A line through the origin, rotates around the origin in such a way that the angle, ¦È, between the line and the positive x-axis changes at the rate of d¦È/dt for time t¡Ý0. Which expression gives the rate at which the slope of the line is changing?
a) d¦È/dt
b) cos¦È*d¦È/dt
c) -sin¦È*d¦È/dt
d) (1/(cos¦È)^2)*d¦È/dt
e) tan¦È*d¦È/dt
that weird 'e' symbol is supposed to be a theta sign
To find the rate at which the slope of the line is changing, we need to differentiate the equation of the line with respect to ¦È and then multiply it by the rate of change of ¦È with respect to time (d¦È/dt).
The equation of the line passing through the origin can be written as y = mx, where m is the slope of the line.
Differentiating this equation with respect to ¦È, we get dy/dx = d(mx)/d¦È.
Now, we can express dx/d¦È and dy/d¦È in terms of dx/dt and dy/dt, respectively, using the chain rule.
dx/d¦È = dx/dt * dt/d¦È = dx/dt * (1/(d¦È/dt))
dy/d¦È = dy/dt * dt/d¦È = dy/dt * (1/(d¦È/dt))
Since the line passes through the origin, dy/dx is equal to the slope of the line which is m.
So, m = dy/dx = (dy/d¦È) / (dx/d¦È)
Substituting the expressions for dx/d¦È and dy/d¦È, we get m = (dy/dt * (1/(d¦È/dt))) / (dx/dt * (1/(d¦È/dt)))
Simplifying, m = dy/dt / dx/dt.
Therefore, the rate at which the slope of the line is changing is equal to dy/dt / dx/dt.
From the given options, the expression that matches this is option b) cos¦È * d¦È/dt.
So, the correct answer is option b) cos¦È * d¦È/dt.
To find the rate at which the slope of the line is changing, let's start by observing how the slope of the line relates to the angle ¦È.
The slope of the line passing through the origin, let's call it m, is equal to the tangent of the angle ¦È. So we have:
m = tan(¦È)
Now, let's differentiate both sides of this equation with respect to time t:
dm/dt = d/dt(tan(¦È))
Using the chain rule, the derivative of tangent function is:
dm/dt = sec^2(¦È) * d¦È/dt
But we need the rate of change of the slope, so we want to solve for d(dm/dt)/dt. Differentiating both sides of the equation again, we have:
d(dm/dt)/dt = d/dt(sec^2(¦È) * d¦È/dt)
Now, let's simplify this expression. The derivative of sec^2(¦È) with respect to time is zero because it does not depend on t. The product rule applies to the remaining part:
d(dm/dt)/dt = d/dt(d¦È/dt)
= d^2¦È/dt^2
Since d(dm/dt)/dt equals the second derivative of ¦È with respect to t, the correct answer is (d) (1/(cos^2(¦È)))*d¦È/dt.