Math Problem Statement
Solution
Let's walk through the problem step by step to identify the mistakes and correct them. The task is to find the Taylor polynomial for about .
Step 1: Calculating
The function is . Plugging in :
So, , which is correctly marked.
Step 2: Calculating
Using the chain rule, we differentiate:
Now, plug in :
This is correct, and marked correctly.
Step 3: Calculating
Differentiate again:
Now, plug in :
This is correct, and marked correctly.
Step 4: Calculating
Differentiate again:
Now, plug in :
This is also correct, and marked correctly.
Step 5: Constructing the Taylor Polynomial
The formula for the Taylor polynomial of degree 3 is:
Now, substitute the values:
Errors in the Expression
The last two terms in the Taylor polynomial are incorrect:
- The coefficient for the term should be , but it's given as .
- The term coefficient is correct.
Correct Taylor Polynomial:
Let me know if you need more details on any specific part!
Here are 5 questions that expand on this:
- How do you compute higher-order derivatives of ?
- Why do we divide by when constructing Taylor polynomials?
- How does the radius of convergence affect the accuracy of a Taylor polynomial?
- How would the polynomial change if we expanded around a different point, say ?
- Can you verify that the approximation for using this Taylor polynomial is accurate?
Tip: When expanding Taylor series, always check the factorial in the denominator, as it affects higher-order terms significantly.
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Math Problem Analysis
Mathematical Concepts
Calculus
Taylor Series
Polynomial Approximation
Derivatives
Formulas
Taylor Polynomial of degree 3: T3(x) = f(a) + f'(a)(x - a) + f''(a)/2!(x - a)^2 + f'''(a)/3!(x - a)^3
f(x) = sqrt(x + 3)
Theorems
Taylor's Theorem
Suitable Grade Level
College Calculus
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