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Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25

UNIT 2 Differentiation: Definition and Fundamental Properties Chapters align with the AP ®
Calculus units.
Every chapter boldly displays the book’s
CHAPTER
2 The Derivative and chapter title as well as the unit title in

the AP ® course description. This way, you
know that the book covers the material
Its Properties
that you need to know to prepare for
the exam.

2.1 Rates of Change
and the Derivative
2.2 The Derivative as a
Function;
Differentiability Immediate engagement.
2.3 The Derivative of a Each chapter opens with a brief story
Polynomial Function; that relates the calculus covered in it
The Derivative of
x
y = e and y = ln x to real-world scenarios in fields such as
2.4 Differentiating the biology, engineering, environmental
Product and the
Quotient of Two sciences, technology, and space travel.
Functions;
Higher-Order
Derivatives
Michael Collins, Apollo 11, NASA
2.5 The Derivative of the The project in Chapter 2 is an
Trigonometric The Apollo Lunar Module
exploration of how physics is
Functions “One Giant Leap for Mankind” apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25
Sullivan
04
Chapter Project used to maneuver the Lunar
On May 25, 1961, in a special address to Congress, U.S. president John F. Kennedy
Chapter Review proposed the goal “before this decade is out, of landing a man on the Moon and returning Module.
AP Review Problems: him safely to the Earth.” Roughly eight years later, on July 16, 1969, a Saturn V rocket
R
Chapter 2 launched from the Kennedy Space Center in Florida, carrying the Apollo 11 spacecraft and Chapter 2 • Chapter Review 227
three astronauts—Neil Armstrong, Buzz Aldrin, and Michael Collins—bound for the Moon.
AP Cumulative Review The Apollo spacecraft had three parts: the Command Module with a cabin for the
R
spacecraft had three parts: the Command Module with a cabin for the
Problems: Chapters three astronauts; the Service Module that supported the Command Module with
three astronauts; the Service Module that supported the Command Module with
1–2 propulsion, electrical power, oxygen, and water; and the Lunar Module for landing on the The Apollo Lunar Module
CHAPTER 2 PROJECT
propulsion, electrical power, oxygen, and water; and the Lunar Module for landing on the
Moon. After its launch, the spacecraft traveled for three days until it entered into lunar
Moon. After its launch, the spacecraft traveled for three days until it entered into lunar
orbit. Armstrong and Aldrin then moved into the Lunar Module, which they landed in the
orbit. Armstrong and Aldrin then moved into the Lunar Module, which they landed in the
flat expanse of the Sea of Tranquility. After more than 21 hours, the first humans to touch This Project may be done 2. What is the reference acceleration a ref (t)?
flat expanse of the Sea of Tranquility. After more than 21 hours, the first humans to touch
Michael Collins, Apollo 11, NASA
the surface of the Moon crawled into the Lunar Module and lifted off to rejoin the individually or as part of a team.
the surface of the Moon crawled into the Lunar Module and lifted off to rejoin the
Command Module, which Collins had been piloting in lunar orbit. The three astronauts
Command Module, which Collins had been piloting in lunar orbit. The three astronauts 3. The rate of change of acceleration is called jerk. Find the
then headed back to Earth, where they splashed down in the Pacific Ocean on July 24. The Lunar Module (LM) was a reference jerk J ref (t).
then headed back to Earth, where they splashed down in the Pacific Ocean on July 24.
In 2022, NASA initiated the Artemis I program, designed to pave the way for the first small spacecraft that detached from
In 2022, NASA initiated the Artemis I program, designed to pave the way for the first
crewed Orion mission and eventually for the return of NASA astronauts to the surface of of
crewed Orion mission and eventually for the return of NASA astronauts to the surface 4. The rate of change of jerk is called snap. Find the reference
the Moon and then on to Mars in the 2030s. the Apollo Command Module and snap S ref (t).
Explore some of the physics at work that allowed engineers and pilots to successfully maneuver the
Explore some of the physics a t work tha t allowed engineers and pilots to successfully maneuver the was designed to land on the Moon. 5. Evaluate r ref (t), v ref (t), a ref (t), J ref (t), and S ref (t) when t = 0.
Lunar Module to the Moon’s surface in the Chapter 2 Project on page 227. Fast and accurate computations
Chapter 2 Project on page 227.
