Question

If r(t) = 2e3t, 2e−3t, 3te3t , find T(0), r''(0), and r'(t) · r''(t).

T(0) =
r''(0) =
r'(t) · r''(t) =

Answer 1

To compute T(0), r''(0), and r'(t) · r''(t) for the vector function r(t), one must differentiate r(t) to obtain the first and second derivatives, evaluate these derivatives at t=0, and calculate the dot product of the first and second derivatives, respectively.

Step-by-step explanation:

To find T(0), r''(0), and the dot product r'(t) · r''(t) for the vector function r(t) = <2e3t, 2e−3t, 3te3t>, we first need to calculate the first and second derivatives of r(t).

• The first derivative r'(t) represents the velocity vector, found by differentiating each component of r(t) with respect to t.
• The second derivative r''(t) represents the acceleration vector, found by differentiating each component of r'(t) with respect to t again.
• To find T(0), simply evaluate r'(t) at t = 0.
• To find r''(0), evaluate r''(t) at t = 0.
• To compute r'(t) · r''(t), take the dot product of r'(t) and r''(t).

After performing the differentiation, we can substitute t = 0 into the derived expressions to find T(0) and r''(0). To find r'(t) · r''(t), we multiply the corresponding components of r'(t) and r''(t) and sum them up for the dot product. These calculations are standard in the study of vector functions in mathematics, particularly within the field of calculus.