Final answer:
a. The dot product r'(t) · r''(t) for the vector-valued function r(t) is 400t^7 + 64t. b. The derivative of the vector-valued function r(t) is (e^t + te^t)i + (ln(t) + 1)j + (2cos(2t))k. c. The limit of r(t) as t approaches infinity is (0)i + (∞)j + (5)k.
Step-by-step explanation:
a. To find r'(t) · r''(t), we first find the derivatives of r(t). The first derivative r'(t) is -10t^4i + 5j + 8tk. The second derivative r''(t) is -40t^3i + 8k. We can then find the dot product by multiplying the corresponding components and summing the result. In this case, r'(t) · r''(t) = (-10t^4)(-40t^3) + (5)(0) + (8t)(8) = 400t^7 + 64t.
b. To compute the derivative of the vector-valued function r(t) = te^t i + t ln(t)j + sin(2t)k, we differentiate each component with respect to t. The derivative of te^t is e^t + te^t, the derivative of t ln(t) is ln(t) + 1, and the derivative of sin(2t) is 2cos(2t). Therefore, the derivative of r(t) is (e^t + te^t)i + (ln(t) + 1)j + (2cos(2t))k.
c. To find the limit lim t→∞ r(t) for the vector-valued function r(t) = (51e^-tcos(t))i + (51e^-tsin(t))j + (5 - 5e^-t)k, we can analyze the behavior of the individual components as t approaches infinity. The first component approaches zero, the second component oscillates between positive and negative infinity, and the third component approaches 5. Therefore, the limit of r(t) as t approaches infinity is (0)i + (∞)j + (5)k.