To analyze the concavity of the given curve, we first need to establish the equations for
dydx and d²ydx². The given parametric equations are:
x = 2sin(t) y = 3cos(t)
1. **Finding dydt**: We must first find the derivatives of x and y with respect to t:
dx/dt = 2cos(t) dy/dt = -3sin(t)
2. **Finding dydx**: The derivative
dydx can be found using the chain rule:
dy/dx = (dy/dt) / (dx/dt) = (-3sin(t)) / (2cos(t)) = - rac{3}{2}tan(t)
3. **Finding d²ydx²**: Now, we need to determine d²ydx². We can find it by applying the quotient rule to
dydx:
d²y/dx² = d/dt(dy/dx) / (dx/dt)
Calculating each part:
d/dt(dy/dx) = rac{d}{dt}igg(- rac{3}{2} an(t)igg) = - rac{3}{2}sec²(t)
So:
d²y/dx² = rac{- rac{3}{2}sec²(t)}{2cos(t)} = - rac{3sec²(t)}{4cos(t)}
4. **Concavity Condition**: The curve is concave upward when d²y/dx² > 0:
- rac{3sec²(t)}{4cos(t)} > 0
Since sec²(t) is always positive, the sign of the expression depends on cos(t). Therefore, we have:
cos(t) < 0
This inequality is satisfied in the intervals where t is in the second quadrant and the third quadrant:
t ∈ (π/2, 3π/2)
Summary: The curve defined by the parametric equations x = 2sin(t) and y = 3cos(t) is concave upward for:
t ∈ (π/2, 3π/2)
That is, for the range of values of t between π/2 and 3π/2, the curve exhibits concave upward behavior.