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Chapter 10: Problem 27

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=2^{x}, a=1$$

Short Answer

Expert verified
Question: Find the Taylor series of the function \(f(x) = 2^x\) centered at \(a = 1\) and write the power series using summation notation. Answer: The first four nonzero terms of the Taylor series for the function \(f(x)=2^x\) centered at \(a=1\) are: $$f(x) \approx 2 + 2\ln(2)(x-1) + \frac{1}{2} 2\ln^2(2) (x-1)^2 + \frac{1}{3!} 2\ln^3(2) (x-1)^3$$ The power series representation of the function \(f(x)=2^x\) in summation notation centered at \(a=1\) is: $$f(x) \approx \sum_{n=0}^{\infty} \frac{2 \ln^n(2)(x-1)^n}{n!}$$

Step by step solution

01

Determine the derivatives of the function

We need to find the first four derivatives of the function \(f(x) = 2^x\). 1. First derivative: \(f'(x) = 2^x\ln(2)\) 2. Second derivative: \(f''(x) = 2^x\ln^2(2)\) 3. Third derivative: \(f^{(3)}(x) = 2^x\ln^3(2)\) 4. Fourth derivative: \(f^{(4)}(x) = 2^x\ln^4(2)\)
02

Evaluate derivatives at the center, "a"

Evaluate the derivatives at \(a=1\): 1. \(f(1) = 2^1 = 2\) 2. \(f'(1) = 2^1\ln(2) = 2\ln(2)\) 3. \(f''(1) = 2^1\ln^2(2) = 2\ln^2(2)\) 4. \(f^{(3)}(1) = 2^1\ln^3(2) = 2\ln^3(2)\)
03

Construct the Taylor series

Now, we can construct the Taylor series using the values of the derivatives at \(a=1\): $$f(x) \approx 2 + 2\ln(2)(x-1) + \frac{1}{2} 2\ln^2(2) (x-1)^2 + \frac{1}{3!} 2\ln^3(2) (x-1)^3$$ This is the first four nonzero terms of the Taylor series for the function \(f(x)=2^x\) centered at \(a=1\).
04

Express the Taylor series in summation notation

Finally, we can write the Taylor series in summation notation: $$f(x) \approx \sum_{n=0}^{\infty} \frac{2 \ln^n(2)(x-1)^n}{n!}$$ This is the power series representation of the function \(f(x)=2^x\) in summation notation centered at \(a=1\).

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Most popular questions from this chapter

Prove that if \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges on the interval \(I,\) then the power series for \(x^{m} f(x)\) also converges on \(I\) for positive integers \(m\)

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{-0.35}^{0.35} \cos 2 x^{2} d x$$

Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan x \approx x$$

Teams \(A\) and \(B\) go into sudden death overtime after playing to a tie. The teams alternate possession of the ball and the first team to score wins. Each team has a \(\frac{1}{6}\) chance of scoring when it has the ball, with Team \(\mathrm{A}\) having the ball first. a. The probability that Team A ultimately wins is \(\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}\) Evaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is \(\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1} .\) Evaluate this series.

Assume that \(f\) has at least two continuous derivatives on an interval containing \(a\) with \(f^{\prime}(a)=0 .\) Use Taylor's Theorem to prove the following version of the Second Derivative Test: a. If \(f^{\prime \prime}(x) > 0\) on some interval containing \(a,\) then \(f\) has a local minimum at \(a\) b. If \(f^{\prime \prime}(x) < 0\) on some interval containing \(a,\) then \(f\) has a local maximum at \(a\)
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