GATE ECE 2008 | Question: 28

Question

In the Taylor series expansion of $\exp (x)+\sin (x)$ about the point $x=\pi$, the coefficient of $(x-\pi)^{2}$ is

• A. $\exp (\pi)$
• B. $0.5 \exp (\pi)$
• C. $\exp (\pi)+1$
• D. $\exp (\pi)-1$

Answer 1

B

Answer 2

The question asks for the coefficient of

(x - π)²

in the Taylor series expansion of

exp(x) + sin(x)

about the point x = π.

To find this coefficient, we need to calculate the second derivative of

exp(x) + sin(x), evaluate it at

x = π, and then divide by 2!.

The second derivative of

exp(x) + sin(x) is:

exp(x) - sin(x), and its value at

x = π

is:

exp(π) - sin(π) = exp(π). 

Therefore, the coefficient of

(x - π)²

in the Taylor series expansion of

exp(x) + sin(x)

about the point

x = π

is: exp(π)/2! = e^π/2.

Therefore, the answer is:

0.5e^π