Give the parametric equations x et cost and y et sint for

Give the parametric equations x = et cost and y = et sint for 0 t 2pi, find the values of t where the slope of the tangent line to the curve has a value of -1. [Hint: you need to find dy/dt / dx/dt.] the arc length of this curve; only set up the integral, do not evaluate it.

Solution

slope = dy/ dx =(dy/dt)/(dx/dt)

(dy/dt) = e^t sint +e^t cost

(dx/dt) = -e^t sint +e^t cost
acording to given condition

(sint + cost)/(sint -cost) =-1

2sint = 0

t =0 , ,2

for dx and dy thickness arc lenth is(dx²+dy²)

so integration (dx²+dy²)

but dx = (-e^t sint +e^t cost )dt

and dy=(e^t sint +e^t cost)

integration (dx²+dy²) = integration e^t *2 dt for t varying from 0 to 2