Part 6: Application to Instantaneous velocity and Instantaneous Acceleration If

Question

Part 6: Application to Instantaneous velocity and Instantaneous Acceleration If you have a function of position x with time t. that is a function x(t), you can calculate the instantaneous velocity and the instantaneous acceleration. Exercise 8: The instantaneous velocity is the derivative of the position function. Suppose our position function is x(t) = 8t^2+ 4. Calculate the instantaneous velocity. Instead of [f(x + delta x) - f(x)]/delta x and then taking delta x right arrow 0 use [x(t + delta t) - x(t)] delta xt and then take delta x right arrow 0. Your result is the instantaneous velocity function v(t). The calculus notation for what you just did is v(t) = dx/dt = the derivative of x with respect to t = the rate of Change of position x with lime t. What is the instantaneous velocity at time t = 10?

Explanation / Answer

GIVEN,

v(t) = dx/dt

x(t) = 8t2 +4

now,( f(x+delta x) - f(x) )/ delta x

(x(t+delta t) - x(t) ) / delta t

applying limit at delta t-->0, we get

(8 (t+delta t)2 + 4 - 8t2 -4 ) / delta t

(8 (2t delta t + (delta t)2 ) ) delta t

v= (16 t delta t )/delta t + (delta t)2/delta t

v= 16t

Now at t=10,

v(10)=16*10 =160 m/sec

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