Math Problem Statement
f(x)=2xe^(e^(3x)^2), a=0; taylor series
Solution
To find the Taylor series of around , we need to compute the derivatives of at and use the Taylor series formula:
Let's proceed by evaluating the function and its derivatives at .
Step 1: Evaluate
Since , the constant term in the Taylor series is zero.
Step 2: Compute the First Derivative,
Using the product rule for derivatives:
To differentiate with respect to , we use the chain rule:
Therefore,
Now, evaluate :
So, the coefficient of in the Taylor series is .
Step 3: Compute the Second Derivative,
This calculation involves differentiating and becomes quite complex. However, for practical purposes, you might only need the first few terms.
After computing higher derivatives, you
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Math Problem Analysis
Mathematical Concepts
Taylor Series
Derivatives
Exponential Functions
Product Rule
Chain Rule
Formulas
Taylor series formula: f(x) = f(0) + f'(0)x + f''(0)/2!x^2 + ...
Product Rule: (fg)' = f'g + fg'
Chain Rule: d/dx(e^g(x)) = e^g(x) * g'(x)
Theorems
Taylor Series Expansion
Suitable Grade Level
Undergraduate (Calculus)