Can someone help me out here with these two problem? I have know idea where to start.
1)The equation: 10(x-1)(x-2)(x-3)=1,
has three real solutions a<b<c
where
a=____, b=___ , and c=___.
Enter your answers with at least six correct digits beyond the decimal point.
Hint: Ask what the solutions are if the right hand side is 0 instead of 1, and use Newton's Method.
2) Find the smallest positive value of which satisfies -- x=4.100cos(2.800x)
Give the answer to four places of accuracy._________
Remember to calculate the trig functions in radian mode.
Mr. Pursley has answered both questions.
If you had read the links, you would be able to come up with preliminary, if not final answers.
If you encounter difficulties, post in detail what they are.
I'll give you a headstart.
Rewrite the equation in the form:
f(x)=0
For example
f(x)=x²-2=0
Calculate f'(x)=2x
Start with an approximation, say x0=1.
Calculate
x1=x0-f(x0)/f'(x0)
=1-(1-2)/2
=1.5
Repeat until the desired accuracy is obtained:
x2=x1-f(x1)/f'(x1)
=1.5-(2.25-2)/(2*2.25)
=1.444
x3=x2-f(x2)/f'(x2)
=1.444-(0.085136)/2.888
=1.41452
...
Notice that the number of accurate figures of the solution doubles with every iteration.
Try the method with the given problems and post if you have difficulties.
Sure, I can help you with both of these problems.
Problem 1:
To find the solutions of the equation 10(x-1)(x-2)(x-3)=1, we can use Newton's Method. However, before applying Newton's Method, let's first consider the equation if the right-hand side is 0.
If we set 10(x-1)(x-2)(x-3)=0, we get three solutions, say p, q, and r. These solutions will help us in applying Newton's Method to find the solutions for the equation when the right-hand side is 1.
Now, let's apply Newton's Method to find the solution for 10(x-1)(x-2)(x-3)-1=0.
1. Start with an initial guess, let's say x = p.
2. Calculate the value of the function at this point: f(p) = 10(p-1)(p-2)(p-3)-1.
3. Calculate the derivative of the function: f'(p) = 10[(p-2)(p-3)+(p-1)(p-3)+(p-1)(p-2)].
4. Update the guess using the formula: p - f(p)/f'(p).
5. Repeat steps 2-4 until you reach a desired level of accuracy, or until the solution converges.
Repeat steps 1-5 using the initial guesses x = q and x = r to find the other two solutions. Remember to use a calculator or software to perform the calculations involved.
The solutions for the equation 10(x-1)(x-2)(x-3)=1, where a<b<c, can then be found using Newton's Method with the initial guesses p, q, and r.
Problem 2:
To find the smallest positive value of x which satisfies the equation x=4.100cos(2.800x), we can use an iterative approach.
1. Start with an initial guess for x, let's say x = 0.
2. Evaluate the right-hand side of the equation: 4.100cos(2.800x).
3. Update the guess using the formula: x = 4.100cos(2.800x).
4. Repeat steps 2-3 until you reach the desired accuracy or until the solution converges.
Continue this iterative process until you find a value of x that satisfies the equation. Remember to perform the cosine calculation in radian mode.
Once you obtain a value for x that satisfies the equation x=4.100cos(2.800x), round it to the desired accuracy (in this case, four decimal places) and provide the answer.
Remember to use a calculator or software to perform the necessary calculations accurately.