ends, is a function of length x given by T . Prove that the rate of change of temperature at the midpoint of the rod is zero. We are given that, T . Hence, the rate of change at any distance from one end is given by dT .
The mid point of the rod is at x = . Substituting x = , we get dT dx = . Example . A person learnt words for an English test.
The number of words the person remembers in t days after learning is given by W t ) , × ≤≤ . What is the rate at which the person forgets the words days after learning? We have, d dt W t ( ) = − × ) t . Therefore at t d dt W t = , ( ) = − .
That is, the person forgets at the rate of words after days of studying. - - Applications of Differential Calculus Example . The distance travelled by a moving particle in time t is given by s t ( ) = . Find the time when the velocity and acceleration are zero.
Distance moved in time ' ' t is s = t + . Velocity at time ' ' t is v t ( ) = ds dt . Acceleration at time ' ' t is a t ( ) = dV dt . Therefore, the velocity is zero when t , that is t = .
The acceleration is zero when t − . That is at time t = . Example . A particle is fired straight up from the ground to reach a height of s feet in t seconds,where s t ( ) = .
(i) Compute the maximum height of the particle reached. (ii) What is the velocity when the particle hits the ground? (i) At the maximum height,