Find All Solutions of the Equation Sin Z 100
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Subsections
- Introduction
- Finding roots of a function or an expression
- Solving equations
- Exercises
Solving Equations with Maple
Introduction
The purpose of this lab is to locate roots and find solutions to one equation.Finding roots of a function or an expression
There are several different methods for finding the roots or the zeros of an expression. For instance, if it is possible, you could factor the expression and set each factor equal to zero. This can be done by using the Maple factor command. Another method for finding roots is to plot the expression and estimate the zeros by looking at where the graph intersects the
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> j := 2*x^3-5*x^2-2*x+5; > factor(j); > plot(j,x=-100..100); > plot(j,x=-3..3);Note that there is only one argument that is necessary for the factor command. The plot command is used to verify that there are exactly three roots for this expression. As the two plot commands show, it is sometimes difficult to see exactly how many roots there are based on the
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> factor(sin(x)+3); > plot(sin(x)+3,x=-4*Pi..4*Pi); > factor(x*sin(x)-1); > plot(x*sin(x)-1,x=-50..50);When an expression is already in factored form or cannot be factored, the Maple output is the same expression that was entered. Remember, not every expression has roots. That is, some expressions, when plotted, don't intersect with the
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Solving equations
Finding roots of an expression or a function
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> solve(equation,variable);The following example illustrates how we can find the roots of the function
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> f := x-> 2*x^3-5*x^2-2*x+5; > solve(f(x)=0,x); > solve(2*x^3-5*x^2-2*x+5=0,x);Here the ``='' sign is used in the equation, not ``:='' which is used for assignment. If you forget to type in an equation and only type in an expression without setting it equal to zero, Maple automatically sets the expression equal to zero. The solve command is not only used for solving for zeros, it can be used to solve other equations as well. In the examples below, you can see some of the solving capabilities of Maple.
> solve(sin(x)=tan(x),x); > solve(x^2+2*x-1=x^2+1,x);Unfortunately, many equations cannot be solved analytically. For example, we can use the quadratic formula to find the roots of any quadratic polynomial. There also exist formulas for finding roots of cubic and quartic (fourth order) equations, but they are so complicated that they are hardly ever used. However, it can be proven that there is no general formula for the roots of a fifth or higher order polynomial. Once we get away from polynomial equations, the situation is even worse. For example, even the relatively simple equation sin(x) = x/2 has no analytical solution.
If an equation cannot be solved analytically, then the only possibility is to solve it numerically. In Maple, the command to use is fsolve. The syntax for fsolve is very similar to that of solve. Sometimes when the solve command is used, the output looks like:
> solve(sin(x)=x/2,x); RootOf(_Z-2sin(_Z))This is not incorrect, as some of the zeros of a function may be imaginary and others may be real. What you need to do is read the output carefully looking for real solutions (each solution is separated by a comma) or use the fsolve command instead. A simple example will show how we can find solutions to the equation
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> fsolve(sin(x) = x/2, x);Note that the result is a decimal approximation and is not exact. Also, a plot of both equations on the same graph will show that this solution is not complete. There are two other intersection points that the fsolve command did not output. The fsolve command allows us to solve the equation in a range of
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> plot({sin(x),x/2},x=-2*Pi..2*Pi); > fsolve(sin(x) = x/2, x=-3..-1); > fsolve(sin(x) = x/2, x=-1..1); > fsolve(sin(x) = x/2, x=1..3);Once you have solved an equation, you may want to use the output or the solution later. In order to label the output to a solution, you need to assign a label in the same line as the solve or fsolve command. For example,
> expr2 := x^2 + 2*x - 5; > answer := solve(expr2=0,x); > evalf(subs(x=answer[1], expr2));Here, an expression was defined first and then the solution was assigned to the label ``answer''. Note that there was more than one solution. In order to substitute the answer that was listed first back into the expression, the subs command was used and [1] was added onto the variable name answer to distinguish the first solution from the second.
Exercises
- Given the expression
,
- Plot the expression and state how many roots there are.
- Use the Maple factor command to factor the expression.
- Use the Maple solve command to find roots of the expression.
- Use the Maple fsolve command to find roots of the expression.
- Given the function
,
- Plot the function over the interval
.
- Find all roots using the fsolve command and label the output.
- Substitute each root back into the function to show that the answer is zero.
- Plot the function over the interval
- Find all points where the functions
and
intersect each other. A plot of both functions on the same graph may be necessary to ensure that you have found all intersection points. Once you have found the
coordinate(s), substitute the solution(s) back into either function to get the corresponding
coordinate(s).
Next: About this document ... Up: lab_template Previous: lab_template Dina Solitro
2004-09-13
Find All Solutions of the Equation Sin Z 100
Source: https://www.math.wpi.edu/Course_Materials/MA1021A04/solve/node1.html