Question

1. (5 points) if $g(x) = \sin(x)$, find $g^{(33)}(x)$.

Explanation:

Step1: Find derivative cycle of sin(x)

First derivative: $g'(x)=\cos(x)$; Second: $g''(x)=-\sin(x)$; Third: $g'''(x)=-\cos(x)$; Fourth: $g^{(4)}(x)=\sin(x)$. Cycle length = 4.

Step2: Calculate remainder of 33 ÷ 4

$33 = 4×8 + 1$, remainder = 1.

Step3: Match remainder to cycle

Remainder 1 corresponds to first derivative: $\cos(x)$.

Answer:

$\cos(x)$