To find the first and second derivatives of the functions defined by x = cos²(t) and y = cos(t), we will start by applying differentiation rules known as the chain rule and product rule in calculus.
Step 1: Finding the first derivative:
For x = cos²(t), we will differentiate using the chain rule:
dx/dt = 2*cos(t)*(-sin(t)) = -2*cos(t)*sin(t)For y = cos(t), differentiating gives:
dy/dt = -sin(t)Step 2: Equating dy/dx:
Now, we can find dy/dx using the formula:
dy/dx = (dy/dt)/(dx/dt) = (-sin(t))/(-2*cos(t)*sin(t)) = 1/(2*cos(t))Step 3: Finding the second derivative:
The second derivative can be found by differentiating dy/dx again with respect to t:
d²y/dx² = d/dt(1/(2*cos(t))) = (-1/2) * (0 - sin(t)*(-1))/(cos²(t)) = sin(t)/(2*cos²(t))Now we have:
- First Derivative:
dy/dx = 1/(2*cos(t)) - Second Derivative:
d²y/dx² = sin(t)/(2*cos²(t))
Conclusion: Therefore, for the functions defined above, the first and second derivatives over the interval 0 ≤ t ≤ π are given by the expressions derived. You can substitute specific values of t within that interval if you need numerical results.