find the rate of change of the distance between the origin and a moving point on the graph of y=x^2+2 if ds/dt=5 centimeters per second.
I will assume you meant
dx/dt = 5, or else the answer to your question is given
s = √(x^2 + y^2)
= (x^2 + (x^2+2)^2)^.5
= (x^4 + 5x^2 + 4)^.5
ds/dt = (1/2)(x^4 + 5x^2 + 4)(4x^3 + 10x)(dx/dt)
so after you plug in the value, not much else can be done
Something fishy about the question, was there an x value given?
forgot the exponent ... should say'
ds/dt = (1/2)(x^4 + 5x^2 + 4)^(-1/2)(4x^3 + 10x)(dx/dt)
To find the rate of change of the distance between the origin and a moving point on the graph of y = x^2 + 2, we can use the concept of the Pythagorean Theorem.
Let's consider the point (x, y) on the graph of y = x^2 + 2. The distance between the origin (0, 0) and this point can be found using the formula:
distance = sqrt(x^2 + y^2)
Let's differentiate both sides of this equation with respect to time (t):
d(distance)/dt = d(sqrt(x^2 + y^2))/dt
Using the chain rule, we can break down this derivative. Let's denote the differentiation with respect to t using prime notation ('):
d(distance)/dt = (1/2) * (x^2 + y^2)^(-1/2) * (2x*dx/dt + 2y*dy/dt)
Since we know ds/dt = 5 cm/s, we need to solve for dx/dt and dy/dt.
From the equation y = x^2 + 2, we can find dy/dt by differentiating implicitly with respect to t:
dy/dt = d/dt(x^2 + 2)
= 2x * dx/dt
Now, we substitute this value of dy/dt back into the equation for d(distance)/dt:
d(distance)/dt = (1/2) * (x^2 + y^2)^(-1/2) * (2x*dx/dt + 2y*(2x*dx/dt))
Substituting y = x^2 + 2:
d(distance)/dt = (1/2) * (x^2 + (x^2 + 2)^2)^(-1/2) * (2x*dx/dt + 2(x^2 + 2)(2x*dx/dt))
Simplifying this equation further:
d(distance)/dt = (1/2) * (x^2 + x^4 + 4x^2 + 4)^(-1/2) * (2x*dx/dt + 4x(x^2 + 2)dx/dt)
Now, we plug in the given value of ds/dt:
5 cm/s = (1/2) * (x^2 + x^4 + 4x^2 + 4)^(-1/2) * (2x*dx/dt + 4x(x^2 + 2)dx/dt)
We need to solve this equation for dx/dt in terms of x. Additionally, we should have a specific value for x to find the rate of change at that particular point.
Please provide a specific value for x to proceed further.