Final answer:
The question seeks f'(3) for f(t) = u(t) · v(t) with given values for u(3) and u'(3). However, due to the lack of clarity regarding whether these functions are scalar or vector-valued and the incomplete vector information provided, a precise answer cannot be given. The student is encouraged to provide detailed information for each component of the vector functions for adequate assistance.
Step-by-step explanation:
The student is asking to find the derivative f'(3) of the function f(t), which is defined as the product of two other functions, u(t) and v(t). The values of u(3) and u'(3) are given, as well as the form of v(t). To find f'(3), we will apply the product rule of differentiation, which states that if f(t) = u(t) · v(t), then f'(t) = u'(t) · v(t) + u(t) · v'(t). These derivatives are evaluated at t = 3.
The student's question, unfortunately, includes a vector notation which suggests that u(t) and v(t) might be multi-dimensional, which complicates the differentiation process. However, since we do not have enough information to deal with vectors, we will assume the functions are scalar and continue accordingly. We need to find v'(t) by differentiating the components of v(t) = (t, t^2, t^3). The derivatives of these components with respect to t are (1, 2t, 3t^2). We then evaluate these at t = 3 to get (1, 6, 27). The final step is to combine these with the given values of u(3) and u'(3) using the product rule.
Since the question is not constructed well to cover the vector case, and sufficient information is not provided to compute f'(3) for vector functions, we will not attempt to provide a solution that could be inaccurate. The student should clarify whether u(t) and v(t) are scalar or vector-valued functions and provide complete information for each component if vector valued.