Math Problem Statement
Solution
The problem asks us to find the linear approximation of at and then use it to estimate .
Step 1: Linear Approximation Formula
The linear approximation of a function around a point is given by:
In this case, and .
Step 2: Calculate
Since , we have:
Step 3: Calculate
The derivative of is:
Thus, .
Step 4: Form the Linear Approximation
Using the formula , we substitute , , and :
Thus, the linear approximation is:
Step 5: Estimate
To estimate , we substitute into the linear approximation :
Thus, the estimated value of is approximately .
Final Answers:
- The linear approximation .
- The estimated value of .
Would you like further details or have any questions?
Here are 5 related questions for further exploration:
- What is the error between the actual value of and the linear approximation?
- How does linear approximation change if we expand around a different point, say ?
- Why does linear approximation work well for values of close to 1?
- How does the linear approximation compare with higher-order approximations (e.g., Taylor series)?
- What is the second derivative of and how does it impact the curvature near ?
Tip: Linear approximation is only accurate for values close to the point of approximation. For better accuracy, consider using more terms in the Taylor series.
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Math Problem Analysis
Mathematical Concepts
Calculus
Linear Approximation
Derivatives
Natural Logarithm
Formulas
L(x) = f(a) + f'(a)(x - a)
f(x) = ln(x)
f'(x) = 1/x
Theorems
Linear Approximation Formula
Suitable Grade Level
Grades 11-12 (High School Calculus)
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