Velocity and Other Rates of Change - PowerPoint Presentation

Slide1

Velocity and Other Rates of Change

Chapter 3.4Slide2

Instantaneous Rates of Change

An important goal for this course is to understand, not only how to find derivatives, for example, but also to understand the derivative as a concept

It’s easy to know how to find the derivative of, say,

; it’s more challenging and more useful to also know how to interpret the derivative of (in general and in specific instances)In this section we will develop the concept of instantaneous rate of change by first looking at position, velocity, and acceleration as archetypes of this conceptAn archetype is “a statement, pattern of behavior, or prototype which other statements, patterns of behavior, and objects copy or emulate”*

*https://en.wikipedia.org/wiki/Archetype

2Slide3

Instantaneous Rates of Change

By “understanding instantaneous rate of change conceptually” is meant that you are able to understand and interpret

The derivative of a function analytically (i.e., when you take a derivative)

The derivative of a function from its graph (or the function from a graph of its derivative)The derivative of a function from a table of dataThe derivative of a function from a verbal description of the data3Slide4

Instantaneous Rate of Change

We earlier defined instantaneous rate of change of a function

at some

as

This is the same as finding the derivative of

at a function, then evaluating the derivative at

The result tells us how the function value is changing with respect to

at

It is also the same as the slope of the tangent line at

4Slide5

Instantaneous Velocity

DEFINITION:

The

instantaneous velocity is the derivative of the position function with respect to time, . At time the instantaneous velocity is

5Slide6

Displacement and Instantaneous Velocity

DEFINITIONS:

Let

be a position function. Then:The displacement of a moving object over the time interval from to is

The

average velocity

of the object over the time interval is

6Slide7

Instantaneous Velocity

Comparison: if

is a position function, then

The displacement over a time interval from to is

The

average velocity

over a time interval from

to

is

The instantaneous velocity

at a time

is

7Slide8

Instantaneous Velocity

To find the velocity of the object at an exact instant

, we take the limit of the average velocity as

, which is just the derivativeNote that the units for are units of position (inches, feet, meters, miles, and so on)The units for are units of time (seconds, minutes, hours, days, and so on)

So the units for both average velocity and instantaneous velocity are units of position per unit of time (feet per second, miles per hour, etc.)

8Slide9

Example 2: Vertical Motion

Suppose that an object is thrown vertically from initial position

feet with initial velocity

feet per second. Since the acceleration due to gravity is approximately feet per second per second, then the position function for the object is

. What is the average velocity over the interval

? What is the instantaneous velocity at

seconds?

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Example 2: Vertical Motion

10Slide11

Speed

Velocity is a vector quantity, meaning that it has both magnitude and direction

When movement is either vertical or horizontal (with respect to some axes), then velocity is either positive or negative

With our usual axes, velocity is positive when movement is upward and negative when movement is downwardHorizontally, velocity is positive when movement is to the right and negative when movement is to the leftThe speed of an object is the value of the velocity without regard to direction11Slide12

Speed

DEFINITION:

Speed

is the absolute value of velocity

12Slide13

Example 3: Reading a Velocity Graph

A student walks around in front of a motion detector that records her velocity at 1-second intervals for 36 seconds. She stores the data in her graphing calculator and uses it to generate the time-velocity graph shown below. Describe her motion as a function of time by reading the velocity graph. When is her

speed

a maximum?13Slide14

Example 3: Reading a Velocity Graph

She walks forward (away from the detector) for the first 14 seconds, moves backward for the next 12 seconds, stands still for 6 seconds, and then moves forward again. Her maximum speed occurs at about 20 seconds, while walking backward.

14Slide15

Acceleration

DEFINITION:

Acceleration

is the derivative of velocity with respect to time. If a body’s velocity at time is , then the body’s acceleration at time

is

or

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Acceleration Due to Gravity

The Earth’s gravity “pulls” objects towards its center with a constant (near the surface) acceleration

The variable

is often used to represent this accelerationIn English units: feet per second per second;

In metric units:

meters per second per second;

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Example 4: Modeling Vertical Motion

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 feet per second. It reaches a height of

after

seconds.How high does the rock go?

What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? on the way down?What is the acceleration of the rock at any time

during the flight?When does the rock hit the ground?

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Example 4: Modeling Vertical Motion

How high does the rock go?

You should recognize that the position function is a quadratic function and its graph is a parabola opening down. This means that it must have a maximum value which occurs at the vertex and we could use the vertex formula to find the time at which it reaches its maximum height. But it is also the case that, at this instant, the rock has zero velocity. So using calculus we can find the velocity function:

Now,

seconds

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Example 4: Modeling Vertical Motion

What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? on the way down?

Find the times when the rock is 256 feet above the ground by solving for

:

On the way up the velocity is

feet per second; speed 96 ft/sec

On the way down the velocity is

feet per second; speed 96 ft/sec

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Example 4: Modeling Vertical Motion

What is the acceleration of the rock at any time

during the flight?

