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04 apcalc4e 45342 ch02 166 233 3pp
June 19, 2023
Sullivan
Chapter 2 • Chapter Review
The Apollo Lunar Module
CHAPTER 2 PROJECT
This Project may be done
2.
What is the reference acceleration a ref (t)?
Michael Collins, Apollo 11, NASA
individually or as part of a team.
3.
The rate of change of acceleration is called jerk. Find the
The Lunar Module (LM) was a
reference jerk J ref (t).
small spacecraft that detached from
4.
The rate of change of jerk is called snap. Find the reference
the Apollo Command Module and
snap S ref (t).
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.
Fast and accurate computations
were needed to bring the LM from an orbiting speed of about
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
tracked its deviation from the preassigned path at each moment. But small variations in propulsion, mass, and countless other 227
1
1
1
2
3
Whenever the LM strayed from the reference trajectory, control r(t) = R T + V T t + 2 A T t + 6 J A t + 24 S A t 4 (2)
thrusters were fired to reposition it. In other words, the LM’s
position and velocity were adjusted by changing its acceleration. We know the target parameters for position, velocity, and
9:25
04 apcalc4e 45342 ch02 166 233 3pp
June 19, 2023
Sullivan
The reference trajectory for each phase was specified by the acceleration. We need to find the actual parameters for jerk and
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 228 (1) Chapter 2 • The Derivative and Its Properties
2 6 24 7. Find the actual acceleration a = a(t) of the LM.
Use equation (2) and the actual velocity found in Problem 6 to
8.
The reference trajectory given in equation (1) is a fourth-degree If a function f has a derivative at a number c,
• Theorem
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 then f is continuous at c. (p. 184) d f g − f d g
Review, Review, Review: Gage your progress. dx g f = dx g 2 dx
d
parameters to satisfy all the mission criteria. Now we see that the
Use the results of Problems 7 and 8 to express the actual
9.
• Quotient Rule
If a function f is discontinuous at a number c,
• Corollary
parameters R T =r ref (0), V T = v ref (0), A T = a ref (0), J T = J ref (0),
acceleration a = a(t) in terms of R T , V T ,A T , r(t), and v(t).
then f has no derivative at c. (p. 185)
and S T = S ref (0). The five parameters in equation (1) are referred to (p. 206) �
�
as the target parameters, since they provide the path the LM The result found in Problem 9 provides the acceleration (force) f = f g − fg �
2.3 The Derivative of a Polynomial Function;
required to keep the LM in its reference trajectory.
should follow. The Derivative of y = e and y = ln x g g 2
x
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
provided g(x) �= 0
d
d
at time t before the end of the landing phase. The engineers• Leibniz notation dy = the elevator stops at a floor is jerk. Would you want jerk to be d g
f (x) (p. 192)
y =
dx
dx
specified the end of the landing phase to take place at t = 0, so that dx small or large in an elevator? Explain. Would you want jerk to Rule d 1 =− dx
• Reciprocal
• Basic derivatives
during the phase, t was always negative. Note that the LM was be small or large on a roller coaster ride? Explain. How would dx g g 2
(p. 207)
landing in three dimensions, so there were actually three equations you explain snap? d � g �
d
1
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
like (1). Since each of those equations had this same form, we will A = 0 A is a constant (p. 192) dx x = 1 (p. 193) g =− g 2
Summaries organized for ease of use.
dx
work in one dimension, assuming, for example, that r represents the d 1
d
x
x
The Chapter Review briefly recaps the
distance of the LM above the surface of the Moon. dx e = e (p. 198) Things to Know provided g(x) �= 0
(p. 199)
ln x =
dx
x
∗
d
A. R. Klumpp, “Apollo Lunar-Descent Guidance,” MIT Charles Stark
n
main ideas and key concepts of the chapter. Draper Laboratory, R-695, June 1971, • Simple Power Rule dx x = nx n − 1 , n an integer (p. 208)
1.
If the LM follows the reference trajectory, what is the reference
contains a detailed list of definitions,
d
velocity v ref (t)? • Simple Power Rule dx x = nx n − 1 , n ≥ 1, an integer
n
http://www.hq.nasa.gov/alsj/ApolloDescentGuidnce.pdf
• Higher-order derivatives
(p. 193) formulas, and theorems with page references See Table 3 (p. 209)
• Position Function
Properties of Derivatives so they can be found easily in the chapter. s = s(t) (p. 210)
d
d
d
Chapter Review • Sum Rule dx [ f + g] = dx f + dx g • Velocity v = v(t) = ds (p. 210)
dt
(p. 195)
2
( f + g) = f + g � dv d s
�
�
• Acceleration a = a(t) = dt = dt 2 (p. 210)
THINGS TO KNOW d d d
• Difference Rule [ f − g] = f − g
(p. 196) dx dx dx 2.5 The Derivative of the Trigonometric Functions
2.1 Rates of Change and the Derivative ( f − g) = f − g � �
�
�
• Rate of change of a function If y = f (x), the derivative f (c)
Basic Derivatives
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• Constant Multiple Rule (p. 194) If k is a constant,
