Absolutely, let's break this down step-by-step. Our goal is to calculate the value of the polynomial function [tex]\( y = x(x-1)(x-0.5) \)[/tex] at various values of [tex]\( x \)[/tex].
Here is a detailed step-by-step solution:
### Step 1: Identify the Polynomial Function
First, we need to recognize that we are working with the polynomial function:
[tex]\[ y = x(x-1)(x-0.5) \][/tex]
### Step 2: Apply the Function to Each [tex]\( x \)[/tex] Value
We will substitute each [tex]\( x \)[/tex] value into the polynomial and perform the computations.
### Step 3: Perform the Calculations
- For [tex]\( x = -0.1 \)[/tex]:
[tex]\[ y = (-0.1)((-0.1)-1)((-0.1)-0.5) \][/tex]
[tex]\[ y = (-0.1)(-1.1)(-0.6) \][/tex]
[tex]\[ y = -0.066 \][/tex]
- For [tex]\( x = -0.01 \)[/tex]:
[tex]\[ y = (-0.01)((-0.01)-1)((-0.01)-0.5) \][/tex]
[tex]\[ y = (-0.01)(-1.01)(-0.51) \][/tex]
[tex]\[ y = -0.005151 \][/tex]
- For [tex]\( x = -0.001 \)[/tex]:
[tex]\[ y = (-0.001)((-0.001)-1)((-0.001)-0.5) \][/tex]
[tex]\[ y = (-0.001)(-1.001)(-0.501) \][/tex]
[tex]\[ y = -0.000501501 \][/tex]
- For [tex]\( x = 0.0 \)[/tex]:
[tex]\[ y = (0.0)((0.0)-1)((0.0)-0.5) \][/tex]
[tex]\[ y = 0.0(0.0)(0.0) \][/tex]
[tex]\[ y = 0.0 \][/tex]
- For [tex]\( x = 0.001 \)[/tex]:
[tex]\[ y = (0.001)((0.001)-1)((0.001)-0.5) \][/tex]
[tex]\[ y = (0.001)(-0.999)(-0.499) \][/tex]
[tex]\[ y = 0.000498501 \][/tex]
- For [tex]\( x = 0.01 \)[/tex]:
[tex]\[ y = (0.01)((0.01)-1)((0.01)-0.5) \][/tex]
[tex]\[ y = (0.01)(-0.99)(-0.49) \][/tex]
[tex]\[ y = 0.004851 \][/tex]
- For [tex]\( x = 0.1 \)[/tex]:
[tex]\[ y = (0.1)((0.1)-1)((0.1)-0.5) \][/tex]
[tex]\[ y = (0.1)(-0.9)(-0.4) \][/tex]
[tex]\[ y = 0.036 \][/tex]
### Step 4: Compile the Results
Now we can compile all the results together into our table:
[tex]\[
\begin{array}{|l|l|l|l|l|l|l|l|}
\hline
x & -0.1 & -0.01 & -0.001 & 0.0 & 0.001 & 0.01 & 0.1 \\
\hline
y & -0.066 & -0.005151 & -0.000501501 & 0.0 & 0.000498501 & 0.004851 & 0.036 \\
\hline
\end{array}
\][/tex]
Thus, these are the calculated values of [tex]\( y = x(x-1)(x-0.5) \)[/tex] for the given [tex]\( x \)[/tex] values.
