Microsoft Word Free Math Add-In - Gonzaga University
PrecalculusConic SectionsExample 1: Graph an ellipse. x24+y216=1Toggles graph to be in and out of proportion.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:Show grid lines (yes/no)Diplay units on the perimeter (yes,no)Note that there is new labeling on the axes. The Display row can furthermore be used to alter how the graph is displayed. Insert a desired domain and/or range.Display axes (yes/no)Example 2: Graph a hyperbola. x216-y24=1Select Plot in 2D.Alternately, you can use the command:PlotEq(x216-y24=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 PrecalculusGraphing Trigonometric FunctionsSet 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:3sin(2 x)The input will appear in blue as shown:Right click and yield the graph:Consider other examples.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 =sinxx .The graph is:sinx/xUse 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.PrecalculusSolving Trigonometric FunctionsThis will bring up the Basic Math feature.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:sin2x=0Select Solve for x to yield:x=π N12Example 2: x=cosxThe answer in radians is:x≈0.7390851332152Example 3: x=tanxThe output is:x≈0The 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=0Example 5: Solve for the indicated variable. Solve for c. c=arctan(12)The Solve for c option brings up:c=tan-112This does not help. Erase the “c =”. The option, Calculate, will appear.tan-112Select Calculate to give the following answer in radian mode:0.4636476090008Select Calculate to give the following answer when the Math Preferences are set to Degrees:26.565051177078PrecalculusComplex NumbersExample 1: Find the modulus of a complex number.-3+2i.The command is abs. Right-click and select Calculate.abs-3+2iThe modulus of the complex number is:13Example 2: Find the quotient of two complex numbers. 2-4i3+5iEnter as a fraction using: The letter i is recognized equal to -1. Right-click and select Calculate. The answer is the complex number:-717-1117iExample 3: Raise a complex number to an exponent that is a natural number.2+2i6Right-click and select Calculate to give:-64iExample 4: Simplify using two different approaches (Zill and Cullen, 2006 , p. 802). 1-3i3Right-click and select Calculate to give:-8Apply DeMoivre’s Theorem, r cosθ+r isinθn =(cosnθ+isinnθ)rn , for another approach. Let r=2 and θ=-π/3 and obtain the same answer.2cos-π3+2isin-π33-8Example 5: Given a complex number, convert it to the polar form, z=reθi.Use the command topolar. Input the complex number, a + bi, following the command. Below is an example input and output. topolar (0+1i)eπ i2The 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.13 etan-123 i3+2iReferenceZill, D. and Cullen, M. Advanced Engineering Mathematics, third edition, Jones and Bartlett, Sudbury, Massachusetts, 2006).PrecalculusSeriesWhen 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=110k Right-click and select Simplify to yield:55Example 2: Find a formula to sum the first n positive integers. k=1nk The output is:n2+n2Example 3: Find the sum of the square of n integers.k=1nk2 The output is:n33+n22+n6Use the command Factor out the previous output.factor(n33+n22+n6)The Simplify command yields:n n+1 2 n+16Example 4: Find the sum of cube of n integers. k=1nk3 The output is:n44+n32+n24Using the Factor command with this output yields:n2 n+124Example 5: Use the seriessum command.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:225PrecalculusOverview with ExamplesExample 1: Solve an equation for x. x3-x2-4=0The Solve for x command yields:x=34 219+5527+3-4 219+5527+13To give a numerical solution, use the nsolve command.nsolve(x3-x2-4=0)Solve for x or Simplify yields:x≈2Example 2: Solve an equation for x that will yield an answer containing a non-real number.x2+4=0Under Math Preferences, select Complex Numbers and click OK.x2+4=0The output is: x=2i x=-2iThe output if the number field selected is Real Numbers is:Example 3: Solve an equation with a degree 3 polynomial.x3-6x2-12x-6=0The Solve for x command yields:x=317+23+323-17+2To give a numerical solution use the nsolve command. Nsolve(x3-6x2-12x-6=0)The output is:x≈7.667178637832The 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(xsinx=14,x,0,π)Output:x≈0.5111022402679Input:nsolve(xsinx=14,x,-π,0)Output:x≈-3.0597966989996The next example will use a target place to look for the solution. An interval is not needed.Input:nsolve(xsinx=14,x,6π)Output:x≈18.8628099023054Example 5: Use the nsolve command and specify an interval for a variable. Use more than one equation.Input:nsolve({xsiny=14,x-y=8},{x,y,0,2π})Output: x&≈8.0311338842625 y&≈0.0311338842625Example 6: Multiply polynomials.Input:x-1*x-2*(x-3)The Expand command yields:x3-6 x2+11 x-6Example 7: Find the integer roots of a polynomial.Input:x3-6 x2+11 x-6=0The Solve for x command finds the answer: x=3 x=1 x=2Example 8: Solve for an indicated variable.Input:x2+6y2=4The screen will show:The option, Solve for y, gives: y=-x26+23 y=--x26+23The option, Solve for x, gives: x=-6 y2+4 x=--6 y2+4Example 9: Simplify an expression.Input:x3y5x2y7The option, Simplify, will yield:xy2Example 10: Solve a trigonometric equation. Find all solutions.Input:Cos2x=0The output where N1is an integer is:x=π N12+π4Example 11: Evaluate a limit where the answer is e.Input:limn→501+1nnThe answer is:eTo get an approximation, select Calculate.Output:2.7182818284591Example 12: Evaluate a limit. Input:limx→∞sinx/xOutput:0Example 13: Simplify an expression.Input:x2*y3+3y-x+82y-12.6yOutput:x2 y3+362 y5-xExample 14: Create an example that will require the paid version of Microsoft Math.Input:max0≤x≤5x2-3x 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: n=1∞1n!n=1∞2nn!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=151n!Select Simplify to yield:10360The partial sums for our problem are:13253412410360Consider the original problem:n=1∞1n!Select Simplify to get:e-1Select Calculate to have a decimal approximation:1.7182818284591Right-click and select Calculate for each output for a partial sum. The answers are: 11.51.66666666666671.70833333333331.7166666666667Consider the input for the second example:n=1∞2nn!Substitute 5 to get: n=152nn!Right-click and select Simplify. The output is:8 63+8 3015+2 2+2Select Calculate to receive:14.2815867455289Similarly, substitute 1, 2, 3, and 4, also. The answer for the partial sums is:22 2+24 63+2 2+28 63+2 2+28 63+8 3015+2 2+2Select Calculate for each output and the answer as a decimal for the partial sums is:24.82842712474628.094413448457111.36039977216814.2815867455289Consider the original problem:n=1∞2nn!Right-click and select Simplify to get:n=1∞2nn!The answer is not given. However, a substitution of a value for n n=1602nn!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+2Select Calculate to get:21.8586197886637Example 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=rsin θThere 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&≈3.5227829907617 θ&≈0.6037493333974The executable line using nsolve is:nsolve({2.9=rcosθ, 2=rsinθ})Select Simplify to yield:r&≈3.5227829907617 θ&≈0.6037493333974Example 17: Solve the following system of polar equations by giving an approximate solution for x, y, r, and θ. r=3+sinθr=2/sinθInsert 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, and θ. ................
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