4. Applications of Derivatives

If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 2?

Assume that y = 5x and dx/dt = 2. Find dy/dt

If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?

A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)

Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

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 $5\operatorname{\mathrm{cm}}$?

A right tringle has a base of $10\operatorname{\mathrm{cm}}$ and a height of $12\operatorname{\mathrm{cm}}$. The height of the right triangle is decreasing at a rate of $0.4\operatorname{\frac{\mathrm{cm}}{s}}$, at what rate is the area of the triangle decreasing?

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.

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)?

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 $3pm$.

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

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

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

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

Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.

c. Find the elasticity function on the domain of the demand function.

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

c. Find the rate at which water flows from the tank and plot the flow rate function. 

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.

b. Graph the volume function and verify that the tank is empty when t=1000 min.