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Chapter 11: Problem 3

Determine the third Taylor polynomial of the given function at \(x=0.\) $$f(x)=5 e^{2 x}$$

Short Answer

Expert verified
The third Taylor polynomial is \( T_3(x) = 5 + 10x + 10x^2 + \frac{20}{3}x^3 \).

Step by step solution

01

Understand the Taylor Series

The Taylor series of a function around a point is given by: \[ T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n \] where \(a\) is the point of expansion, in this case \(a=0\).
02

Compute the Derivatives

We need to find the first three derivatives of the function \(f(x) = 5 e^{2x} \). The derivatives are: \( f(x) = 5 e^{2x} \) \( f'(x) = 10 e^{2x} \) \( f''(x) = 20 e^{2x} \) \( f'''(x) = 40 e^{2x} \)
03

Evaluate the Derivatives at \(x = 0\)

Now, evaluate each derivative at \(x=0 \): \( f(0) = 5 e^{2(0)} = 5 \) \( f'(0) = 10 e^{2(0)} = 10 \) \( f''(0) = 20 e^{2(0)} = 20 \) \( f'''(0) = 40 e^{2(0)} = 40 \)
04

Write the Taylor Polynomial

Substitute the evaluated derivatives into the Taylor series formula up to the third degree: \[ T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 \] So, we get: \[ T_3(x) = 5 + 10x + \frac{20}{2}x^2 + \frac{40}{6}x^3 \] \[ T_3(x) = 5 + 10x + 10x^2 + \frac{20}{3}x^3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor Series
The Taylor series is a powerful tool in mathematics used to approximate more complex functions with polynomials.
It expands a function into an infinite sum of terms calculated from its derivatives at a single point.
The general form of the Taylor series around the point \(a\) is:
T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \frac{f^{(n)}(a)}{n!}(x-a)^n + \text{...}

Here, \(f(a)\) is the function value at \(a\), \(f'(a)\) is the first derivative at \(a\), and so forth.
By using these tailored polynomial approximations, we make complex functions easier to handle.
Derivatives
Derivatives are fundamental in finding our Taylor series.
They provide a measure of how a function changes as its input changes. Knowing the derivatives of a function at a point is essential to form the Taylor polynomial.
For our problem, we start with the given function \(f(x) = 5e^{2x}\) and proceed by calculating its derivatives:
  • First derivative: \(f'(x) = 10e^{2x}\)


  • Second derivative: \(f''(x) = 20e^{2x}\)


  • Third derivative: \(f'''(x) = 40e^{2x}\)

These steps outline the systematic way to compute necessary derivatives before evaluating them at the desired point, in this case, \(x=0\).
Function Evaluation
After finding the necessary derivatives, the next step involves evaluating them at the point of interest, which is \(x=0\) in this problem.
These values are crucial for building the Taylor polynomial:
  • Function at \(x=0\): \(f(0) = 5e^{2(0)} = 5\)


  • First derivative at \(x=0\): \(f'(0) = 10e^{2(0)} = 10\)


  • Second derivative at \(x=0\): \(f''(0) = 20e^{2(0)} = 20\)


  • Third derivative at \(x=0\): \(f'''(0) = 40e^{2(0)} = 40\)

With these values, we can now substitute them back into the Taylor series formula. This gives us the third-degree Taylor polynomial:
\(T_3(x) = 5 + 10x + 10x^2 + \frac{20}{3}x^3\),
providing a simplified polynomial function that approximates \(f(x)\) around \(x=0\).

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Most popular questions from this chapter

Use the Taylor series expansion for \(\frac{x}{(1-x)^{2}}\) to find the func- tion whose Taylor series is \(1+4 x+9 x^{2}+16 x^{3}+25 x^{4}+\cdots\)

Determine the sums of the following geometric series when they are convergent. $$1+\frac{1}{6}+\frac{1}{6^{2}}+\frac{1}{6^{3}}+\frac{1}{6^{4}} \cdots$$

Determine the sums of the following infinite series: $$\sum_{k=1}^{\infty}\left(\frac{1}{3}\right)^{2 k}$$

Find the Taylor series at \(x=0\) of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at \(x=0\) of \(\frac{1}{1-x}, e^{x},\) or \(\cos x .\) These series are derived in Examples 1 and 2 and Check Your Understanding Problem 2. $$\cos 3 x$$

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) $$\sum_{k=1}^{\infty} \frac{1}{e^{2 k+1}}$$
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