Final answer:
To find the length of the curve 2t i + e^t j + e^(-t) k from t=0 to t=2, one must compute the derivative of the curve, find the magnitude to get the speed function, and then integrate this speed over the interval [0,2].
Step-by-step explanation:
The length of the curve given by the vector function 2t i + e^t j + e^(-t) k, for the parameter range 0 ≤ t ≤ 2, can be found using the integral of the speed function with respect to time. The speed function is obtained by finding the derivative of the curve function with respect to time (t), then taking the magnitude of that derivative vector. The formula for the length of a curve is the integral from the lower to the upper bound of the parameter of the magnitude of the derivative of the curve function, which in mathematical terms is presented as ∫ |ℓr(t)/dt| dt, where ℓr(t)/dt is the derivative vector.
To solve this problem, we first compute the derivative of the given vector function with respect to t, yielding dr/dt = (2, e^t, -e^(-t)). Then we find the magnitude of this derivative which will give us the speed function as a function of time. To find the curve's length, we integrate the speed function over the given interval of t from 0 to 2.
As an example provided, the length of a curve given by a function dr(t) = 4.0t i + 3.0 j + 5.0 k, with respect to time t, is obtained by first computing the speed at time t, Speed (t), which is the magnitude (denoted by square roots) of the derivative of the position vector, and then integrating this speed over the desired interval.