Question

"Find (dy / dx) for the parametric curve. Problem 1. (x, y) = (te^t, t + sin(t)).

Answer 1

The derivative dy/dx for the parametric equations (x, y) = (te^t, t + sin(t)) is found by first computing dx/dt and dy/dt, and then dividing dy/dt by dx/dt.

Step-by-step explanation:

Finding the derivative dy/dx for a parametric curve:

The question asks to find dy/dx for the parametric equations x = te^t and y = t + sin(t). To accomplish this, one must first find dx/dt and dy/dt and then divide dy/dt by dx/dt to get dy/dx.

1. Find dx/dt: Differentiating x = te^t with respect to t using the product rule gives dx/dt = e^t + te^t.
2. Find dy/dt: Differentiating y = t + sin(t) with respect to t gives dy/dt = 1 + cos(t).
3. Find dy/dx: Now, divide dy/dt by dx/dt to get dy/dx. This results in dy/dx = (1 + cos(t)) / (e^t + te^t).

That is the slope of the tangent to the curve at any point for a given t.