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.
http://www.math.montana.edu/frankw/ccp/calculus/numerical/newton/learn.htm
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To find the smallest positive value of x that satisfies the equation x = 4.100cos(2.800x), we can use an iterative method like the Newton-Raphson method. Here's how you can do it:
1. Start by guessing a value of x, let's say x0 = 0.
2. Calculate the value of f(x0) = x0 - 4.100cos(2.800x0).
3. Calculate the derivative of f(x) with respect to x: f'(x) = 1 + 11.480sin(2.800x).
4. Use the formula x1 = x0 - f(x0)/f'(x0) to calculate a better approximation of x.
5. Repeat steps 2-4 until the desired level of accuracy is achieved.
Now let's proceed with the calculation and find the smallest positive value of x that satisfies the equation:
Step 1: Guess x0 = 0.
Step 2: f(x0) = 0 - 4.100cos(2.800 * 0) = -4.100
Step 3: f'(x) = 1 + 11.480sin(2.800 * 0) = 1
Step 4: Using x1 = x0 - f(x0)/f'(x0), we have x1 = 0 - (-4.100)/1 = 4.100.
Step 5: Let's repeat the process with x1 as our new approximation.
Step 1: Guess x0 = 4.100.
Step 2: f(x0) = 4.100 - 4.100cos(2.800 * 4.100) = -0.350
Step 3: f'(x) = 1 + 11.480sin(2.800 * 4.100) = 5.456
Step 4: Using x1 = x0 - f(x0)/f'(x0), we have x1 = 4.100 - (-0.350)/5.456 ≈ 4.167.
Step 5: Repeat the process until the desired level of accuracy is achieved.
By continuing this process, you will eventually find the smallest positive value of x that satisfies the equation to the desired level of accuracy.