Final answer:
To find the derivative of the vector function, differentiate each component with respect to time. To find the position vector at a certain time, substitute that time into the vector function. The derivative of the position vector at a certain time can also be found by substituting time into the derivative.
Step-by-step explanation:
The given vector function is r(t) = (5+t)i + t3j. To find the derivative r'(t), we differentiate each component of the vector with respect to 't' to get r'(t) = i + 3t2j. If we want to find the position vector r(t₀) at a particular time t₀, just substitute t₀ into the original vector function.
For example, if t₀ = 2, then r(2) = (5+2)i + (23)j = 7i + 8j.
To find r'(t₀), again substitute t₀ into the derivative of the vector function, r'(t). In case of t₀ = 2, r'(2) = 1i + 3(22)j = i + 12j. Remember that vector differentiation works component-wise in a very similar manner to normal differentiation.
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