Acceleration is the second derivative of the position function:

The acceleration is a constant 32 feet per second per second towards the Earth.

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Example 4: Modeling Vertical Motion

When does the rock hit the ground?

At ground level, the position function equals zero (and this must occur twice).

The rock returns to the ground after 10 seconds.

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Example 5: Studying Particle Motion

A particle moves along the

-axis so that its position at any time

is given by the function

, where

is measured in meters and

is measured in seconds.

Find the displacement of the particle during the first 2 seconds.

Find the average velocity of the particle during the first 4 seconds.

Find the instantaneous velocity of the particle when

.

Find the acceleration of the particle when .Describe the motion of the particle. At what values of does the particle change directions? 22Slide23

Example 5: Studying Particle Motion

Find the displacement of the particle during the first 2 seconds.

We have defined displacement to be the value

, where

for this problem. So the displacement is

Note that displacement gives the distance and direction (by means of the sign) that the particle moved

from its start position

. A separate but related question that we will encounter later is, what is its new position on the

-axis?

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Example 5: Studying Particle Motion

Find the average velocity of the particle during the first 4 seconds.

By definition

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Example 5: Studying Particle Motion

Find the instantaneous velocity of the particle when

.

Instantaneous velocity is the derivative of at

so

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Example 5: Studying Particle Motion

Find the acceleration of the particle when

.

By definition

The acceleration is constant at 2 meters per second per second

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Example 5: Studying Particle Motion

Describe the motion of the particle. At what values of

does the particle change directions?

The graph of is a parabola that opens up. So the particle will move downward until it reaches the vertex point, then it will continue upward. At the vertex, it will have stopped moving for an instant, so its velocity there must be zero. We can find the vertex point by solving for in the velocity function:

So for

, the particle moves to the left, and for

it moves right.

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Example 6: Enlarging Circles

Find the rate of change of the area

of a circle with respect to its radius

.Evaluate the rate of change of at and at

If is measured in inches and

is measured in square inches, what units would be appropriate for

?

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Example 6: Enlarging Circles

Find the rate of change of the area

of a circle with respect to its radius

.The wording above is important: you are being asked to find the derivative of with respect to , or

29Slide30

Example 6: Enlarging Circles

Evaluate the rate of change of

at

and at Find and

Note that the rate of change increases as the radius increases. The same change in

produces a correspondingly greater change in

. That is, the area is increasing much faster than the radius.

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Example 6: Enlarging Circles

31Slide32

Example 6: Enlarging Circles

32Slide33

Example 6: Enlarging Circles

If

is measured in inches and

is measured in square inches, what units would be appropriate for ?Notice the analogy here to velocity versus position. That is, given a position function, the independent values are units of time while the dependent (function) values are those of distance. So the units of the derivative are units of distance per unit of time. By analogy in this case, the independent values are those of length while the dependent (function) values are of area. So the units for the derivative are square inches of area per inch of radius.

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Derivatives in Economics

The previous example showed how we can use the relationship between position and velocity to understand the relationship between area and radius in an expanding circle

What follows next is a similar analogy, this time in economics

In a manufacturing operation, the cost of production, , is a function of , the number of units producedThe marginal cost of production is the rate of change of cost with respect to level of production (that is, with respect to the number of units produced)What should be the units of marginal cost of production?

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Derivatives in Economics

Let’s develop the idea of marginal cost in the same way we developed instantaneous velocity (we could call marginal cost “instantaneous cost”)

Economists often think of marginal cost as the cost of producing one more item (because individual items are not infinitely divisible)

So, if is the cost of producing items, then is the cost of producing

itemsThe average cost of producing one more item is thus

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Derivatives in Economics

The average cost of producing one more item is thus

Marginal cost is actually a way of approximating the producing of “one more item” by using the derivative of

when

is much larger than 1

That is,

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Example 7: Derivatives in Economics

Suppose a company has estimated that the cost (in dollars) of producing

items is

. What is the marginal cost of producing 500 items? What is the actual cost of producing 1 more item (i.e., the 501st item)?

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Example 7: Derivatives in Economics

Since marginal cost is the derivative of the cost function, then

The cost of producing 1 more item is

We see that the marginal cost is close to the actual cost.

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Linear Density

If a rod is homogeneous, then its mass is distributed equally along its length

Its linear density is uniform and defined as mass per unit length,

, and the units are kilograms per meter (the symbol is the Greek letter rho)If the rod is not homogeneous, then its mass changes along its length (think of a conical rod) and the mass from its left end to some point is

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Linear Density

If we pick two points along the rod at distances

and

from the left (with

, then the average density for this section of the rod is

If we now let

(that is, let

), then we can find the linear density at

Note that the units are still in mass per unit distance from the left

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Example 8: Linear Density

Suppose that the mass of a rod varies with distance from its left end such that

, where

is in meters and

is kilograms. What is the average density of the rod for

? What is the density at

meter?

41Slide42

Example 8: Linear Density

The

average density for the interval

is

The linear density at

meters is

So

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Exercise 3.4

Online exercise

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