Let's analyze the solution step by step to determine if there was any mistake made by Jacob.
Step 4:
The given expression to evaluate is:
[tex]\[ \left(2^{\left(\frac{3}{2}\right)} - 4\right) - \left(3^{\left(-\frac{1}{2}\right)} - 2\right) \][/tex]
We need to calculate this expression.
Perform the calculations for each component:
1. [tex]\( 2^{\left(\frac{3}{2}\right)} \)[/tex] = [tex]\( \sqrt{2^3} \)[/tex] = [tex]\( \sqrt{8} \)[/tex] = [tex]\( 2\sqrt{2} \approx 2.828 \)[/tex]
2. [tex]\( 3^{\left(-\frac{1}{2}\right)} \)[/tex] = [tex]\( \frac{1}{\sqrt{3}} \approx 0.577 \)[/tex]
So, the expression becomes:
[tex]\[ (2.828 - 4) - (0.577 - 2) \][/tex]
[tex]\[ \approx (-1.172) - (-1.423) \][/tex]
[tex]\[ \approx (-1.172 + 1.423) \][/tex]
[tex]\[ \approx 0.251 \][/tex]
The evaluation of the expression indeed results in approximately 0.251, confirming that this numerical step is correct.
Step 5:
Based on the evaluation value being positive (0.251), Jacob decides to adjust the bounds. Here are the given bounds:
- Previous lower bound: [tex]\( x = \frac{3}{2} \)[/tex]
- Previous upper bound: [tex]\( x = 2 \)[/tex]
According to the problem, since the evaluation value in Step 4 is positive, we should update the bounds. The rule for updating bounds generally revolves around setting the new bounds such that the interval containing the root is narrowed down. Since the evaluated expression was positive at [tex]\( x = \frac{3}{2} \)[/tex], it indicates that the expression increases from negative to positive as [tex]\( x \)[/tex] increases. Therefore, the zero root should lie between these bounds.
Jacob's error can be identified here. Jacob updated the lower bound instead of the upper bound:
- The new lower bound should remain [tex]\( x = \frac{3}{2} \)[/tex] since the positive value indicates that the zero root is likely between [tex]\( \frac{3}{2} \)[/tex] and 2.
Therefore, the corrected conclusion for step updating bounds should be setting [tex]\( x = \frac{3}{2} \)[/tex] as the new upper bound instead.
So, the correct answer is:
B. Jacob made a mistake at step 5. He should have used [tex]\( x = \frac{3}{2} \)[/tex] as the new upper bound.
