Question

To find the antiderivative of the vector-valued function r′(t)=⟨e−t,e−5t,t62⟩ that satisfies the initial condition r(1)=⟨e−1,−e,2⟩, you will need to integrate each component separately.

Let's find the antiderivative for each component of μ f{r}'(t):
For the x-component: ∫e−tdt=−e−t+C1.

Answer 1

To find the antiderivatives of a vector-valued function, integrate each component separately and use the given initial conditions to determine the constants of integration for the x-, y-, and z-components.

Step-by-step explanation:

When finding the antiderivative of a vector-valued function such as r'(t)=\langle e^{-t},e^{-5t},t^{6/2}\rangle, you indeed integrate each component of the function separately. The general antiderivative for the x-component, which is e^{-t}, will be -e^{-t} + C_1. This represents the integration of an exponential function. For the initial condition r(1)=\langle e^{-1},-e,2\rangle, you would substitute t=1 into the antiderivative to solve for the constants C_1, C_2, and C_3 associated with each component.

To find the antiderivatives for the y- and z-components, you would similarly integrate e^{-5t} and t^{6/2} (which simplifies to t^3), respectively. The resulting antiderivatives will also include constants of integration, which can be determined using the provided initial conditions.