If r(t) = 4e^3t, 2e^-3t, 4te^3t , find t(0), r''(0), and r'(t) · r''(t).

Solution:

Given, r(t) = 4e³^t, 2e^-3t, 4te^3t -------------- (1)

We have to find t(0), r''(0), and r'(t) · r''(t).

To find r(0),

Put t = 0 in (1),

r(0) = 4e³⁽⁰⁾, 2e^-3(0), 4(0)e³⁽⁰⁾

r(0) = (4, 2, 0)

Therefore, at t(0) the value of r(0) is (4, 2, 0).

r’(t) = 3(4e^3t), -3(2e^-3t), 4e^3t+3(4te^3t)

r’(t) = 12e^3t, - 6e^-3t, 4e^3t+12te^3t

Put t = 0 in r’(t),

r’(0) = 12e³⁽⁰⁾, - 6e^-3(0), 4e³⁽⁰⁾+12(0)e³⁽⁰⁾

= 12, -6, 4

Therefore, r’(0) = (12, -6, 4)

r’’(t) = 3(12e^3t), -3(-6e^-3t), 3(4e^3t) + 12e^3t + 3(12te^3t)

r’’(t) = 36e^3t, 18e^-3t, 12e^3t+12e^3t+36te^3t

r’’(t) = 36e^3t, 18e^-3t, 24e^3t+36te^3t

Put t = 0 in r’’(t),

r’’(0) = 36e³⁽⁰⁾, 18e^-3(0), 24e³⁽⁰⁾+36(0)e³⁽⁰⁾

r’’(0) = (36, 18, 24)

Now, r’(0).r’’(t) = (12, -6, 4).(36, 18, 24)

= 12(36) - 6(18) + 4(24)

= 432 - 108 + 96

= 528 - 108

= 420

Therefore, r’(0).r’’(0) = 420

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If r(t) = 4e^3t, 2e^-3t, 4te^3t , find t(0), r''(0), and r'(t) · r''(t).

Summary:

If r(t) = 4e^3t, 2e^-3t, 4te^3t , at t(0) the value of r(0) = (4, 2, 0), r''(0) = (36, 18, 24), and r'(t) · r''(t) = 420.