Find the instantaneous rate of change of r with respect to theta for r = (theta/2)- (1/theta).
r = (1/2)Ø - 1/Ø
dr/dØ = 1/2 + 1/Ø^2
Well, let me tell you, the rate of change for this function is like a roller coaster ride. We're going up, then we're going down, then we're going back up again. Hang on tight!
To find the instantaneous rate of change, we need to take the derivative of the function r with respect to theta. So, let's put on our mathematical seatbelts and get ready for some calculus!
First, let's differentiate the function r = (theta/2) - (1/theta) with respect to theta. The first term, (theta/2), gives us 1/2, and the second term, (1/theta), gives us -1/theta^2.
So, the derivative of r with respect to theta is (1/2) - (1/theta^2).
And there you have it! The instantaneous rate of change of r with respect to theta is (1/2) - (1/theta^2). Just remember to keep your hands and feet inside the derivative at all times!
To find the instantaneous rate of change of r with respect to theta, we need to find the derivative of r with respect to theta.
Given the equation r = (theta/2) - (1/theta), we can use the quotient rule of differentiation.
The quotient rule states that if we have a function in the form f(theta)/g(theta), the derivative of this function with respect to theta is given by:
(d/dtheta) [f(theta)/g(theta)] = (g(theta) * f'(theta) - f(theta) * g'(theta)) / [g(theta)]^2
Let's find the derivative of r using the quotient rule:
f(theta) = (theta/2)
f'(theta) = 1/2 (derivative of theta/2 with respect to theta)
g(theta) = 1/theta
g'(theta) = -1/theta^2 (derivative of 1/theta with respect to theta)
Now, we can substitute the values into the quotient rule formula:
(d/dtheta) [r] = [(1/theta) * (1/2) - (theta/2) * (-1/theta^2)] / [(1/theta)^2]
Simplifying further:
(d/dtheta) [r] = [1/(2theta) + theta/(2theta^2)] / (1/theta^2)
To simplify the expression inside the brackets, we can combine the terms over a common denominator:
(d/dtheta) [r] = [(1 + theta)/(2theta^3)] / (1/theta^2)
To divide by a fraction, we can multiply by its reciprocal:
(d/dtheta) [r] = [(1 + theta)/(2theta^3)] * (theta^2/1)
Cancel out the common terms:
(d/dtheta) [r] = (1 + theta)/ (2theta)
Therefore, the instantaneous rate of change of r with respect to theta is given by:
(d/dtheta) [r] = (1 + theta)/ (2theta)
Please let me know if I can help you with anything else.
To find the instantaneous rate of change of r with respect to theta for the given equation, we need to take the derivative of r with respect to theta.
Let's begin by differentiating the equation r = (theta/2) - (1/theta) term by term:
d/ dtheta [r] = d/ dtheta [(theta/2) - (1/theta)]
Now, we will differentiate each term separately using the rules of differentiation.
For the first term (theta/2), we can use the power rule. The power rule states that if we have a function of the form f(theta) = theta^n, then the derivative of f(theta) is given by d/dtheta [f(theta)] = n * theta^(n-1).
In this case, n = 1, so differentiating the first term, we get:
d/ dtheta [(theta/2)] = 1 * (theta^(1-1)) = 1 * theta^0 = 1 * 1 = 1
For the second term (1/theta), we can use the power rule again. This time, n = -1:
d/ dtheta [(1/theta)] = -1 * (theta^(-1-1)) = -1 * theta^(-2)
Combining both terms, we get:
d/ dtheta [r] = 1 - (1 * theta^(-2))
Simplifying further, we have:
d/ dtheta [r] = 1 - (1/theta^2)
Therefore, the instantaneous rate of change of r with respect to theta is given by:
d/ dtheta [r] = 1 - (1/theta^2)