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Derivatives and Rate of Change

(1.) A particle moves along the x-axis so that at any time that t is greater than or equal to zero, its position is given by x(t)= t^3-12t+5.
a.) Find the velocity of the particle at any time t.
b.) Find the acceleration of the particle at any time t.
c.) Find all values of t for which the particle is at rest.
d.) Find the speed of the particle when its acceleration is zero.
e.) Is the particle moving toward the origin or away from the origin when t=3? Justify answer.

(2.) The volume V of a cone, V=(1/3)(pi)r^2h, is increasing at the rate of 4pi cubic inches per second. At the instant when the radius of the cone is 2 inches, its volume is 8pi cubic inches and the radius is increasing at (1/3) inches per second.
a.) At the instant when the radius of the cone is 2 inches, what is the rate of change of the area of its base?
b.) At the instant when the radius of the cone is 2 inches, what is the rate of change of its height h?
c.) At the instant when the radius of the cone is 2 inches, what is the instantaneous rate of change of the area of its base with respect to its height?

Solution Preview

(1)
a)
v(t) = d x(t)/dt = d/dt(t^3-12t+5)=3t^2-12
b)
a(t)=d v(t)/dt = d/dt(3t^2-12)=6t
c)
v(t)=0 =>3t^2-12=0=>t^2=4=>t=2 (t must ...

Solution Summary

Rate of change problems are solved using derivatives.

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