find the rate of change of the distance between the orgin and a moving point on the graph of y=x^2+1 idf dx/dt=2 centimeters per second.
if the distance is s,
s^2 = x^2 + y^2
2s ds/dt = 2x dx/dt + 2y * 2x dx/dt
2s ds/dt = (2x^2 + 2x + 2) dx/dt
s ds/dt = (x^2+x+1)(2)
ds/dt = 2(x^2+x+1)/sqrt(x^4 + 3x^2 + 1)
If you know x, plug it in
To find the rate of change of the distance between the origin and a moving point on the graph of y=x^2+1, we need to use the Pythagorean theorem.
The distance between the origin and a point on the graph is given by the equation:
distance = √(x^2 + y^2)
Since the graph equation is y = x^2 + 1, we can substitute this into the distance equation:
distance = √(x^2 + (x^2 + 1)^2)
To find the rate of change of the distance, we need to differentiate the distance equation with respect to time (t), using the chain rule.
Let's calculate the derivative step by step:
1. Take the derivative of the distance equation:
d(distance)/dt = d(√(x^2 + (x^2 + 1)^2))/dt
2. Apply the chain rule:
d(distance)/dt = (1/2)(x^2 + (x^2 + 1)^2)^(-1/2) * d(x^2 + (x^2 + 1)^2)/dt
3. Simplify the derivative:
d(distance)/dt = (1/2)(x^2 + (x^2 + 1)^2)^(-1/2) * (2x + 2(x^2 + 1) * dx/dt)
Now, substitute the given value of dx/dt = 2 centimeters per second into the equation:
d(distance)/dt = (1/2)(x^2 + (x^2 + 1)^2)^(-1/2) * (2x + 2(x^2 + 1) * 2)
Simplifying further:
d(distance)/dt = (x + (x^2 + 1)) / √(x^2 + (x^2 + 1)^2)
Therefore, the rate of change of the distance between the origin and a moving point on the graph when dx/dt = 2 centimeters per second is (x + (x^2 + 1)) / √(x^2 + (x^2 + 1)^2).
To find the rate of change of the distance between the origin (0,0) and a moving point on the graph of y = x^2 + 1, given dx/dt = 2 centimeters per second, we can use the concept of the derivative.
1. Let's denote the distance between the origin and the moving point as D(t).
2. The distance, D(t), can be represented by the formula: D(t) = sqrt(x^2 + y^2), where (x, y) are the coordinates of the moving point.
3. To find the rate of change of D(t), we need to differentiate the equation D(t) with respect to time, t.
4. So, we'll differentiate D(t) = sqrt(x^2 + y^2) with respect to t using the chain rule.
d(D(t))/dt = d(sqrt(x^2 + y^2))/dt
5. To apply the chain rule, we need to find dx/dt and dy/dt.
Given dx/dt = 2 cm/s, we can differentiate the equation y = x^2 + 1 with respect to t.
dy/dt = d(x^2 + 1)/dt = 2x * dx/dt
6. Now, we can substitute dx/dt and dy/dt into the expression obtained in step 4.
d(D(t))/dt = d(sqrt(x^2 + y^2))/dt
= (1/2)*(x^2 + y^2)^(-1/2) * (2x * dx/dt + 2y * dy/dt)
7. Since we are given the equation y = x^2 + 1, we can substitute y into the expression:
d(D(t))/dt = (1/2)*(x^2 + (x^2 + 1)^2)^(-1/2) * (2x * 2 + 2(x^2 + 1) * 2x)
8. Simplifying the expression:
d(D(t))/dt = (1/2)*(x^2 + (x^4 + 2x^2 + 1))^(-1/2) * (4x + 4x^3 + 4x)
d(D(t))/dt = (1/2)*(x^4 + 3x^2 + 1)^(-1/2) * (8x + 4x^3)
9. Now, substitute dx/dt = 2 cm/s into the expression and calculate the rate of change of the distance.
rate of change = d(D(t))/dt * dx/dt
= (1/2)*(x^4 + 3x^2 + 1)^(-1/2) * (8x + 4x^3) * 2 cm/s
Note: To calculate the exact rate of change at a specific point on the graph, you need to substitute the value x or y at that point into the final expression and solve the equation.