A particle moves along a path described by y=x^2. At what point along the curve are x and y changing at the same rate? Find this rate if at time t we have x=sin t and y= sin^2t.
I solved the first part and got (1/2, 1/4), but I have no idea how to tackle the second part. Help is much appreciated.
dy/dt = dy/dx * dx/dt
but
dy/dx = 2x
so
dy/dt = 2 x dx/dt
so when 2 x = 1 (when the slope = 1 of course)
x = 1/2 then y = 1/4
x = sin t
when is sin t = 1/2
when t = 30 degrees or pi/6 radians
dx/dt = cos t = cos 30 = (sqrt 3 )/ 2
dy/dt = 2 sin t cos t = 2(1/2)(sqrt 3/2) sure enough
Thank you very much. I understand it now. :-)
Ah, the joys of calculus and the art of finding rates of change! Let's see if I can clown around with this question and help you out.
To find the rate at which x and y are changing, we can use the chain rule. But before we dive into that, let's differentiate y with respect to x to find dy/dx:
Given y = x^2,
dy/dx = 2x.
Now, let's find dx/dt and dy/dt and plug them into the chain rule formula for dx/dt:
Given x = sin(t),
dx/dt = cos(t).
Given y = sin^2(t),
dy/dt = 2sin(t) * cos(t).
Now, we can use the chain rule:
dy/dt = (dy/dx)(dx/dt) = (2x)(cos(t)).
Since we're looking for the point where x and y are changing at the same rate, we set the rates of change equal to each other:
2x * cos(t) = 2sin(t) * cos(t).
Now, substitute x = sin(t) into the equation:
2sin(t) * cos(t) = 2sin(t) * cos(t).
And voila! We see that the equation holds true for any value of t. So, x and y are changing at the same rate for all values of t.
As for finding the rate at which x and y are changing, well, they change at the same rate, so the rate of change is equal for both x and y. In other words, the rate of change is simply 2 times the cosine of t.
Keep in mind that this is just clowning around with math, so make sure to verify the solution with your own calculations. Happy clowntegration!
To find the point along the curve where x and y are changing at the same rate, we can use the concept of the rate of change or derivative.
Let's start with the first part:
1. Given the equation y = x^2, we want to find the point along the curve where dx/dt = dy/dt.
2. To find dx/dt and dy/dt, we need to differentiate both x and y with respect to t. Since x = sin(t) and y = sin^2(t), we can differentiate them using the chain rule.
dx/dt = d/dt (sin(t)) = cos(t)
dy/dt = d/dt (sin^2(t)) = 2sin(t) * cos(t)
3. Now we equate dx/dt and dy/dt and solve for t:
cos(t) = 2sin(t) * cos(t)
4. We can divide both sides of the equation by cos(t):
1 = 2sin(t)
5. Divide both sides by 2:
sin(t) = 1/2
6. Now we solve for t by taking the inverse sine (or arcsine) of both sides:
t = pi/6 or t = 5pi/6
7. Substitute the values of t back into x = sin(t) and y = sin^2(t) to find the corresponding points:
For t = pi/6:
x = sin(pi/6) = 1/2
y = sin^2(pi/6) = 1/4
For t = 5pi/6:
x = sin(5pi/6) = 1/2
y = sin^2(5pi/6) = 1/4
Therefore, the points where x and y are changing at the same rate are (1/2, 1/4) and (1/2, 1/4). This confirms the solution you obtained.
Now, let's move on to the second part of the question: finding the rate of change at time t when x = sin(t) and y = sin^2(t).
To find the rate of change, we need to differentiate y with respect to x (dy/dx). Here's how we do it:
1. Find dx/dt and dy/dt as we did before:
dx/dt = d/dt (sin(t)) = cos(t)
dy/dt = d/dt (sin^2(t)) = 2sin(t) * cos(t)
2. Now, since the question provides x and y in terms of t, we need to differentiate y with respect to x using the chain rule:
dy/dx = (dy/dt)/(dx/dt)
3. Substitute the values of dx/dt and dy/dt:
dy/dx = (2sin(t) * cos(t))/cos(t) = 2sin(t)
4. Simplify the expression:
dy/dx = 2sin(t)
5. To find the rate of change at a specific time, substitute t into the expression:
For t = pi/6:
dy/dx = 2sin(pi/6) = 2(1/2) = 1
For t = 5pi/6:
dy/dx = 2sin(5pi/6) = 2(-1/2) = -1
Therefore, the rate of change at time t, when x = sin(t) and y = sin^2(t), is 1 for t = pi/6 and -1 for t = 5pi/6.