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.

Answer 1

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).

Answer 2

compute lim t -> 2 (t ^ 2 + 3t - 10)/(t ^ 2 - 2t)

Answer 3

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.