4. Applications of Derivatives

Given $x=e^t$ and $y=t{e}^{-t}$, which of the following is the correct value of $\frac{dy}{dx}$ at $t=1$?

Two cars leave the same intersection and drive in perpendicular directions. Car A travels east at a speed of $60\(\frac{mi}{hr}\)$, Car B travels north at a speed of $40\(\frac{mi}{hr}\)$. Car A leaves the intersection at $2pm$, while Car B leaves at $2:30pm$. Determine the rate at which the distance between the two cars is changing at $3pm$.

Which of the following gives the equations of both lines through the point $\left(2,,,,-,3\right)$ that are tangent to the parabola $y=x^2+x$?

A 15-foot plank leans against a vertical pole. The top of the plank begins to slide down the pole at a steady speed of 2 inches per second. How fast is the bottom of the plank moving away from the pole when it is 8 feet away from the base of the pole (in inches per second)?

Given the equation $x+3y{y}^{13}$=$y$, what is $\frac{dy}{dx}$ at the point $(2,8)$?

The perimeter of a rectangle is fixed at . If the length $L$ is increasing at a rate of , for what value of $L$ does the area start to decrease? Hint: the rectangle's area starts to decrease when the rate of change for the area is less than 0.

Given the equation below, find $dy/dt$ when $dx/dt=12$ and $x=\frac{15}{2}$.

$y=\sqrt{2x+1}$

Given the equation $e^x-y=x{y}^3+e^2-18$, what is the value of $\frac{dy}{dx}$ at the point $(2,2)$?

Given the equation below, find $dz/dt$ when $dx/dt=3$, $dy/dt=2$, $x=2$, $y=4,$ and $z=1$.

$x^2+y^2-3z^2=125$

A right triangle has a base of and a height of . The height of the right triangle is decreasing at a rate of , at what rate is the area of the triangle decreasing?

A sphere is growing at a rate of $50\(\frac{\operatorname{\mathrm{cm}\)}^3}{s}$. At what rate is the radius of the sphere increasing when the radius is $5\(\operatorname{\mathrm{cm}\)}$?