4-An object moving along a curve in the xy-plane has dy/dt=3+e^sin(t). The derivative dx/dt is not explicitly given. A

Question

4-An object moving along a curve in the xy-plane has dy/dt=3+e^sin(t). The derivative dx/dt is not explicitly given. At time t=2. The value of dy/dx=-5.4 find the value of dx/dt when t=2

Answer 1

Hello Gahij!

We want to know things about [dx/dt] when t = 2 but we only know a function for [dy/dt] and only know a single point of data for [dy/dx] when t = 2.

The big trick is that by carefully multiplying or dividing our known derivatives, we can create an expression for our desired unknown derivative. Let me try and explain.

If we take [dy/dt] and multiply it by [dy/dx], we can do fraction multiplication to see that this would equal [(dy*dy)/(dt*dx)], which is messy and not really useful...

BUT what if instead we took [dy/dt] and DIVIDED it by [dy/dx]? We would be dividing a "fraction" by a "fraction" (which can be thought of as multiplying by the reciprocal of [dy/dx] instead)... leaving us with...

[dy/dt] / [dy/dx] = [dy/dt] * [dx/dy] = [(dy*dx)/(dt*dy)] (Note: we have a "dy" on top and bottom, they CANCEL)

[dy/dt] / [dy/dx] = [dx/dt]

This is actually the exact quantity that we want to solve for here [dx/dt]! Now let's solve it out numerically. We are only interested in knowing [dx/dt] specifically when t = 2, so we don't need to solve for [dx/dt] as a function, and we can just plug in numbers for [dy/dt] and [dy/dx] when t = 2.

We are given that when t = 2, [dy/dx] = -5.4

We need to solve for the value of [dy/dt] when t = 2 though....

[dy/dt] = 3+e^sin(t)

[dy/dt] = 3+e^sin(2)

[dy/dt] = 3+e^0.909

[dy/dt] = 5.483 when t = 2

Now plug into our earlier relationship to solve for dx/dt when t = 2...

[dx/dt] = [dy/dt] / [dy/dx]

[dx/dt] = [5.483] / [-5.4]

[dx/dt] = -1.02

Hope that helps! Let me know if you have other questions, I'd be happy to help!

--Zach