• Physical If the signed distance s from the origin at time t d
d
of an object moving on a line is given by the position sin x = cos x (p. 218) dx sec x = sec x tan x (p. 221)
d
d
dx
[kf ] = k
f (x) − f (c) function s = f (t), the derivative f (t 0 ) is the velocity of the
f
�
�
Form (1) f (c) = lim dx dx d d
x→c x − c object at time t 0 . (p. 173) cos x =−sin x (p. 219) csc x =−csc x cot x (p. 221)
(kf ) = k · f
�
�
dx dx
provided the limit exists. (p. 173) 2.4 Differentiating the Product and the Quotient of d d
2.2 The Derivative as a Function
2
2
Three Interpretations of the Derivative Two Functions; Higher-Order Derivatives dx tan x = sec x (p. 221) dx cot x =−csc x (p. 221)
• Definition of a derivative function
• Geometric If y = f (x), the derivative f (c) Properties of Derivatives
�
is the slope of the tangent line to the graph of f at the • Product Rule d d Form (2) d f (x) = lim f (x + h) − f (x)
�

point (c, f (c)). (p. 173) ( fg) = f f g h→0 h
(p. 204) dx dx g + dx
provided the limit exists. (p. 179)
( fg) = fg + f g
�
�
�
Preparing for the
OBJECTIVES AP Exam
R
AP Review Problems
R
Section You should be able to ... Examples Review Exercises
2.1 1 Find equations for the tangent line and the normal line 1 67–70 7, 10
to the graph of a function (p. 168)
2 Find the rate of change of a function (p. 169) 2, 3 1, 2, 73 (a) 6
The Objectives table 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25 4, 5 71(a), (b); 72(a), (b) 5, 11
12
3 Find average velocity and instantaneous velocity (p. 170)
Sullivan
6–9
3–8, 75
4 Find the derivative of a function at a number (p. 173)
displays section-by-section 2.2 1 Define the derivative function (p. 179) 1–3 9–12 2, 13
lists of the objectives and 2 Graph the derivative function (p. 181) 4, 5 9–12, 15–18 4
13, 14, 75
6–8
3 Identify where a function is not differentiable (p. 182)
Chapter 2 • Review Exercises
the worked examples of the 4 Explain the relationship between differentiability 9, 10 13, 14, 75 4 229
chapter with page references. and continuity (p. 184)
1 Differentiate a constant function (p. 192)
1
2.3
It also includes references 2.4 1 Differentiate the product of two functions (p. 203) 1, 2 27, 28, 36, 46, 47–50, 53–56, 60 6, 7, 9
2 Differentiate a power function; the simple
19–22
2, 3
power rule (p. 192)
to Review Exercises and 2 Differentiate the quotient of two functions (p. 206) 3–6 29–35, 37–43, 57–59, 68, 73, 74 3, 10 6, 8, 12
3 Differentiate the sum and the difference of two
23–26, 33, 34, 40, 51, 52, 67
4–6
AP ® Review Problems that 3 Find higher-order derivatives (p. 208) x 9 7, 8 61–66, 71, 72 8, 12
functions (p. 195)
4 Find the acceleration of an object moving on a
71, 72
8, 12
pertain to each objective. line (p. 210) 4 Differentiate the exponential function y = e 7, 8 44, 45, 53, 54, 56, 59, 69 6, 7, 9
and the natural logarithm function y = ln x (p. 197)
2.5 1 Differentiate trigonometric functions (p. 218) 1–6 49–60, 70 1, 9, 11
REVIEW EXERCISES
In Problems 1 and 2, use a definition of the derivative to find the rate 17. Use the information in the graph of y = f (x) to sketch the graph
Review Exercises of change of f at the indicated numbers. of y = f (x).
�
offer an opportunity 1. f (x) = √ x at (a) c = 1 (b) c = 4 y
to return to the key 2 (c) c any positive real number (0, 3) 4 (2, 4)
concepts of the 2. f (x) = x − 1 at (a) c = 0 (b) c = 2 (�4, 2) y � f (x)
chapter for each (c) c any real number, c �= 1 (�6, 0) (�2, 1) 1 (5, 0)
objective. In Problems 3–8, use a definition of the derivative to find the derivative �6 �4 �2 2 4 6 x
of each function at the given number.
2
3. F(x) = 3x + 6 at −2 4. f (x) = 8x + 1 at −1 f �(�6) f �(�4) f �(�2) f �(0) f �(2) f�(5)
3 � 2 � 0 � 0 � 1 � 0 � �3
2
5. f (x) = 3x + 5x at 0 6. f (x) = at 1
x
√ x + 1 18. Match the graph of y = f (x) with the graph of its derivative.
7. f (x) = 4x + 1at0 8. f (x) = at 1
2x − 3 y
8 y � f (x)
In Problems 9–12, use a definition of the derivative to find the 6 (5, 8)
derivative of each function. Graph f and f on the same set of axes. (�3, 6)
�
2 (0, 3)
9. f (x) = x − 6 10. f (x) = 7 − 3x 2 x
1 �5 �3 �1 1 3 5
11. f (x) = 12. f (x) = π �4
2x 3
In Problems 13 and 14, determine whether the function f has a xxv
xv
x
© 2024 BFW Publishers PAGES NOT FINAL
derivative at c. If it does, find the derivative. If it does not, explain why. y y
For Review Purposes Only, all other uses prohibited
Graph each function. 3 3
3
13. f (x) =|x − 1| at c = 1 Do Not Copy or Post in Any Form. (3, 3)
2 (5, 1)
4 − 3x if x ≤ −1 1 1
14. f (x) = 3 at c =−1
−x if x > −1
�5 �3 �1 1 3 5 x �5 �3 1 3 5 x
01_apcalc4e_45342_fm_i_xxix_3pp.indd 25 �1 �1 10/11/23 2:39 PM
In Problems 15 and 16, determine whether the graphs represent a
function f and its derivative f . If they do, indicate which is the graph
�
of f and which is the graph of f . � �3 �3
15. 16.
y (A) (B)
y 6
6 y y
4
5 (5, 5) 5
4
2
2 3 3 (5, 1)
�1 1 x 1 �3 1 1
�2 �1 1 2 x �2
�5 �3 �1 (0, 0) 3 5 x �5 (0, 0) 3 5 x
�2 (�3, �1)
�4
�3 �3
�5 �5
(C) (D)