Question

Use the properties of the derivative to find the following.

r(t) = ti + 4tj + t²k,
u(t) = 4ti + t²j + t³k
(a) r'(t)

Answer 1

To find the derivative r'(t) of the vector function r(t) = ti + 4tj + t²k, differentiate each component with respect to t, giving r'(t) = i + 4j + 2tk.

Step-by-step explanation:

The question asks us to find the derivative of the vector function r(t). This vector function is expressed as ti + 4tj + t²k, where i, j, and k are the unit vectors in three-dimensional space, and t represents time.

Using the properties of derivatives for vector functions, we find the derivative component by component. For the derivative r'(t), we apply this to each component of r(t):

1. For the component ti, the derivative with respect to time is 1i or simply i.
2. For the component 4tj, the derivative is 4j.
3. For the component t²k, the derivative is 2tk.

Therefore, the derivative r'(t) is given by:

r'(t) = i + 4j + 2tk.