The given compound is \( CH_2 = CHCH_2NHCH_3 \). To assign the correct IUPAC name, we need to:
1. Identify the longest carbon chain that includes the double bond, which is 5 carbon atoms long. This gives the base name "pentene."
2. Number the chain from the end nearest the double bond, which is at position 1, so the compound is pent-1-ene.
3. Identify the substituents. The amino group (-NH2) is located at position 4 on the chain.
4. The methyl group (-CH3) is attached to the nitrogen atom, which is implied in the name "N-methyl" (indicating the methyl group is attached to the nitrogen atom).
Thus, the correct IUPAC name is 4-aminopent-1-ene.
Therefore, the correct answer is option (C) 4-aminopent-1-ene.