Find the exact value of the slope of the line which is tangent to the curve given by the equation r = 2 + cos θ at θ=pi/2. You must show your work.
My answer is "-1" but my friend has "1/2". Which answer is correct?
r = 2+cosθ
r' = -sinθ
r'(pi/2) = -1
The slope of the line is dy/dx
y = rsinθ = 2r + cosθsinθ = 2r + 1/2 sin2θ
dy/dθ = 2r' + cos2θ
dy/dθ at pi/2 = -1
x = rcosθ = 2r + cos^2θ
dx/dθ = 2r' + 2cosθ(-sinθ)
dx/dθ at pi/2 = -2
so, dy/dx = (dy/dθ)/(dx/dθ) at pi/2 = 1/2
Maybe next time you could show your work. Or else ...
To find the slope of the line tangent to the curve given by the polar equation r = 2 + cos θ at θ = π/2, we can use the concept of derivative in polar coordinates.
1. First, we need to express the equation in Cartesian form using the conversion formulas:
x = r * cos θ
y = r * sin θ
Substituting the given equation r = 2 + cos θ:
x = (2 + cos θ) * cos θ
y = (2 + cos θ) * sin θ
2. Differentiate both x and y with respect to θ, treating them as functions of θ. Apply the chain rule as follows:
dx/dθ = -(2 + cos θ) sin θ - sin θ * cos θ
dy/dθ = (2 + cos θ) cos θ - sin θ * sin θ
3. Evaluate dx/dθ and dy/dθ at θ = π/2:
dx/dθ = -(2 + cos(π/2)) sin(π/2) - sin(π/2) * cos(π/2)
= -(2 + 0) * 1 - 1 * 0
= -2
dy/dθ = (2 + cos(π/2)) cos(π/2) - sin(π/2) * sin(π/2)
= (2 + 0) * 0 - 1 * 1
= -1
4. Calculate the slope of the tangent line using the formula:
slope = dy/dx = (dy/dθ)/(dx/dθ)
slope = (-1)/(-2)
Simplifying the expression, we get:
slope = 1/2
Therefore, your friend's answer of 1/2 is correct.