State the domain of r determine the direction and magnitude

Question

1. Arc Length:

a. Consider the following vector function:

r (t) = (t, 2t², 8/3t³)

i. Determine the direction and magnitude of the vector derivative at t = 1.

ii. Determine the length of the curve from 0 ≤ t ≤ 1.

b. Consider the following vector function:

r(t) = (e^t, e^-t, √2t).

i. State the domain of r.

ii. Determine the direction and magnitude of the vector derivative at t = ln(2).

iii. Determine the length of the curve from 0 ≤ t ≤ 1.

c. Reparametrize the following curves by arc length, starting from t = 0:

i. r(t) = (l+2t, 5-t, 3t)

ii. r(t) = (sin (t), 2t, cos (t))

iii. r(t) = (e^3t sin (3t), e^3t cos (3t)).

2. Multivariate Functions:

a. Consider the following multivariate function:

f(x. y) = y/(x²- 1)

i. State the domain of f.

ii. Sketch the contour plot of f in the (x, y)-plane.

iii. Determine the direction and magnitude of steepest ascent at the point (0,2).

iv. Is the function continuous at the point (1. 0)? Justify your answer.