On what intervals of t is the curve described by the given parametric equation concave up? Concave down?
x=t^2; y=t^(3) + 3t
I am a bit confused on how to solve this...any help/explanations are welcome!! (& greatly appreciated!)
To determine the intervals of t on which the curve is concave up or concave down, we need to analyze the second derivative of y with respect to x, which is denoted as d^2y/dx^2.
Step 1: Find dx/dt and dy/dt
Given:
x = t^2
y = t^3 + 3t
Differentiating both x and y with respect to t, we get:
dx/dt = 2t
dy/dt = 3t^2 + 3
Step 2: Find dy/dx
Next, we need to find dy/dx, which is the derivative of y with respect to x. This can be calculated by dividing dy/dt by dx/dt:
dy/dx = (dy/dt) / (dx/dt)
Substituting the derivatives we found earlier, we have:
dy/dx = (3t^2 + 3) / (2t)
dy/dx = (3/2)(t + 1/t)
Step 3: Find d^2y/dx^2
To determine the concavity, we need to find the second derivative, d^2y/dx^2. This can be calculated by taking the derivative of dy/dx with respect to x:
d^2y/dx^2 = d/dx(dy/dx)
Using the quotient rule, the derivative of dy/dx is obtained as follows:
d^2y/dx^2 = [(2t)(2/t) - (3/2)(t - 1/t^2)] / (2t)^2
d^2y/dx^2 = (4 - 3(t - 1/t^2) / (4t^2)
d^2y/dx^2 = (4 - 3t + 3/t^2) / (4t^2)
Step 4: Determine concavity
Now that we have the second derivative, we can analyze where the curve is concave up or concave down.
For concave up, we need d^2y/dx^2 > 0.
For concave down, we need d^2y/dx^2 < 0.
To find the intervals of t, where the curve is concave up, we set d^2y/dx^2 > 0:
(4 - 3t + 3/t^2) / (4t^2) > 0
Next, we consider the intervals where the numerator and denominator have the same sign. This occurs when:
Numerator: 4 - 3t + 3/t^2 > 0 and Denominator: 4t^2 > 0
Solving these two inequalities will give us the intervals of t where the curve is concave up. Similarly, for the curve to be concave down, we set d^2y/dx^2 < 0 and solve the inequality.
It is worth noting that in parametric equations, concavity is determined based on d^2y/dx^2 rather than d^2y/dt^2, which is used in Cartesian equations.
I hope this explanation helps in understanding how to find the intervals of concavity for a parametric equation.