Approximate the greatest real zero of the function g(x)= x^3-3x+1 to the nearest tenth.
I know that there is a zero between -2 and -1, and another between 0 and 1 but do not know how to find it to the nearest tenth. The only example shown in my book uses a calculator and mine does not have instructions for the same functions as the one in the book. I have checked for online calculators and cannot find one to do the calculations either.
Any help would be great as I have to get finished with math this week.
Thanks
Try solving it iteratively: if g(X)=0, then X^3-3X+1=0, so rearrange the equation to read
X = (3X-1)^(1/3)
Not put X = 2, and evaluate the function. You'll get about 1.71. Feed that into the equation again, and you'll get about 1.60. Keep going for a few more iterations until it settles down. Then try feeding it into the original equation and see if it works.
To approximate the greatest real zero of the function g(x) = x^3 - 3x + 1 to the nearest tenth without using a calculator or online resources, we can use a numerical method called the bisection method. Here's how it works:
1. First, we need to find an interval [a, b] that contains the zero we are interested in. You correctly mentioned that there is a zero between -2 and -1, and another between 0 and 1. Let's start with the interval [-2, -1].
2. Calculate g(a) and g(b) to determine if the zero lies in the interval. Evaluate g(-2) and g(-1) to check the signs. If g(a) and g(b) have different signs, then by the Intermediate Value Theorem, we know there is a zero in the interval [a, b]. In this case, g(c) = 0, where c is some value between a and b.
3. Divide the interval in half by finding the midpoint c = (a + b)/2.
4. Calculate g(c) to determine its sign. If g(c) = 0, then c is the zero we are looking for. Otherwise, if g(c) and g(a) have the same sign, replace a with c. Otherwise, replace b with c.
5. Repeat steps 3 and 4 until the interval becomes very small or until you reach the desired accuracy.
Let's apply these steps:
1. Let's start with the interval [-2, -1]. Calculate g(-2) and g(-1):
g(-2) = (-2)^3 - 3(-2) + 1 = -13
g(-1) = (-1)^3 - 3(-1) + 1 = 3
Since g(-2) is negative and g(-1) is positive, we know there is a zero in the interval [-2, -1].
2. Calculate the midpoint c = (-2 + (-1))/2 = -1.5.
3. Calculate g(-1.5):
g(-1.5) = (-1.5)^3 - 3(-1.5) + 1 = -3.375
Since g(-1.5) is negative, we replace a with -1.5.
4. Repeat steps 2 and 3:
New midpoint c = (-2 + (-1.5))/2 = -1.75
g(-1.75) = (-1.75)^3 - 3(-1.75) + 1 = -0.859375
Since g(-1.75) is negative, we replace a with -1.75.
5. Repeat steps 2 and 3 again:
New midpoint c = (-1.75 + (-1.5))/2 = -1.625
g(-1.625) = (-1.625)^3 - 3(-1.625) + 1 = -0.388671875
Since g(-1.625) is negative, we replace a with -1.625.
Continue repeating steps 2 and 3 until the interval becomes very small or until you reach the desired accuracy. Keep in mind that this method requires a significant number of iterations to reach high precision.
By following these steps, you can approximate the greatest real zero of the function g(x) = x^3 - 3x + 1 to the nearest tenth.