He is desperate to pass the exam.
The employees are eagerly working on the development of the new product.
He is desperately trying to improve his grades.
The employees worked frantically to increase sales.
熱心 means keen and enthusiastic in a calm, positive sense, while 躍起 implies a more agitated, single-minded intensity that can look desperate.
冷静 means calm and composed, the opposite of the frantic, worked-up state described by 躍起.
Standard kanji spelling; the only form given.
The kanji 躍 (leap, dance) and 起 (rise, get up) combine to suggest a state of being roused to action, figuratively 'leaping up' in eagerness or desperation. The exact historical derivation is uncertain.