were needed to bring the LM from an orbiting speed of about But small variations in propulsion, mass, and countless other
5500 ft/s to a speed slow enough to land it within a few feet of a variables cause the LM to deviate from the predetermined path. To
designated target on the Moon’s surface. The LM carried a 70-lb correct the LM’s position and velocity, NASA engineers apply a
computer to assist in guiding it successfully to its target. The force to the LM using rocket thrusters. That is, they changed the
approach to the target was split into three phases, each of which acceleration. (Remember Newton’s second law, F = ma.)
followed a reference trajectory specified by NASA engineers. ∗ Engineers modeled the actual trajectory of the LM by
The position and velocity of the LM were monitored by sensors that
The end-of-chapter project, suitable for tracked its deviation from the preassigned path at each moment.
group or individual work, takes over where Whenever the LM strayed from the reference trajectory, control r(t) = R T + V T t + 1 A T t + 1 J A t + 1 S A t 4 (2)
2
3
the opening story leaves off. Answer the thrusters were fired to reposition it. In other words, the LM’s 2 6 24
questions to shed light on how the calculus position and velocity were adjusted by changing its acceleration. We know the target parameters for position, velocity, and
techniques you have learned can be applied The reference trajectory for each phase was specified by the acceleration. We need to find the actual parameters for jerk and
in these different fields. engineers to have the form snap to know the proper force (acceleration) to apply.
1 2 1 3 1 4 6. Find the actual velocity v = v(t) of the LM.
r ref (t) = R T + V T t + A T t + J T t + S T t (1)
2 6 24 7. Find the actual acceleration a = a(t) of the LM.
The reference trajectory given in equation (1) is a fourth-degree
xx © 2024 BFW Publishers PAGES NOT FINAL 8. Use equation (2) and the actual velocity found in Problem 6 to
express J A and S A in terms of R T , V T , A T , r(t), and v(t).
polynomial, the lowest degree polynomial that has enough free
For Review Purposes Only, all other uses prohibited 9. Use the results of Problems 7 and 8 to express the actual
parameters to satisfy all the mission criteria. Now we see that the
Do Not Copy or Post in Any Form. acceleration a = a(t) in terms of R T , V T ,A T , r(t), and v(t).
parameters R T =r ref (0), V T = v ref (0), A T = a ref (0), J T = J ref (0),
and S T = S ref (0). The five parameters in equation (1) are referred to
as the target parameters, since they provide the path the LM The result found in Problem 9 provides the acceleration (force)
01_apcalc4e_45342_fm_i_xxix_3pp.indd 20 10/11/23 2:38 PM
should follow. required to keep the LM in its reference trajectory.
The variable r ref in (1) represents the intended position of the LM 10. When riding in an elevator, the sensation one feels just before
at time t before the end of the landing phase. The engineers the elevator stops at a floor is jerk. Would you want jerk to be
specified the end of the landing phase to take place at t = 0, so that small or large in an elevator? Explain. Would you want jerk to
during the phase, t was always negative. Note that the LM was be small or large on a roller coaster ride? Explain. How would
landing in three dimensions, so there were actually three equations you explain snap?
like (1). Since each of those equations had this same form, we will
work in one dimension, assuming, for example, that r represents the
distance of the LM above the surface of the Moon.
∗ A. R. Klumpp, “Apollo Lunar-Descent Guidance,” MIT Charles Stark
1. If the LM follows the reference trajectory, what is the reference Draper Laboratory, R-695, June 1971,
velocity v ref (t)? http://www.hq.nasa.gov/alsj/ApolloDescentGuidnce.pdf
Chapter Review
THINGS TO KNOW
2.1 Rates of Change and the Derivative • Rate of change of a function If y = f (x), the derivative f (c)
�
is the rate of change of f with respect to x at c. (p. 173)
• Definition Derivative of a function f at a number c
• Physical If the signed distance s from the origin at time t
of an object moving on a line is given by the position
f (x) − f (c)
�
�
Form (1) f (c) = lim function s = f (t), the derivative f (t 0 ) is the velocity of the
x→c x − c object at time t 0 . (p. 173)
provided the limit exists. (p. 173)
2.2 The Derivative as a Function
Three Interpretations of the Derivative • Definition of a derivative function
• Geometric If y = f (x), the derivative f (c) f (x + h) − f (x)
�
is the slope of the tangent line to the graph of f at the Form (2) f (x) = lim
�
point (c, f (c)). (p. 173) h→0 h
provided the limit exists. (p. 179)