# Calculus 1 : How to find rate of change

## Example Questions

### Example Question #2921 : Calculus

Find the rate of change of a function  from  to .

Explanation:

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

.

Solving for  at our given points:

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

### Example Question #12 : How To Find Rate Of Change

At what time  does the function  have a slope of ? Round to the nearest hundredth.

Explanation:

First, we want the slope, so we have to take a derivative of . We will need to use the power rule which is,

on the first term. We need to also recall that the derivative of  is.

Applying these rules we get the following derivative.

We're looking for the time the slope is , so we have to set the derivative (which gives you slope) equal to

.

At this point you can use a graphing calculator to graph the function , and trace the graph to find the x value that results in a y value of . The positive solution rounded to the nearest hundredth is .

### Example Question #13 : How To Find Rate Of Change

A rectangle has a length of four feet and a width of six feet. If the width of the rectangle increases at a rate of , how fast is the area of the rectangle increasing?

The area of the rectangle does not change.

Explanation:

In this problem we are given the length and width of a rectangle as well as the rate at which the width is increasing. We are asked to find the rate of change of the area of a rectangle. The equation for finding the area of a rectangle is given as

.

By taking the derivative of this equation with respect to time, we can find how the area changes with respect to time.  To take the derivative of an equation with two variables, we must use the product rule,

.

Applying the product rule to the equation we obtain

.

Because the width of the rectangle is increases at a rate of

Since the length of the rectangle does not change with respect to time, .

and  are given to us as 4 feet and 6 feet respectively .

Therefore the area of this rectangle changes at a rate of  when the width of the rectangle is increasing by .

### Example Question #11 : How To Find Rate Of Change

Find the rate of change of a function  from  to .

Explanation:

We can solve by utilizing the formula for the average rate of change: .  Solving for  at our given points:

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

.

### Example Question #15 : How To Find Rate Of Change

Find the rate of change of a function  from  to .

Explanation:

We can solve by utilizing the formula for the average rate of change: .  Solving for  at our given points:

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

.

### Example Question #16 : How To Find Rate Of Change

Find the rate of change of a function  from  to .

Explanation:

We can solve by utilizing the formula for the average rate of change: .  Solving for  at our given points:

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

.

### Example Question #17 : How To Find Rate Of Change

You are looking at a balloon that is   away. If the height of the balloon is increasing at a rate of  , at what rate is the angle of inclination of your position to the balloon increasing after  seconds?

Explanation:

Using right triangles we know that

.

Solving for  we get

.

Taking the derivative, we need to remember to apply the chain rule to  since the height depends on time,

.

We are asked to find . We are given and since  is constant, we know that the height of the balloon is given by .

Therefore, at  we know that the height of the balloon is  .

Plugging these numbers into  we find

### Example Question #18 : How To Find Rate Of Change

Boat  leaves a port at noon traveling  . At the same time, boat  leaves the port traveling east at  . At what rate is the distance between the two boats changing at ?

Explanation:

This scenario describes a right triangle where the hypotenuse is the distance between the two boats. Let  denote the distance boat  is from the port,  denote the distance boat  is from the port,  denote the distance between the two boats, and  denote the time since they left the port. Applying the Pythagorean Theorem we have,

.

Implicitly differentiating this equation we get

.

We need to find  when .

We are given

which tells us

Plugging this in we have

.

Solving we get

.

### Example Question #2931 : Calculus

Find  if the radius of a spherical balloon is increasing at a rate of  per second.

Explanation:

The volume function, in terms of a radius , is given as

.

The change in volume over the change in time, or

is given as

and by implicit differentiation, the chain rule, and the power rule,

.

Setting  we get

.

As such,

.

### Example Question #20 : How To Find Rate Of Change

Find the rate of change of a function  from  to .