variables x and y are related by the equation y=2 - 4/x. given that x and y are functions of t that dx/dt = 10, find dy/dt in terms of x.
To find dy/dt in terms of x, we need to use the chain rule.
First, let's rewrite the equation y = 2 - 4/x as y = 2 - 4x^(-1).
Now, we can find dy/dt.
dy/dt = dy/dx * dx/dt.
Since y = 2 - 4x^(-1), we can differentiate both sides with respect to x:
dy/dx = d/dx (2 - 4x^(-1)) = 0 - (-4)(-1)x^(-2) = 4x^(-2).
Now, we know that dx/dt = 10.
Therefore, we have dy/dt = dy/dx * dx/dt = 4x^(-2) * 10.
Simplifying this expression, we get:
dy/dt = 40x^(-2).
So, dy/dt in terms of x is 40x^(-2).
compute lim t -> 2 (t ^ 2 + 3t - 10)/(t ^ 2 - 2t)
To compute the limit as t approaches 2 of the expression (t^2 + 3t - 10)/(t^2 - 2t), we can directly evaluate the expression at t = 2, and the denominator is not equal to zero at t = 2:
(t^2 + 3t - 10)/(t^2 - 2t) = (2^2 + 3(2) - 10)/(2^2 - 2(2))
= (4 + 6 - 10)/(4 - 4)
= 0/0 [indeterminate form]
Since this is an indeterminate form, we need to further simplify the expression before evaluating the limit.
Factoring the numerator and denominator:
(t^2 + 3t - 10)/(t^2 - 2t) = ((t - 2)(t + 5))/(t(t - 2))
Now we can cancel out the common factor (t - 2):
((t - 2)(t + 5))/(t(t - 2)) = (t + 5)/t
Now, we can evaluate the limit as t approaches 2:
lim t -> 2 (t^2 + 3t - 10)/(t^2 - 2t) = lim t -> 2 (t + 5)/t
Substituting t = 2:
lim t -> 2 (t + 5)/t = lim t -> 2 (2 + 5)/2 = 7/2
Therefore, the limit as t approaches 2 of the expression (t^2 + 3t - 10)/(t^2 - 2t) is 7/2.
To find dy/dt in terms of x, we need to use the chain rule of differentiation. First, let's differentiate the equation y = 2 - 4/x with respect to t on both sides:
d/dt(y) = d/dt(2 - 4/x)
Now, according to the chain rule, d/dt(f(g(t))) = d(f(g(t)))/dx * d(g(t))/dt. Applying this to our equation, we have:
dy/dt = d(2 - 4/x)/dx * dx/dt
Now, let's differentiate (2 - 4/x) with respect to x:
d(2 - 4/x)/dx = 0 - (-4/x^2)
Simplifying this, we have:
d(2 - 4/x)/dx = 4/x^2
Substituting this back into the equation for dy/dt:
dy/dt = (4/x^2) * dx/dt
We are given that dx/dt = 10, so:
dy/dt = (4/x^2) * 10
Finally, we can simplify this expression further if desired.