Question

Find r(t) if r'(t) = t⁴ i + eᵗ j + 4te⁴t k and r(0) = i + j + k?

Answer 1

To find the position vector r(t), we integrate the given velocity function r'(t) component-wise and apply the initial condition r(0) = i + j + k to determine the constants of integration.

Step-by-step explanation:

To find the position vector r(t), we must integrate the given velocity vector function r'(t) = t⁴ i + eᵗ j + 4te⁴t k. Given that r(0) = i + j + k, this information serves as our initial condition for the integration.

Integrating each component of the velocity vector separately, we get the following:

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Finally, don't forget to add the constants of integration, which can be determined using the initial condition r(0) = i + j + k. This will serve to adjust each component so that the initial condition is satisfied.

Once the integration is complete and the constants have been calculated, we will have the specific function r(t) representing the position of the particle at time t.