We can solve by utilizing the formula for the average rate of change: .

Solving for f(x) at our given points:

Plugging our values into the average rate of change formula, we get:

.

The volume of a sphere is given by 

 , where r is the radius of the sphere.

The surface area of a sphere is given by  (r is the radius of the sphere). The rate of change of the volume of the sphere is found by taking the first derivative of the function for volume: 

, where  is the change of the radius with respect to time.

This derivative was found by using the power rule 

. 

The rate of change of the surface area of the sphere is found the same way, instead using the function for surface area: 

 .

The balloon's volume is increasing at a constant rate of 0.1 cubic meters per second when the radius of the balloon is 1 meter, thus plugging in these values into the  equation gives us the following: 

. Solving for , we get  meters per second. Since we are solving for the rate of change of surface area of the balloon, we plug this value into its equation along with the radius we want (1 m). Solving for the rate of change of the surface area we get: 

 square meters per second.

First, find the rate of increase in the radius :

Next, take the derivative of the area of a circle () by using the Power Rule ():

Lastly, plug in the givens and simplify:

The average rate of change of a function f(x) on an interval [a,b] is given by

 while the instantaneous rate of change at a is given by

 .

Finding each of these on the interval [-2, 2] for the equations given: 

We can see that the two rates are equal for f(x) and g(x), but not h(x).

To evaluate, first take the derivatives of both sides of the equation. This gives

By factoring and simple algebraic simplification, the equation becomes

The volume of a sphere is given by 

To find the change in volume, differentiate the volume with respect to time.

Simply plug in the values given in the problem to find the rate of change of the volume.

To calculate  using implicit differentiation, we differentiate everything in the relation  and solve for :

So we have .

If b(t) models the rate at which a library buys and sells books, at what rate was the library gaining or losing books when ?

We need to find an instantaneous rate of change. Rates of change can be found with derivatives. Start by finding b'(t), and then using our given t value to find our final answer:

So the library was gaining 195.76 books per day.

To solve this problem, first understand what it's asking. When it talks about the distance between the center of the basketball and the point of connection, it's talking about a radius, and when it asks about the movement of this point, it's talking about a change in the radius.

First, write an equation for the radius with respect to the volume:

Afterwards, derive each side with respect to time, and plug in known values.

Another approach is to derive the original volume equation:

However, since , this also reduces to:

A cylinder's volume is defined in terms of its radius and height as:

Therefore, it's rate of volume expansion is found by taking the derivative of each side with respect to time:

Since there's zero growth in height, the second term goes to zero, and we're left with:

What an unnatural cylinder.