Question

find the function r that satisfies the given condition. r'(t) = (eᵗ, sin t, sec² t): r(0) = (2, 2, 2) r(t) = ()

Answer 1

To find the function r that satisfies the given condition, we need to integrate the given derivative r'(t) with respect to t.

Step-by-step explanation:

To find the function r that satisfies the given condition, we need to integrate the given derivative r'(t) with respect to t. The given derivative is (eᵗ, sin t, sec² t). Integrating each component separately, we get:

r(t) = (eᵗ + C₁, -cos t + C₂, tan t + C₃)

Using the initial condition r(0) = (2, 2, 2), we can solve for the constants C₁, C₂, and C₃:

C₁ = 2 - 1 = 1

C₂ = 2 + 1 = 3

C₃ = 2 - tan 0 = 2

Substituting these constants back into the function, we get the final function:

r(t) = (eᵗ + 1, -cos t + 3, tan t + 2)