He started from a graph where he found the solution to be between 1 and 2. Using the lower and upper bounds for the following work for the first iteration.
\begin{tabular}{|l|l|}
\hline Step 1 & Rewrite the equation so that it equals zero on one side.\
$\begin{aligned}
2^x-4 & =3^{-x}-2 \\
\left(2^x-4\right)-\left(3^{-x}-2\right) & =0
\end{aligned}$ \\
\hline Step 2 & Evaluate the rewritten equation at the lower and upper bounds. To find the solution that lies between 1 and 2 , set these values as the lower and upper bounds while finding the solution.\
$\left(2^{(1)}-4\right)-\left(3^{(-1)}-2\right) \approx-0.333$
$\left(2^{(2)}-4\right)-\left(3^{(-2)}-2\right) \approx 1.889$ \\
\hline Step 3 & Take the average of the lower and upper bounds.\
$\frac{1+2}{2}=\frac{3}{2}$ \\
\hline Step 4 & Evaluate the rewritten equation at $x=\frac{3}{2}$.\
$\left(2^{\left(\frac{3}{2}\right)}-4\right)-\left(3^{\left(-\frac{1}{2}\right)}-2\right) \approx 0.636$ \\
\hline \multicolumn{2}{|l|}{Where did Jacob make a mistake, and what was the error?} \\
\hline
\end{tabular}
A. Jacob made a mistake at step 2. The actual evaluation of the rewritten equation at $x=2 is 3$
B. Jacob made a mistake at step 5. He should have used $x=\frac{3}{2}$ as the new upper bound
C. Jacob did not make any mistakes in the calculation process
D. Jacob made a mistake at step 4. The actual evaluation of the rewritten equation is approximately -1.636
Answer (2)
Verify each step of Jacob's calculations.
Step 1: Correctly rewrites the equation.
Step 2: Correctly evaluates the equation at x=1 and x=2.
Step 4: Correctly evaluates the equation at x=3/2.
Conclude that Jacob did not make any mistakes: C .
Explanation
Problem Analysis We are given a problem where Jacob is trying to find the solution to the equation 2 x − 4 = 3 − x − 2 using an iterative method, starting with the interval [ 1 , 2 ] . We need to identify if there's a mistake in his calculations and, if so, pinpoint the error.
Verifying the Calculations Let's verify the calculations in each step.
Step 1: Rewrite the equation so that it equals zero on one side. 2 x − 4 = 3 − x − 2 becomes ( 2 x − 4 ) − ( 3 − x − 2 ) = 0 . This step is correct.
Step 2: Evaluate the rewritten equation at the lower and upper bounds. For x = 1 : ( 2 1 − 4 ) − ( 3 − 1 − 2 ) = ( 2 − 4 ) − ( 3 1 − 2 ) = − 2 − ( 3 1 − 3 6 ) = − 2 − ( − 3 5 ) = − 2 + 3 5 = − 3 6 + 3 5 = − 3 1 ≈ − 0.333 . This is correct. For x = 2 : ( 2 2 − 4 ) − ( 3 − 2 − 2 ) = ( 4 − 4 ) − ( 9 1 − 2 ) = 0 − ( 9 1 − 9 18 ) = 0 − ( − 9 17 ) = 9 17 ≈ 1.889 . This is also correct.
Step 3: Take the average of the lower and upper bounds. 2 1 + 2 = 2 3 = 1.5 . This is correct.
Step 4: Evaluate the rewritten equation at x = 2 3 .
( 2 2 3 − 4 ) − ( 3 − 2 3 − 2 ) = ( 2 2 − 4 ) − ( 3 3 1 − 2 ) ≈ ( 2 × 1.414 − 4 ) − ( 3 × 1.732 1 − 2 ) ≈ ( 2.828 − 4 ) − ( 5.196 1 − 2 ) ≈ − 1.172 − ( 0.192 − 2 ) ≈ − 1.172 − ( − 1.808 ) ≈ − 1.172 + 1.808 ≈ 0.636 . This is also correct.
Conclusion Comparing our calculations with the values provided in the table, we see that all the calculations are correct. Therefore, Jacob did not make any mistakes in the calculation process.
Final Answer Therefore, the correct answer is C. Jacob did not make any mistakes in the calculation process.
Examples
Consider a scenario where you're trying to find the optimal temperature for a chemical reaction. You know the reaction works between two temperature bounds. This problem demonstrates how you can iteratively narrow down the temperature range to find the ideal point by repeatedly evaluating the reaction's outcome at different temperatures and refining your search. This method is applicable in various scientific and engineering optimization problems.
Jacob followed the calculation steps correctly for solving the equation, with no errors identified during the evaluation of the steps. Every calculation matched the expected results for the equation components. Therefore, the conclusion is that Jacob did not make any mistakes: C .
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