# Complex Numbers Questions and Answers – Expansion of Trigonometric Functions

«
»

This set of Complex Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Expansion of Trigonometric Functions”.

1. The Taylor series for f(x)=7x2-6x+1 at x=2 is given by a+b(x-2)+c(x-2)2. Find the value of a+b+c.
a) -1
b) 0
c) 17
d) 46

Explanation: We know
$$f(x)=7x^2-6x+1$$
$$f'(x)=14x-6$$
$$f”(x)=14$$
$$f”'(x)=0$$
Thus for n>=3, the derivative of the function is 0.
As per the Taylor Series,
$$7x^2-6x+1=\sum_{n=0}^{\infty} \frac{f^n (2)(x-2)^n}{n!}$$
$$7x^2-6x+1=f(2)+f'(2)(x-2)+\frac{1}{2} f”(2) (x-2)^2+0$$
$$7x^2-6x+1=17+22(x-2)+7(x-2)^2$$
Thus, a=17, b=22, c=7
a+b+c=46

2. Find the Taylor Series for the function $$f(x)=e^{-6x}$$ about x=-4.
a) $$\sum_{n=0}^{\infty} \frac{(-6)^n}{n!} e^{12} (x+4)^n$$
b) $$\sum_{n=0}^{\infty} \frac{(-6)^n}{n!} e^{24} (x-4)^n$$
c) $$\sum_{n=0}^{\infty} \frac{(-6)^n}{n!} e^{24} (x+4)^n$$
d) $$\sum_{n=0}^{\infty} \frac{(-4)^n}{n!} e^{24} (x+4)^n$$

Explanation: We start by finding the derivative of the given function,
$$f(x)=e^{-6x}$$
$$f'(x) = -6e^{-6x}$$
$$f”(x) = 36e^{-6x}$$
$$f”'(x) = -216e^{-6x}$$
$$f””(x) = 1296e^{-6x}$$
Thus we take derivative of maximum to the fourth order.
Thus according to formula of Taylor series about x=-4
$$e^{-6x}=\sum_{n=0}^{\infty} \frac{f^n (-4)}{n!} (x+4)^n$$
$$e^{-6x}=\sum_{n=0}^{\infty} \frac{(-6)^n}{n!} e^{24} (x+4)^n$$
Thus the Taylor Series is given by
$$e^{-6x}=\sum_{n=0}^{\infty} \frac{(-6)^n}{n!} e^{24} (x+4)^n$$.

Sanfoundry Global Education & Learning Series – Complex Analysis.

To practice all areas of Complex Analysis, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs! 