Math Problem Statement
Solution
To approximate the value of using differentials, we follow these steps:
Step 1: Define the function
Let , so we want to approximate .
Step 2: Choose a point close to 63
We'll choose because , which is an easy value to compute and close to 63.
Step 3: Find the derivative
We need the derivative of . Using the power rule: At , this becomes:
Step 4: Use the linear approximation formula
The differential approximation formula is: Substituting , , and , we get:
Conclusion:
Using differentials, the approximate value of is .
Would you like further details or have any questions?
Related Questions:
- What is the purpose of differentials in calculus approximations?
- How can we apply differentials to approximate values for higher powers, such as ?
- Why is a better choice than other numbers close to 63 for this problem?
- What happens if we try to approximate using differentials?
- How does the error in a differential approximation compare to other approximation methods?
Tip:
When using differentials for approximation, always choose a point near the target value where the function is easy to evaluate exactly.
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Math Problem Analysis
Mathematical Concepts
Differentials
Approximations
Derivatives
Formulas
f(x) = x^{1/3}
f'(x) = (1/3) * x^{-2/3}
f(x) ≈ f(x_0) + f'(x_0) * (x - x_0)
Theorems
Linear Approximation Theorem
Suitable Grade Level
Grades 11-12 (or early university calculus)
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