Velocity and Other Rates of Change Chapter 3.4. Instantaneous Rates of Change 2 *.

Velocity and Other Rates of Change

Chapter 3.4

*https://en.wikipedia.org/wiki/Archetype 2

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 concept• An archetype is “a statement, pattern of behavior, or prototype which other

statements, patterns of behavior, and objects copy or emulate”*

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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 data• The derivative of a function from a verbal description of the data

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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

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Instantaneous Velocity

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

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• Suppose an object is moving along a straight line, say the horizontal -axis, such that we know its position on the axis at any time (• The displacement of the object over the time interval from to is

• The average velocity of the object over the time interval is

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• To find the velocity of the object at an exact instant , we take the limit of the average velocity as , which is just the derivative• Note 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.)

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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

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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 downward• Horizontally, velocity is positive when movement is to the right and

negative when movement is to the left• The speed of an object is the value of the velocity without regard to

direction

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Speed

DEFINITION:Speed is the absolute value of velocity

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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?

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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.

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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 acceleration• In 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.a) How high does the rock go?b) 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?c) What is the acceleration of the rock at any time during the flight?d) When does the rock hit the ground?

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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.a) 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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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.b) 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/secOn the way down the velocity is feet per second; speed 96 ft/sec

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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.c) 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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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.d) 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.a) Find the displacement of the particle during the first 2 seconds.b) Find the average velocity of the particle during the first 4 seconds.c) Find the instantaneous velocity of the particle when .d) Find the acceleration of the particle when .e) Describe the motion of the particle. At what values of does the

particle change directions?

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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.a) 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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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.b) Find the average velocity of the particle during the first 4 seconds.By definition

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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.c) Find the instantaneous velocity of the particle when .Instantaneous velocity is the derivative of at

so

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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.d) Find the acceleration of the particle when .By definition

The acceleration is constant at 2 meters per second per second

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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.e) 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 1: Enlarging Circles

a) Find the rate of change of the area of a circle with respect to its radius .

b) Evaluate the rate of change of at and at

c) If is measured in inches and is measured in square inches, what units would be appropriate for ?

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a) 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

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b) 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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c) 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 produced• The 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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• 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

items• The average cost of producing one more item is thus

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• 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

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)?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• 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 it

• 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?

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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?The average density for the interval is

The linear density at meters is

So

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Exercise 3.4

Velocity and Other Rates of Change Chapter 3.4. Instantaneous Rates of Change 2 *.

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