(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.

Water is running out of a conical funnel at the rate of 1 cubic inch per second. If the radius of the top of the funnel is 4 inches and the height is 8 inches, find the rate at which the water level is dropping when it is 2 inches from the top.

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1.) A fugitive is running along a wall at 4.0m/s. A searchlight 20m from the wall is trained on him. How fast is the searchlight rotating at the instant when he is 10m from the point on the wall nearest the searching?
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(Chart is in attachment)
a) Use the date from the table to find an approximation for W'(12). Show the computations that lead to you

1. Functions f, g, and h are continuous and differentiable for all real numbers, and
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x f (x) g(x) h(x) f'(x) g'(x) h'(x)
0 1 -1 -1 4 1 -3
1 0 3 0 2 3 6
2 3 2

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a. Find the equation of the tangent line to the particle's trajectory at the point r(1).
b. The particle flies off on tangent at t0 = 2 and moves along the tangent line to its trajectory with the same velocity that it had at time 2. (Note: