Parametric Equations. Eliminating the Parameter. 1). 2). 11.2 Slope and Concavity. For the curve given by Find the slope and concavity at the point (2,3). At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up. Horizontal and Vertical tangents.
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For the curve given by
Find the slope and concavity at the point (2,3)
At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up
A horizontal tangent occurs when dy/dt = 0 but dx/dt0.
A vertical tangent occurs when dx/dt = 0 but dy/dt0.
x2 + y2 = r2
tan θ = y/x
x = r cos θ
y = r sin θ
Graph r2 = 4 cos θ
Third point does not show up.
On r = 1, point is (1, π)
On r = 1-2 cos θ, point is (-1, 0)
Where x = r cos θ = f(θ) cos θ
And y = r sin θ = f(θ) sin θSlope of a polar curve
Horizontal tangent where dy/dθ = 0 and dx/dθ≠0
Vertical tangent where dx/dθ = 0 and dy/dθ≠0
For r = 1 – cosθ
(a) Find the slope at θ = π/6
(b) Find horizontal tangents
(c) Find vertical tangentsFinding slopes and horizontal and vertical tangent lines
r = 1 – cosθ
Horizontal tangents at:
Vertical tangents at:
r = 2 sin 3θ
r = 2 sin 3θ = 0
3θ = 0, π, 2 π, 3 π
θ = 0, π/3, 2 π/3, π
Find the length of the arc for r = 2 – 2cosθ
2 sin2A =1-cos2A
2 sin2 (1/2θ) =1-cosθ