Answer :
Let's closely analyze each step that Jacob followed to identify where he might have gone wrong.
### Step by Step Analysis
#### Step 1:
Jacob rewrote the given equation:
[tex]\[ 2^x - 4 = 3^{-x} - 2 \][/tex]
as:
[tex]\[ (2^x - 4) - (3^{-x} - 2) = 0 \][/tex]
This step is algebraically correct.
#### Step 2:
Jacob evaluated the rewritten equation at the bounds [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[ (2^1 - 4) - (3^{-1} - 2) \approx -0.333 \][/tex]
This value is correct because:
[tex]\[ (2 - 4) - \left(\frac{1}{3} - 2\right) = -2 - (-1.6667) = -0.3333 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ (2^2 - 4) - (3^{-2} - 2) \approx 1.889 \][/tex]
This value, however, is incorrect. The actual evaluated value given by the solution is:
[tex]\[ (4 - 4) - \left(\frac{1}{9} - 2\right) = 0 - (-1.8889) = 1.8889 \][/tex]
Thus, Jacob's calculated value of [tex]\( 1.889 \)[/tex] at [tex]\( x = 2 \)[/tex] was indeed correct to 3 decimal places. Hence, there isn't an issue in step 2.
#### Step 3:
Jacob took the average of the lower and upper bounds:
[tex]\[ \frac{1 + 2}{2} = \frac{3}{2} = 1.5 \][/tex]
This calculation is also correct.
#### Step 4:
Jacob evaluated the rewritten equation at [tex]\( x = 1.5 \)[/tex]:
[tex]\[ \left(2^{1.5} - 4\right) - \left(3^{-1.5} - 2\right) \approx 0.636 \][/tex]
This value is accurate as the given solution provides:
[tex]\[ (2^{1.5} - 4) - (3^{-1.5} - 2) \approx 0.636 \][/tex]
#### Step 5:
Jacob decided to adjust the bounds based on the midpoint evaluation. Since [tex]\( x = \frac{3}{2} \)[/tex] yielded a positive value ([tex]\( \approx 0.636 \)[/tex]), he should modify the bounds to [tex]\( [1, 1.5] \)[/tex], with the midpoint becoming the new upper bound.
### Conclusion
From the above steps, it becomes clear that all calculations performed in steps 1 through 4 were correct, and Jacob correctly identified that at [tex]\( x = 2 \)[/tex], the function equals approximately [tex]\( 1.889 \)[/tex]. Jacob also correctly evaluated the function at [tex]\( x = 1.5 \)[/tex].
However, if the problem states that Jacob made a mistake, it appears in adjusting the bounds based on the new information. Despite selecting the bounds correctly in step 2, since [tex]\( 1.889 \)[/tex] (at [tex]\( x = 2 \)[/tex]) is positive and [tex]\( -0.333 \)[/tex] (at [tex]\( x = 1 \)[/tex]) is negative, Jacob should continue ranging between these bounds more accurately.
Given that the values were accurate based on step 2, the correct bounding process in step 5 should proceed accurately without an issue. Jacob most probably didn't commit a mistake in these critical steps.
### Final Answer:
C. Jacob did not make any mistakes in the calculation process.
### Step by Step Analysis
#### Step 1:
Jacob rewrote the given equation:
[tex]\[ 2^x - 4 = 3^{-x} - 2 \][/tex]
as:
[tex]\[ (2^x - 4) - (3^{-x} - 2) = 0 \][/tex]
This step is algebraically correct.
#### Step 2:
Jacob evaluated the rewritten equation at the bounds [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[ (2^1 - 4) - (3^{-1} - 2) \approx -0.333 \][/tex]
This value is correct because:
[tex]\[ (2 - 4) - \left(\frac{1}{3} - 2\right) = -2 - (-1.6667) = -0.3333 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ (2^2 - 4) - (3^{-2} - 2) \approx 1.889 \][/tex]
This value, however, is incorrect. The actual evaluated value given by the solution is:
[tex]\[ (4 - 4) - \left(\frac{1}{9} - 2\right) = 0 - (-1.8889) = 1.8889 \][/tex]
Thus, Jacob's calculated value of [tex]\( 1.889 \)[/tex] at [tex]\( x = 2 \)[/tex] was indeed correct to 3 decimal places. Hence, there isn't an issue in step 2.
#### Step 3:
Jacob took the average of the lower and upper bounds:
[tex]\[ \frac{1 + 2}{2} = \frac{3}{2} = 1.5 \][/tex]
This calculation is also correct.
#### Step 4:
Jacob evaluated the rewritten equation at [tex]\( x = 1.5 \)[/tex]:
[tex]\[ \left(2^{1.5} - 4\right) - \left(3^{-1.5} - 2\right) \approx 0.636 \][/tex]
This value is accurate as the given solution provides:
[tex]\[ (2^{1.5} - 4) - (3^{-1.5} - 2) \approx 0.636 \][/tex]
#### Step 5:
Jacob decided to adjust the bounds based on the midpoint evaluation. Since [tex]\( x = \frac{3}{2} \)[/tex] yielded a positive value ([tex]\( \approx 0.636 \)[/tex]), he should modify the bounds to [tex]\( [1, 1.5] \)[/tex], with the midpoint becoming the new upper bound.
### Conclusion
From the above steps, it becomes clear that all calculations performed in steps 1 through 4 were correct, and Jacob correctly identified that at [tex]\( x = 2 \)[/tex], the function equals approximately [tex]\( 1.889 \)[/tex]. Jacob also correctly evaluated the function at [tex]\( x = 1.5 \)[/tex].
However, if the problem states that Jacob made a mistake, it appears in adjusting the bounds based on the new information. Despite selecting the bounds correctly in step 2, since [tex]\( 1.889 \)[/tex] (at [tex]\( x = 2 \)[/tex]) is positive and [tex]\( -0.333 \)[/tex] (at [tex]\( x = 1 \)[/tex]) is negative, Jacob should continue ranging between these bounds more accurately.
Given that the values were accurate based on step 2, the correct bounding process in step 5 should proceed accurately without an issue. Jacob most probably didn't commit a mistake in these critical steps.
### Final Answer:
C. Jacob did not make any mistakes in the calculation process.
