a. The different meanings between 认为【rènwéi】, 以为【yǐwéi】, and 觉得【juéde】
Example 1:
I thought that she loved me.
我【wǒ=I】 以为【yǐwéi=think】她【tā=she】 喜欢【xǐhuān=like, love】 我【wǒ=I】。
我 以为 她 喜欢 我。
The speaker thought that she loved him but he was wrong.
Compared that with example 2:
I believe that she loves me.
我【wǒ=I】 认为【rènwéi=think, believe】 她【tā=she】 喜欢【xǐhuān=like, love】 我【wǒ=I】。
我 认为 她 喜欢 我。
The speaker says with confidence that she loves him.
Compared that with example 3:
I feel that she loves me.
我【wǒ=I】 觉得【juéde=feel】 她【tā=she】 喜欢【xǐhuān=like, love】 我【wǒ=I】。
我 觉得 她 喜欢 我。
The speaker is not sure, just feels that she loves him.
b. The different meanings between 认为【rènwéi】and 觉得【juéde】
Example 1:
He thinks rock music is too noisy.
他【tā=he】 认为【rènwéi=think】 摇滚乐【yáogǔnlè=rock music】 太【tài=very, over】 吵【chǎo=noisy】 了【le】。
他 认为 摇滚乐 太 吵 了。
He says with confidence that live music is too noisy.
Compared that with example 2:
He thinks rock music is too noisy.
他【tā=he】 觉得【juéde= think, feel】 摇滚乐【yáogǔnlè=rock music】 太【tài=very, over】 吵【chǎo=noisy】 了【le】。
He just expresses his perception that rock music is too noisy.
他 觉得 摇滚乐 太 吵 了。