Find d^2y/dx^2 for the curve given by x = 2t + 5 and y= 3t/2
a) 3/(4t+10)
b) (4t+10)/3
c) 1
d) 0
x = 2t + 5
t = (x-5)/2
and y= 3t/2 = (3/2)(x-5)/2 = (3/4)(x - 5)
or
y = (3/4)x - 15/4
so your given parametric equations are simply the straight line
y = (3/4)x - 15/4
dy/dx = 3/4
d(dy/dx) / dx = 0
Since x and y are both linear functions of t,
dy/dx = (dy/dt) / (dx/dt) will be a constant.
d^2y/dx^2 = 0
To find the second derivative of y with respect to x, we need to use the formula:
d^2y/dx^2 = d/dx(dy/dx)
To find dy/dx, we need to differentiate both x and y with respect to t and then divide dy/dt by dx/dt:
dx/dt = 2
dy/dt = 3/2
dy/dx = (dy/dt) / (dx/dt) = (3/2) / 2 = 3/4
Now, we need to differentiate dy/dx with respect to t and divide it by dx/dt:
d(dy/dx)/dt = d(3/4)/dt = 0
Finally, note that dx/dt = 2 is a constant, so its derivative is 0. Therefore, the answer is option d) 0.
To find d^2y/dx^2 for the curve, we need to find dy/dt and dx/dt first.
Given:
x = 2t + 5
y = 3t/2
To find dy/dt, we differentiate y with respect to t:
dy/dt = d/dt(3t/2)
To differentiate (3t/2) with respect to t, we can use the power rule:
dy/dt = (3/2) * d/dt(t)
dy/dt = (3/2) * 1
dy/dt = 3/2
Next, we need to find dx/dt by differentiating x with respect to t:
dx/dt = d/dt(2t + 5)
To differentiate (2t + 5) with respect to t, we can use the constant rule:
dx/dt = d/dt(2t) + d/dt(5)
dx/dt = 2 * d/dt(t) + 0
dx/dt = 2
Now, we can use the chain rule to find d^2y/dx^2:
d^2y/dx^2 = (d/dt(dy/dt)) / (d/dt(dx/dt))
Substituting dy/dt and dx/dt into the equation:
d^2y/dx^2 = (d/dt(3/2)) / (d/dt(2))
To differentiate (3/2) with respect to t, we have a constant, so its derivative is 0:
d^2y/dx^2 = 0 / (d/dt(2))
Differentiating a constant, 2, with respect to t gives us 0:
d^2y/dx^2 = 0 / 0
Therefore, the answer is indeterminate (d) 0.