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example: find the solutions to x^3 + 3x^2 -17x + 9 = 0 1. graphically: draw a graph and observe where it crosses the x-axes 2. numerically: use the Netwton-Raphson method to iterate to each root starting with a guess 3. algebraically: use the formula for cubic equations to find the roots.
the algebraic method is the only one that will give you an "exact" solution
Well, it's mainly my trouble of differentiating between numerically and algebraically because I have to find the solution both ways...the actual problem involves a sinusoid and the line y = 6. I can solve for x in the equation, but I don't know what the numerical way would be.
have you been taught any numerical methods?
Well, the textbook says to use a calculator and use the intersection function, but my teacher says don't use a calculator.
so, just to be clear, you have something to solve and you know how to solve it algebraically but are looking for a way to solve it numerically without using a calculator?
Yes. THat is the problem. lol. I don't exactly know what it means by numerically, so you can see the problem XD
well in this case I would say your teacher may have made a mistake. for a numerical solution you need to use some method that eventually iterates to the correct answer. one such method is the Newton-Raphson method (described here: http://en.wikipedia.org/wiki/Newton%27s_method). some calculators have this built-in, so you can just enter the equation that needs solving and they will execute the most appropriate numerical algorithm to get the answer. does that explain it better?
a numerical method is basically some algorithm that guarantees that it will converge to one or more solutions of a given problem.
each algorithm has its restrictions - so you need to make sure the problem you are solving satisfies the restrictions of the algorithm otherwise it will not converge to a solution.
Well, I guess that I'll just ask my teacher tomorrow since I never learned that method. I think I understand what it means to solve numerically now, but I think that we do indeed need a calculator now...I'll see tomorrow. Thank you. :)