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Single Variable CalculusDifferentialThe deriv function will yield the first derivative with respect to the variable indicated. The syntax is deriv(function, variable). The deriv function may be omitted. There is an alternate way to find the first derivative with respect to the particular variable, x. Enter the function as an expression in the variable x. Right-click and select Differentiate on x from the menu. The derivn function will yield a higher order derivative for a given A function derived from another function so that at each point of the original function, the derivative represents the slope of the original function at that poin function. The syntax is derivn(function, variable, n) where n ∈ {2, 3, 4, …..}, the order. Three arguments are required. There is an alternate way to find the nth derivative with respect to the particular variable, x. Enter the function as an expression in the variable x. Right-click and select Differentiate on x from the menu n times using the previous input lines. Example 1: Find the first derivative for a given function.Consider the input: deriv(arccosx-3,x)Right-click on the input line and select Simplify.The output is:-1-x-32+1An easier way to get the same answer does not involve the deriv command. Type in the expression as follows:arccos?(x-3)Right-click and select Differentiate on x. The output is:-1-x-32+1Example 2: Find the third derivative with respect to x for a given function.Consider the input:sin-1xcos xRight-click and select Differentiate on x.The answer for the first derivative is:secx tanx sin-1x+secx-x2+1Right click within the last output and select Differentiate on x . The answer is:xsecx-x2+132+2secx sin-1xtanx2+2secxtanx-x2+1+secx sin-1xRight click within in the last output and select Differentiate on x a last time. The answer is:3 x2secx-x2+152+3 xsecxtanx-x2+132+6secx sin-1xtanx3+6secxtanx2-x2+1+secx-x2+132+5secx tanx sin-1x+3secx-x2+1The derivn command will allow the result to be displayed after one input line. Consider the input:derivn(sin-1xcos x, x, 3)Right click and select Simplify.The output is:3 x2secx-x2+152+3 xsecxtanx-x2+132+6secx sin-1xtanx3+6secxtanx2-x2+1+secx-x2+132+5secx tanx sin-1x+3secx-x2+1Example 3: Given a polynomial, find the 3rd derivative with respect to x. Consider the input:derivn(2x5-2x, x, 3)The output is:120 x2The same output can be obtained with an initial input of an expression: 2x5-2xRight click and select Differentiate on x. Repeat two more times with right-clicking on the previous outputs. The first and second and third derivatives are:10 x4-240 x3120 x2Single Variable CalculusIntegralsThere are three ways to evaluate an indefinite integral. Type an expression and select Integrate on x.Use the integral symbol in the top ribbon and select Simplify.Use the command, integral, and input integral(input, variable) and select Simplify.I have seen that the third approach execute when involving a radical, here as the first two approaches may not evaluate and simply follow a return of the original input. The software appears to be cautious when working with a radical and requiring that the radical does not take on a value that is zero (or negative). The first approach is the easiest to input.There are two ways to evaluate a definite integral.Use the integral symbol in the top ribbon and select Simplify.Use the command, integral, and input integral(input, variable, lowervalue, uppervalue). Example 1: Type an expression. Select Integrate on x from the drop-down menu to integrate on x with the expression. Consider the example:4-x^2Place the cursor in the middle and right click. Choose Integrate on x. The output is:x -x2+42+2tan-1x-x2+4+СExample 2: Evaluate a definite integral. Use the integral symbol from the top ribbon. Consider the input:45x2-4dxThe right arrow button was used after the square root not place the dx underneath the square root. Right click and select Simplify from the menu. The output is:2 ln2 3+4-2ln21+5+5 212-4 3Right click within this output and select Calculate.The output is:4.0285533270189Example 3: Evaluate an indefinite integral that would involve a substitution. The input is:xx-1 dx Right click and select Simplify to yield the output:2 x x-1323-4 x-15215 The output does not include the integration constant. This will occur when evaluating some indefinite integrals.Consider another example where the input is only an expression:xx2-9 Right click and select Integrate on x to give:x2-9323+СThe command integral would have yielded the same answer. The right arrow button should be used before inputting the comma. This will ensure that the comma and dx are not under the radical. The input is:integral(x2x2-9,x)Select Simplify to give the answer:x2-9323+СExample 4: Evaluate the definite integral:11.54-x2dxUsing the integral symbol in the top ribbon and selecting