Free Math Add-In 4 β6 + 2 β2 + 2 3 8 β6 + 2 β2 + 2 3 8 β6 8 β30 + + 2 β2 + 2 3 15 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: β β π=1 2π βπ! Right-click and select Simplify to get: β β π=1 2π βπ! The answer is not given. However, a substitution of a value for n 60 β π=1 2π βπ! such as n=60 gives the answer (with a little pause for the calculation) for the partial sum to be: ©2009 Nord Page 24 Microsoft Word Free Math Add-In 8 β6 8 β30 16 β5 16 β70 32 β35 64 β7 128 β77 128 β231 256 β858 + + + + + + + + 3 15 15 63 105 315 3465 10395 135135 256 β1430 256 β3003 512 β24310 1024 β12155 2048 β46189 + + + + + 225225 135135 11486475 34459425 654729075 2048 β230945 4096 β176358 4096 β969969 8192 β676039 + + + + 654729075 13749310575 13749310575 225881530875 8192 β4056234 16384 β104006 32768 β44574 + + + 316234143225 1581170716125 14230536445125 32768 β312018 65536 β1292646 65536 β66786710 + + + 14230536445125 412685556908625 63966261320836875 131072 β1077205 131072 β2203961430 262144 β33393355 + + + 2063427784543125 2110886623587616875 63966261320836875 262144 β64822395 1048576 β353452638 + + 2110886623587616875 1640158906527578311875 1048576 β3357800061 2097152 β90751353 + + 1640158906527578311875 44328619095339954375 2097152 β1531628098 2097152 β3829070245 + + 21322065784858518054375 106610328924292590271875 4194304 β134564468610 4194304 β156991880045 + + 13113070457687988603440625 4371023485895996201146875 8388608 β526024740930 8388608 β5786272150230 + + 563862029680583509947946875 563862029680583509947946875 16777216 β105204948186 + 1691586089041750529843840625 33554432 β2287064091 + 1691586089041750529843840625 33554432 β35830670759 + 61836869254970658257624840625 67108864 β71661341518 + 2782659116473679621593117828125 67108864 β107492012277 + 79504546184962274902660509375 134217728 β281132955186 + 141915614940157660701249009234375 134217728 β3654728417418 + 141915614940157660701249009234375 268435456 β14900046624858 + 7521527591828356017166197489421875 536870912 β2483341104143 + 22564582775485068051498592468265625 536870912 β1912172650190110 + 8687364368561751199826958100282265625 ©2009 Nord Page 25 Microsoft Word Free Math Add-In 1073741824 β136583760727865 1241052052651678742832422585754609375 1073741824 β108993841060836270 + 495179769008019818390136611716089140625 2147483648 β1879204156221315 + 495179769008019818390136611716089140625 4294967296 β7391536347803839 + 29215606371473169285018060091249259296875 4294967296 β110873045217057585 + + 2 β2 + 2 29215606371473169285018060091249259296875 + 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 = ππππ π 2 = ππ ππ π 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 π, π. The answer is: { π β 3.5227829907617 π β 0.6037493333974 The executable line using nsolve is: ππ πππ£π({2.9 = π cos π , 2 = π sin π}) Select Simplify to yield: { π β 3.5227829907617 π β 0.6037493333974 Example 17: Solve the following system of polar equations by giving an approximate solution for π₯, π¦, π, and π. ©2009 Nord Page 26 Microsoft Word Free Math Add-In π = 3 + sin π π = 2/ sin π 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 π₯, π¦, π, and π. ©2009 Nord Page 27