Assignment 9

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course Mth 277

7/10 2 pm

Question: Find both F' and F'' for F(t) = (4sin^2 t)i + (9cos^2 t)j + tkYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

F' = 16sin(t)*cos(t) i - 81sin(t)*cos(t) j + k

F'' = 16cos(2t) i - 81cos(2t) j

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Question: Given the position vector of a particle R(t) = (cos t)i + tj + (4 sin t)k, find the particle's velocity and acceleration vectors and then find the speed and direction of the particle at t = pi/2.

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Your solution:

Velocity = (-sin t)i + j + (4 cos t)k

Acceleration = (-cos t)i - (4 sin t)k

Velocity(pi/2) = -1 + 0 + 4*0 = -1 units/second

Acceleration(pi/2) = 0 - 4*1 = -4 units/second^2

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Velocity is a vector with i, j and k components. At least two of the components of this velocity vector will be nonzero.

Acceleration is also a vector.

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Question: Find Int( dt) (Where Int( f(t) dt) is the integral of f with respect to t)

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Your solution:

<-cos t, sin t, t^3/3>

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Question: Find Integral((e^t)* dt)

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Your solution:

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Question: Find the velocity and position vectors given the acceleration vector `A(t) = 4(t^2)i - 2 sqrt(t) j + 5(e^3t)k, initial position R(0) = 2i + j -3k and initial velocity v(0) = 4i + j + 2k.

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Your solution:

Velocity = 4/3 t^3 + c1 i - 4/3 t^(3/2) + c2 j + 5/2 e^3*t^2 + c3 k

Use the initial velocity and substitute 0 for t:

4 i + j + 2k = (0 + c1)i - (4/3*0 + c2) j + (5/2*e^3*t^2 + c3)k

4 i + j + 2k = c1 i - c2 j + c3 k

c1 = 4, c2 = -1, c3 = 2

V = (4/3 t^3 + 4) i - (4/3 t^(3/2) - 1) j + (5/2 e^3*t^2 + 2) k

Position: 1/3 t^4 + 4t + c1 i - 8/15 t^(5/2) - t + c2 j + 5/6*e^3*t^3 + 2t + c3

Use the initial position, substitute 0 for t:

(1/3*0 + 0 + c1)i + (-8/15*0 - 0 + c2)j + (5/6*e^3*0 + 0 + c3)k

Set it equal to the initial:

c1 i - c2 j + c3 k = 4i + j + 2k

c1 = 4, c2 = -1, c3 = 2

S = (1/3 t^4 + 4t + 4) i + (-8/15 t^(5/2) - t - 1) j + (5/6*e^3*t^3 + 2t + 2)k

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Given Solution:

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Question: F(t) = e^(-kt)i + e^(kt)k. Show that F and F'' are parallel.

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Your solution:

F' = -ke^(-kt)i + ke^(kt)k

F'' = k^2*e^(-kt)i + k^2*e^(kt)k

Angle between two parallel vectors is 0.

theta = acos (F dot F'')/(mag(a)*mag(b))

theta = acos (k^2*e^(-2kt) + k^2*e^(2kt))/(k^2*e^(-2kt) + k^2*e^(2kt))

theta = acos(1) = 0

Therefore, F and F'' are parallel.

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Good.

Alternatively, F '' = k^2 * F. Once is a constant scalar multiple of the other so they are parallel.

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Given Solution:

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Question: "

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Very good, except for the second question.

You should revise that and submit a copy of the question with your solution, just to be sure you see what's going on there.

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Incidentally, we sent copies of tests and your proctor's required cover sheets this afternoon. Check to be sure they have been received.

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