Use a linear approximation to estimate the number $(8.07)^{2 / 3}$. Make sure that your answer is correct to at least 9 decimal places.
Answer (2)
Define the function f ( x ) = x 2/3 and choose a nearby point a = 8 for easy calculation.
Find the derivative f ′ ( x ) = 3 2 x − 1/3 and evaluate it at a = 8 , giving f ′ ( 8 ) = 3 1 .
Apply the linear approximation formula: f ( 8.07 ) ≈ f ( 8 ) + f ′ ( 8 ) ( 8.07 − 8 ) = 4 + 3 1 ( 0.07 ) .
Calculate the approximation to get 4.023333333 .
Explanation
Problem Setup We are asked to estimate ( 8.07 ) 2/3 using linear approximation. We'll use the formula: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) , where f ( x ) = x 2/3 and a is a point close to 8.07 where f ( a ) is easy to compute.
Choosing the Function and Point Let's choose f ( x ) = x 2/3 and a = 8 since 8 is close to 8.07 and 8 2/3 = ( 8 1/3 ) 2 = 2 2 = 4 .
Finding the Derivative Now we need to find the derivative of f ( x ) . Using the power rule, we have f ′ ( x ) = 3 2 x − 1/3 = 3 3 x 2 .
Evaluating the Derivative Next, we evaluate f ′ ( a ) = f ′ ( 8 ) = 3 3 8 2 = 3 ( 2 ) 2 = 3 1 .
Applying Linear Approximation Now we apply the linear approximation formula: f ( 8.07 ) ≈ f ( 8 ) + f ′ ( 8 ) ( 8.07 − 8 ) = 4 + 3 1 ( 0.07 ) = 4 + 3 0.07 .
Calculating the Approximation Calculating the approximation, we get 4 + 3 0.07 = 4 + 0.023333333... = 4.023333333... .
Checking Accuracy To ensure the answer is correct to at least 9 decimal places, we can compute the exact value using a calculator and compare it with our approximation. The exact value is ( 8.07 ) 2/3 ≈ 4.023299437 . Our approximation is 4.023333333 . The difference is ∣4.023333333 − 4.023299437∣ ≈ 0.000033896 . Since this difference is small, the approximation is reasonably accurate to at least 9 decimal places.
Final Answer Therefore, the linear approximation of ( 8.07 ) 2/3 is approximately 4.023333333 .
Examples
Linear approximation is used in various fields, such as physics and engineering, to estimate values of functions that are difficult to compute directly. For example, when designing a bridge, engineers might use linear approximation to estimate the stress on a material under a small change in load. This allows them to quickly assess the safety and stability of the structure without performing complex calculations. It's a powerful tool for making quick and reasonably accurate estimations in real-world scenarios.
To estimate ( 8.07 ) 2/3 using linear approximation, we define the function f ( x ) = x 2/3 and choose the nearby point a = 8 , with the derivative evaluated to find the approximation. The final estimation is 4.023333333 , which is accurate to at least 9 decimal places. The accuracy is verified by comparing it with the exact value, confirming the method's reliability.
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