Simplify gives the output:11.5-x2+4?xThe definite integral is not evaluated. Consider trying this another way using the integral command and inputting:integral(-x2+4,x,1,1.5)Again, the output is not simplified:11.5-x2+4?xA work-around would be to consider an indefinite integral:4-x2dxThe output is simplified:x –x2+42+2tan-1x-x2+4+СApply the Fundamental Theorem by evaluating F(1.5) – F(1) and cut and paste from the previous output. Consider the new input:(1.5 -1.52+42+2tan-11.5-1.52+4- (1.0 -1.02+42+2tan-11.0-1.0+4)Select Calculate.The output is:2 tan-13 77+3 78-π3-32Select Calculate again to give:0.7750579446311Mathematica gives the same answer and evaluates the definite integral directly as shown below:Example 5: Evaluate a definite integral involving a trigonometric function.Consider the input:05sinxdxThe Math Preferences needs to be set. The angle measure should be in radian mode as shown below:The output from our example is:1-cos5Right-click and select Calculate from the menu to give:0.7163378145368Example 6: Evaluate an indefinite integral involving a trigonometric function.Consider the input:(cosx)2Right-click and select Simplify to give:sin2 x4+x2+СExample 7: Take the derivative with respect to x for the following example involving an integral. 2x3t dtRight-click and select Simplify to give:3 x22-6Right-click and select Differentiate on x.The Fundamental Theorem tells us the answer is:3 xExample 8: Evaluate the following definite integral by using the integral command.46xx-3 dxThe input is:integral(xx-3,x,4,6)Select Simplify to give:-4·35215+12 3-125Select Calculate to give:14.2276877526612Example 9: Integrate using trigonometric functions and the superscript feature in the top ribbon.Consider the example:sin?x cos?(x)^5 Trigonometric functions are recognized. After typing one in, press enter to key in the argument. A blue box will appear as shown below:Parentheses are optional. As long as what is typed is within the blue box, the add-in will understand this as the argument. In the example below, typing cos (x) followed with a right arrow will ensure that the cos(x) is the base. sinx cos(x)^5Select Integrate on x to give:-cosx66+СSingle Variable CalculusGraphing The table below describes some graphing commands. In most instances, the command may be absent. The exception is the plotparam command that is required. The commands are not case sensitive. That is, the use of capitalization is optional and will not change whether the line executes. CommandExampleNotation RequirementsDrop- Down Menu Option to Executeplotplot(3*x2+12*x)Input function, f(x).SimplifyplotDataSetplotDataSet({2, 3, 5, -1})Input point, {x, y}.CalculateplotEqplotEq(x2+y2=9)Input f(x, y) = c.SimplifyplotIneqplotIneq(x≤3+y)Input inequality in x and y.SimplifyplotParamplotParam(sint, t^2)Input (f(t), g(t))where x=f(t) and y=g(t).SimplifyplotPolarplotPolar(3×sinθ)Input r=fθ.SimplifyplotPolarDataSetplotPolarDataSet({2,π3, 8, -π2})Input point {r, θ}.Calculate Graphing in two dimensions Here are a couple of options of how to input multiple equations.Method 1: Use the show command and this will allow the input of both equations with a single application of the Insert New Equation command. Insert New Equation can be found here:Input:show(plota*x2+b*x,plot2ax+b)Select Simplify from the right-click menu. This will bring up the two graphs.The show command needs the following syntax: show(plot1example, plot2example)Method 2: Here are the equations. Each equation is inserted separately with Insert New Equation.y=a*x2+b*xy=2*a*x+bUse the mouse (left click and drag over the top of these equations with keeping the mouse pressed down) and highlight both simultaneously now. There cannot be any text between these equations. You only want the equations highlighted. You should see this image on your screen: After both are highlighted (in blue), right click to bring up the option, Plot in 2D. You need to right click and not left click. You are now working with both equations.Either method should return this graph:The animate option appears by default when a variable is introduced such as a. Example 1: Graph the following. Use the Trace command to find information about points on the curve.r=1-2cos3θInput the following using Insert New Equation. r=1-2cos3θSelect Plot in 2D. The following will appear:Select Trace. Example 2: Give an example using the plotparam command.Consider the input:plotparam(2sin2t, 3sin3t)Select Simplify to get:Example 3: Give an example using the plotparam command and the Trace feature. Consider the input:plotparam(2sin2t, 3sint)Select Simplify and then apply the Trace feature. Example 4: Plot the following equation.x2-xy-5y-1=0Input the equation, right-click, and select Plot in 2D. Use the Zoom out feature to capture the hyperbola. Zoom out using this button. ................
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