Question

Find dy/dx and d2y/dx2. x = 4 sin(t), y = 5 cos(t), 0 < t < 2π dy dx = correct: your answer is correct. d2y dx2 = correct: your answer is correct. for which values of t is the curve concave upward? (enter your answer using interval notation.)

Answer 1

To find the intervals where the curve is concave upward, we need to determine where the second derivative is positive. By differentiating the given equations, we can find dy/dx and d2y/dx2. Analyzing the sign of the second derivative will allow us to identify the concave upward intervals.

Step-by-step explanation:

To find the concave upward intervals of a curve, we need to determine where the second derivative is positive. In this case, we need to find the expression for d2y/dx2.

We are given the equations x = 4 sin(t) and y = 5 cos(t).

We can differentiate these equations with respect to t and then apply the chain rule to find dy/dx and d2y/dx2.

From the given equations, we can find dx/dt and dy/dt.

Then, we can calculate dy/dx as (dy/dt) / (dx/dt). Similarly, we can find d2y/dx2 as (d/dt(dy/dx)) / (dx/dt).

After finding dy/dx and d2y/dx2, we can determine the intervals of t for which d2y/dx2 is positive. These intervals indicate where the curve is concave upward.