Question

Define a derivative of a vector valued function r(t). Compute the first, second and third derivative function. r(t)=3ti+6lntj+5e⁻³ᵗ k

Answer 1

The derivative of a vector valued function r(t) is a vector that represents the rate of change of the function with respect to time. The first derivative gives the velocity, the second derivative yields the acceleration, and the third derivative can be calculated similarly by differentiating the acceleration component-wise.

Step-by-step explanation:

The derivative of a vector-valued function r(t) represents the rate at which the function's output changes with respect to time. For the given vector r(t) = 3ti + 6lntj + 5e⁻³ᵗ k, the derivatives are found by differentiating each component with respect to t.

First Derivative (Velocity)

The first derivative is the velocity vector v(t), given by:

v(t) = dr(t)/dt = 3i + (6/t)j - 15e⁻³ᵗ k.

Second Derivative (Acceleration)

The second derivative is the acceleration vector a(t), found by differentiating the velocity:

a(t) = d2r(t)/dt2 = 0i - (6/t2)j - 45e⁻³ᵗ k.

Third Derivative

Finally, the third derivative can also be computed by differentiating the acceleration:

d3r(t)/dt3 = 0i + (12/t3)j + 135e⁻³ᵗ k.