91Ó°ÊÓ

The position of a particle moving on a line is given by \(s(t)-2 t^{3}-21 t^{2}+60 t, t \geq 0\), where \(t\) is measured in seconds and \(s\) in feet. (a) What is the velocity after 3 seconds and after 6 seconds? (b) When is the particle moving in the positive direction? (c) Find the total distance traveled by the particle during the first 7 seconds.

Find the first and second derivatives. \(y=x+1\)

A toy rocket fired straight up into the air has height \(s(t)=160 t-16 t^{2}\) feet after \(t\) seconds. (a) What is the rocket's initial velocity (when \(t=0\) )? (b) What is the velocity after 2 seconds? (c) What is the acceleration when \(t=3\) ? (d) At what time will the rocket hit the ground? (e) At what velocity will the rocket be traveling just as it smashes into the ground?

Find the first and second derivatives. \(y=100\)

A helicopter is rising straight up in the air. Its distance from the ground \(t\) seconds after takeoff is \(s(t)\) feet, where \(s(t)=t^{2}+t\). (a) How long will it take for the helicopter to rise 20 feet? (b) Find the velocity and the acceleration of the helicopter when it is 20 feet above the ground.

Write the equation of the tangent line to the graph of \(y=x^{2}\) at the point where \(x=2.5\).

Find the first and second derivatives. \(f(r)=\pi r^{2}\)

Find the first and second derivatives. \(y=\pi^{2}+3 x^{2}\)

Find the derivative of \(f(x)\) at the designated value of \(x\). \(f(x)=\frac{1}{x}\) at \(x=\frac{2}{3}\)

Find the point on the graph of \(y=x^{2}\) where the tangent line is parallel to the line \(3 x-2 y=2\).