Respuesta :
Answer:
θ [tex]0.75\pi[/tex] [tex]1.75\pi[/tex]
r 10.551 244.151
Step-by-step explanation:
The maximum value for [tex]\theta[/tex] is:
[tex]\theta_{max} = \ln r[/tex]
[tex]\theta_{max} = 2.199\pi\,rad[/tex]
The formula for the slope of the tangent line in polar coordinates is:
[tex]m = \frac{r'\cdot \sin \theta + r \cdot \cos \theta}{r' \cdot \cos \theta - r \cdot \cos \theta}[/tex]
Horizontal tangent lines have a slope of zero. So, the following relation must be satisfied:
[tex]r'\cdot \sin \theta + r \cdot \cos \theta = 0[/tex]
[tex]r'\cdot \sin \theta = - r \cdot \cos \theta[/tex]
[tex]\tan \theta = - \frac{r}{r'}[/tex]
[tex]\tan \theta = -\frac{e^{\theta}}{e^{\theta}}[/tex]
[tex]\tan \theta = -1[/tex]
[tex]\theta = \tan^{-1}(-1)[/tex]
[tex]\theta = \frac{3}{4}\pi + i\cdot \pi[/tex], for all [tex]i \in \mathbb{N}_{O}[/tex].
The maximum value of i is:
[tex]i = \frac{\theta_{max}-\frac{3}{4}\pi }{\pi}[/tex]
[tex]i = \frac{2.199-0.75}{1}[/tex]
[tex]i = 1.449[/tex] ([tex]i_{max} = 1[/tex]).
Then, values are listed below:
θ [tex]0.75\pi[/tex] [tex]1.75\pi[/tex]
r 10.551 244.151
Answer:
{ ( 2.193 , π / 4) , ( 10.551 , 3π / 4) , ( 50.754 , 5π / 4) , ( 244.151 , 7π / 4) }
Step-by-step explanation:
Given:-
- The polar curve has the equation:
r = e^θ
- list the points starting with the smallest value of r such that:
1 ≤ r ≤ 1000 , 0 ≤ θ < 2π.
Find:-
List all of the points (r,θ) where the tangent line is horizontal
Solution:-
- We will first transform the polar curve to cartesian coordinate system using the parametric relations:
x = r*cos (θ)
y = r*sin (θ)
- The tangent line is horizontal when the " dy / dθ " = 0 and " dx / dθ " = 0, so:
x = e^θ*cos (θ) , y = e^θ*sin (θ)
dx / dθ = e^θ*cos (θ) - e^θ*sin(θ)
= e^θ*[cos (θ) - sin(θ)]
dx / dθ = e^θ*[cos (θ) - sin(θ)] = 0,
e^θ = 0 , [cos (θ) - sin(θ)] = 0
e^θ ≠ 0 for the given interval 0 ≤ θ< 2π
cos (θ) - sin(θ) = 0 , tan ( θ ) = 1 - (1st quad and 3rd quad)
θ = { π / 4 , 5π / 4 } , 0 ≤ θ< 2π
- Similarly, evaluate dy/dθ = 0;
dy/dθ = e^θ*cos (θ) + e^θ*sin(θ)
= e^θ*[cos (θ) + sin(θ)]
dy / dθ = e^θ*[cos (θ) + sin(θ)] = 0,
e^θ = 0 , [cos (θ) + sin(θ)] = 0
e^θ ≠ 0 for the given interval 0 ≤ θ< 2π
cos (θ) + sin(θ) = 0 , tan ( θ ) = -1 , (2nd quad and 4th quad)
θ = { 3π / 4 , 7π / 4 } , 0 ≤ θ< 2π
- All possibilities of " θ " over the interval satisfying the a horizontal tangent line to the given polar curve:
θ = { π / 4, 3π / 4 , 5π / 4 , 7π / 4 } , 0 ≤ θ < 2π
- We will plug the evaluated list of values of "θ " in the given polar curve and determine the corresponding values of "r":
r = e^θ
θ = π / 4 , r = e^(π / 4) = 2.193
1: ( r , θ ) = ( 2.193 , π / 4)
θ = 3π / 4 , r = e^(3π / 4) = 10.55072
2: ( r , θ ) = ( 10.551 , 3π / 4)
θ = 5π / 4 , r = e^(5π / 4) = 50.754
3: ( r , θ ) = ( 50.754 , 5π / 4)
θ = 7π / 4 , r = e^(7π / 4) = 244.151
4: ( r , θ ) = ( 244.151 , 7π / 4)
