is the rate of change of the velocity, that is, the derivative of velocity

the amount of a function $f (x)$ over an interval $[x, x + h]$ is $f (x + h) - f (x)$

average rate of change

is a function $f (x)$ over an interval $[x, x + h]$ is $\frac{f (a + h) - f (a)}{h}$

is the derivative of the cost function, or the approximate cost of producing one more item

is the derivative of the revenue function, or the approximate revenue obtained by selling one more item

is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item

population growth rate

is the derivative of the population with respect to time

is the absolute value of velocity, that is, $| v (t) |$ is the speed of an object at time $t$ whose velocity is given by $v (t)$

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Calculus Volume 1 · 3.4 Derivatives as Rates of Change

Derivatives as Rates of Change: practice problems

14 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 151
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
$s (t) = 2 t^{3} - 15 t^{2} + 36 t - 10$
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Exercises 153
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
A model rocket is fired vertically upward from the ground. The distance $s$ in feet that the rocket travels from the ground after $t$ seconds is given by $s (t) = - 16 t^{2} + 560 t .$
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Exercises 155
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
The position function $s (t) = t^{2} - 3 t - 4$ represents the position of the back of a car backing out of a driveway and then driving in a straight line, where $s$ is in feet and $t$ is in seconds. In this case, represents the time at which the back of the car is at the garage door, so is the starting position of the car, 4 feet inside the garage.
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Exercises 157
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The position of the potato relative to the ground after $t$ seconds is given by $s (t) = - 16 t^{2} + 100 t + 85.$
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Exercises 159
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
The following graph shows the position $y = s (t)$ of an object moving along a straight line.
Use the graph of the position function to determine the time intervals when the velocity is positive, negative, or zero.
Sketch the graph of the velocity function.
Use the graph of the velocity function to determine the time intervals when the acceleration is positive, negative, or zero.
Determine the time intervals when the object is speeding up or slowing down.
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Exercises 161
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
The price $p$ (in dollars) and the demand $x$ for a certain digital clock radio is given by the price–demand function $p = 10 - 0.001 x .$
Find the revenue function $R (x) .$
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Exercises 163
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
[T] In general, the profit function is the difference between the revenue and cost functions: $P (x) = R (x) - C (x) .$
Suppose the price-demand and cost functions for the production of cordless drills is given respectively by $p = 143 - 0.03 x$ and where is the number of cordless drills that are sold at a price of dollars per drill and is the cost of producing cordless drills.
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Exercises 165
For the following exercises, the given functions represent the position of a particle traveling along a horizontal line; $t \geq 0$ .
Find the velocity and acceleration functions.
Determine the time intervals when the object is slowing down or speeding up.
[T] A culture of bacteria grows in number according to the function $N (t) = 3000 (1 + \frac{4 t}{t ^{2} + 100}),$ where is measured in hours.
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Exercises 167
The following questions concern the population (in millions) of London by decade in the 19th century, which is listed in the following table.
Source: http://en.wikipedia.org/wiki/Demographics_of_London.
Years since 1800 Population (millions)
1 0.8795
11 1.040
21 1.264
31 1.516
41 1.661
51 2.000
61 2.634
71 3.272
81 3.911
91 4.422
[T]
Using a calculator or a computer program, find the best-fit linear function to measure the population.
Find the derivative of the equation in a. and explain its physical meaning.
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Exercises 169
For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table.
Time after dropping (s) Position (m)
0 0
1 −1
2 −2
3 −5
4 −7
5 −14
[T]
Using a calculator or computer program, find the best-fit quadratic curve to the data.
Find the derivative of the position function and explain its physical meaning.
Find the second derivative of the position function and explain its physical meaning.
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Exercises 171
The following problems deal with the Holling type I, II, and III equations. These equations describe the ecological event of growth of a predator population given the amount of prey available for consumption.
[T] The Holling type I equation is described by $f (x) = a x,$ where $x$ is the amount of prey available and $a > 0$ is the rate at which the predator meets the prey for consumption.
Graph the Holling type I equation, given $a = 0.5.$
Determine the first derivative of the Holling type I equation and explain physically what the derivative implies.
Determine the second derivative of the Holling type I equation and explain physically what the derivative implies.
Using the interpretations from b. and c. explain why the Holling type I equation may not be realistic.
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Exercises 173
The following problems deal with the Holling type I, II, and III equations. These equations describe the ecological event of growth of a predator population given the amount of prey available for consumption.
[T] The Holling type III equation is described by $f (x) = \frac{a x ^{2}}{n ^{2} + x ^{2}},$ where is the amount of prey available and is the maximum consumption rate of the predator.
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Review Exercises 389
The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by $w (t) = 1.9 + 2.9 cos (\frac{π}{6} t),$ where t is measured in hours after midnight, and the height is measured in feet.
Find $w^{'} (3) .$ What is the physical meaning of this value?
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Review Exercises 391
The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.
Source: http://news.nationalgeographic.com/news/2005/09/0914_050914_katrina_timeline.html.
Hours after Midnight, August 26 Wind Speed (mph)
1 45
5 75
11 100
29 115
49 145
58 175
73 155
81 125
85 95
107 35
Estimate the derivative of the wind speed at hour 83. What is the physical meaning?
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Key terms · 8

acceleration: is the rate of change of the velocity, that is, the derivative of velocity
amount of change: the amount of a function $f (x)$ over an interval $[x, x + h]$ is $f (x + h) - f (x)$
average rate of change: is a function $f (x)$ over an interval $[x, x + h]$ is $\frac{f (a + h) - f (a)}{h}$
marginal cost: is the derivative of the cost function, or the approximate cost of producing one more item
marginal revenue: is the derivative of the revenue function, or the approximate revenue obtained by selling one more item
marginal profit: is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item
population growth rate: is the derivative of the population with respect to time
speed: is the absolute value of velocity, that is, $| v (t) |$ is the speed of an object at time $t$ whose velocity is given by $v (t)$

Learn 3.4 on the whiteboard

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Read 3.4 in the textbook · This section on openstax.org

Derivatives as Rates of Change: practice problems

14 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 151