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December 21, 2014

December 21, 2014

Posted by **jane** on Thursday, April 2, 2009 at 9:52pm.

A particle moves along the x axis so that its velocity at any time t greater than or equal to 0 is given by v(t) = 1 - sin(2pi*t).

a. Acceleration a(t) of the particle at any time t?

i said that acceleration was the derivative of velocity, so i took the derivative of the velocity function.

v(t) = 1 - sin(2pi*t)

v'(t) = -cos(2pi*t) x 2pi

a(t) = -2pi*cos(2pi*t)

b. Values of t between 0 and 2 where the particle is at rest?

i said it was at rest when velocity = 0.

0 = 1 - sin(2pi*t)

1 = sin(2pi*t)

sin(pi/2) = 1, so...

2pi*t = pi/2

4pi*t = pi

4t = 1

t = 1/4

c. Position x(t) of the particle at any time t if x(0) = 0?

i didn't understand the part about x(0) = 0. i kind of ignored it and it didn't work out. i figured that the position function was the anti-derivative of the velocity function and did:

anti-derivative of 1 - sin(2pi*t)

1/2pi x anti-derivative of 1 - sin(2pi*t)

1/2pi (t + cos(2pi*t)) = x(t)

but that doesn't equal 0 when t = 0. help?

- calc -
**Reiny**, Thursday, April 2, 2009 at 10:15pmin b) you were right to find

t = 1/4

but remember the period of 1 - sin(2pi*t)

is 2pi/2pi = 1

so another solution would be 1/4 + 1 or

t = 5/4

so the times between 0 and 2 when the object is at rest is

t = 1/4 and t = 5/4

for c) if

v(t) = 1 - sin(2pi*t)

then x(t) = t + (1/2pi)cos(2pi*t) + c

but you were given x(0) = 0

0 = 0 + (1/2pi)cos(0) + c

0 = )1/2pi) + c

c = -1/(2pi)

so x(t) = t + (1/2pi)cos(2pi*t) - 1/(2pi)

- haha how do i respond to responses? -
**jane**, Thursday, April 2, 2009 at 10:22pmomg i can't believe i just forgot c in part c. it's even called part c!

and thank you for the help with part b. i completely forgot about that!

- calc -
**Anonymous**, Friday, April 3, 2009 at 10:32amLet R be the region bounded by the y-axis and the graph of y=xcubed divided by 1+xsquared and y=4-2x,as shown inthe figure above.

find the area of R

find the volume of the solid generated when R is revolved about the x-axis

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