N exponential spiral has parametric equationr(t) =〈etcost, etsint〉.(a) Plotr(t). Make sure your plot includeslimt→−[infinity]r(t).(b) Find the unit tangent vectorT(t)tor(t).(c) Find the length ofr(t)for−[infinity]< t≤0.(d) Letf(x, y) =−12ln(x2+y2)−arctanxy. Find∇f(x, y).(e) Compute the angle between∇f(r(t))andr′(t).

please see the answer below

Step-by-step explanation:

we have

(b) the unit tangential vector is computed by using

where we have used cos^2t+sin^2t=1

(c) the arc length is given by

(d)

(e)

hope this helps!!

answer: (c) m∠qpo + (2x + 16)° = 180°

step-by-step explanation:

the opposite angles of a quadrilateral are supplementary.

so, ∠o + ∠q = 180°   and   ∠p + ∠r = 180°

since ∠r = 2x + 16°, we can use substitution as follows:

∠p + ∠r = 180°

∠p + (2x + 16)° = 180°