Question

Find the slope of the curve below at the given points. Sketch the curve along with its tangents at these points. R = sin 2 theta: theta = plusminus pi/4, plusminus 3 pi/4 The slope of the curve at theta = pi/4 is:__________

Answer 1

-1

Explanation:

Given that:

r = sin 2θ , θ = ± π/4 , ± 3π/4

Recall that:

x = r cosθ

y = r sinθ

The differential of y with respect to x

`(dy)/(dx) = ((dy)/(d \theta))/((dx)/(d \theta))`

`(dy)/(dx) =((dy)/(d \theta))/((dx)/(d \theta)) = \frac{ r cos \theta + sin \theta* (dr)/(d \theta) } {(dr)/(d \theta) *cos \theta- sin \theta \ r}`

at θ = π/4 , r = sin π/2

r = 1

`(dy)/(dx) = \frac{ r cos \theta + 2 cos (2 \theta)*sin \ \theta } {2 \ cos (2 \theta) *cos \theta- sin 2 \theta * sin \theta}`

where;

θ = π/4

`(dy)/(dx) = \frac{ 1 * cos ((\pi)/(4)) + 2 cos ((\pi)/(2))*sin ((\pi)/(4)) } {2 \ cos ((\pi)/(2))*cos ((\pi)/(4))- sin ((\pi)/(2)) * sin ((\pi)/(4))}`

`(dy)/(dx) = ((1)/(√(2))+0 )/(0-1 * (1)/(√(2)))`

`(dy)/(dx) = ((1)/(√(2)) )/(- (1)/(√(2)))`

`\mathbf{(dy)/(dx) =-1}`

slope of the curve (dy/dx) at theta(θ) = pi/4 is -1