# The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation ofmotion s = sin 2*pi*t + 3*cos pi*t , where t is measured in seconds.(b)...

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of

motion s = sin 2*pi*t + 3*cos pi*t , where t is measured in seconds.

(b) Estimate the instantaneous velocity of the particle when t = 1.

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The displacement of a particle in centimeters that is moving back and forth along a straight line is given by the equation of motion `s(t) = sin 2*pi*t + 3*cos pi*t` where t is measured in seconds.

In any interval [a, b] the average velocity of the particle is equal to the displacement in the given interval divided by the time.

This is `(s(b) - s(a))/(b - a)`

The instantaneous velocity of the particle at t = a, is the limit `lim_(h = 0)(s(a+h)-s(a))/h` . This is equivalent to the value of the derivative of s(t) at t = a.

Using `s(t) = sin 2*pi*t + 3*cos pi*t` , the instantaneous velocity at t = 1 is

`(2*pi*cos 2*pi*t - 3*pi*sin pi*t)_(t = 1)`

= `2*pi*cos 2*pi - 3*pi*sin pi`

= `2*pi`

The required instantaneous velocity is `2*pi` cm/s