# How do you find the velocity and position vectors if you are given that the acceleration vector is #a(t)= (-4 cos (-2t))i + (-4sin (-2t))j + (-2t)k# and the initial velocity is #v(0)=i+k# and the initial position vector is #r(0)= i+j+k#?

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To find the velocity and position vectors, integrate the acceleration vector with respect to time to obtain the velocity vector, and then integrate the velocity vector to obtain the position vector.

The velocity vector v(t) is given by: v(t) = ∫a(t) dt

Integrating each component of the acceleration vector separately, we get: v(t) = ∫(-4 cos(-2t)) dt * i + ∫(-4 sin(-2t)) dt * j + ∫(-2t) dt * k

Integrating each component: v(t) = (2 sin(-2t)) i + (-2 cos(-2t)) j + (-t^2) k + C

Where C is the constant of integration.

Given that v(0) = i + k, we can solve for C: v(0) = (2 sin(0)) i + (-2 cos(0)) j + (0) k + C v(0) = 2i - 2j + C

Comparing coefficients, we find C = 1 + 2j.

Thus, the velocity vector is: v(t) = (2 sin(-2t)) i + (-2 cos(-2t)) j + (-t^2) k + (1 + 2j)

To find the position vector r(t), integrate the velocity vector: r(t) = ∫v(t) dt

Integrating each component of the velocity vector separately: r(t) = ∫(2 sin(-2t)) dt * i + ∫(-2 cos(-2t)) dt * j + ∫(-t^2) dt * k + ∫(1 + 2j) dt

Integrating each component: r(t) = (-cos(-2t)) i + (-sin(-2t)) j + (-1/3 * t^3) k + (t + 2tj) + C'

Where C' is the constant of integration.

Given that r(0) = i + j + k, we can solve for C': r(0) = (-cos(0)) i + (-sin(0)) j + (-1/3 * 0^3) k + (0 + 2(0)) + C' r(0) = -i + j + C'

Comparing coefficients, we find C' = 1.

Thus, the position vector is: r(t) = (-cos(-2t)) i + (-sin(-2t)) j + (-1/3 * t^3) k + (t + 2tj + 1)

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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