Math Problem Statement
Use differentials to approximate the number sqrt(2.012^2 + 5.972^2 + 8.982^2) . (Round your answer to five decimal places.) dont round throughout the process
Solution
To approximate the value of using differentials, we'll follow these steps:
Step 1: Define the function
Let .
We want to evaluate this function at , since is close to these values.
Step 2: Find the partial derivatives
The differential approximation formula is:
For , the partial derivatives are:
Step 3: Evaluate the partial derivatives at
First, calculate :
Now, the partial derivatives evaluated at are:
Step 4: Calculate the differentials
We have:
Step 5: Calculate
Substituting into : Let's calculate this accurately.
The value of is approximately .
Step 6: Approximate
Using the differential approximation: Let's compute the final value.The approximated value of using differentials is approximately (rounded to five decimal places).
Do you want more details or have any questions?
Here are 5 related questions that might expand your understanding:
- How would the differential approximation change if you chose a different point?
- Why is it useful to approximate functions using differentials in calculus?
- Can we use this method to approximate other functions beyond square roots?
- How accurate is this differential method compared to using a calculator?
- What are partial derivatives and how do they differ from regular derivatives?
Tip: The differential approximation works best when the values you're approximating are very close to the chosen point.
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Math Problem Analysis
Mathematical Concepts
Differentials
Partial Derivatives
Square Roots
Formulas
f(x, y, z) = sqrt(x^2 + y^2 + z^2)
df = (∂f/∂x) dx + (∂f/∂y) dy + (∂f/∂z) dz
Theorems
Differential Approximation
Chain Rule
Suitable Grade Level
Grades 11-12, Early College