| Introduction |
|
1 | (4) |
|
|
|
1 | (1) |
|
|
|
2 | (1) |
|
|
|
3 | (1) |
|
|
|
3 | (1) |
|
|
|
4 | (1) |
|
UNIT 1 AN OVERVIEW OF CALCULUS |
|
|
5 | (20) |
|
Chapter 1 What Is Calculus? |
|
|
7 | (6) |
|
|
|
7 | (1) |
|
So What Is Calculus, Already? |
|
|
8 | (1) |
|
Real-World Examples of Calculus |
|
|
9 | (4) |
|
Chapter 2 The Two Big Ideas of Calculus: Differentiation and Integration -- Plus Infinite Series |
|
|
13 | (8) |
|
|
|
13 | (2) |
|
The derivative is a slope |
|
|
14 | (1) |
|
|
|
15 | (1) |
|
Investigating Integration |
|
|
15 | (2) |
|
Sorting Out Infinite Series |
|
|
17 | (4) |
|
|
|
17 | (1) |
|
|
|
17 | (4) |
|
Chapter 3 Why Calculus Works |
|
|
21 | (4) |
|
The Limit Concept: A Mathematical Microscope |
|
|
21 | (1) |
|
What Happens When You Zoom In |
|
|
22 | (2) |
|
Two Caveats; or, Precision, Preschmidgen |
|
|
24 | (1) |
|
I may lose my license to practice mathematics |
|
|
24 | (1) |
|
What the heck does "infinity really mean? |
|
|
24 | (1) |
|
UNIT 2 WARMING UP WITH CALCULUS PREREQUISITES |
|
|
25 | (92) |
|
Chapter 4 Pre-Algebra, Algebra, and Geometry Review |
|
|
27 | (40) |
|
Fine-Tuning Your Fractions |
|
|
28 | (6) |
|
|
|
28 | (1) |
|
|
|
28 | (1) |
|
|
|
29 | (1) |
|
|
|
29 | (1) |
|
|
|
30 | (1) |
|
|
|
31 | (3) |
|
|
|
34 | (9) |
|
Absolute value -- absolutely easy |
|
|
34 | (1) |
|
|
|
34 | (1) |
|
|
|
35 | (2) |
|
Logarithms -- This Is Not an Event at a Lumberjack Competition |
|
|
37 | (1) |
|
Factoring Schmactoring -- When Am I Ever Going to Need It? |
|
|
38 | (1) |
|
Solving Quadratic Equations |
|
|
39 | (4) |
|
|
|
43 | (9) |
|
Handy-dandy geometry formulas |
|
|
44 | (1) |
|
Two special right triangles |
|
|
45 | (7) |
|
Practice Questions Answers and Explanations |
|
|
52 | (7) |
|
Whaddya Know?
Chapter 4 Quiz |
|
|
59 | (2) |
|
|
|
61 | (6) |
|
Chapter 5 Funky Functions and Their Groovy Graphs |
|
|
67 | (28) |
|
|
|
67 | (4) |
|
The defining characteristic of a function |
|
|
68 | (1) |
|
Independent and dependent variables |
|
|
69 | (1) |
|
|
|
69 | (1) |
|
|
|
70 | (1) |
|
What Does a Function Look Like? |
|
|
71 | (3) |
|
Common Functions and Their Graphs |
|
|
74 | (7) |
|
Lines in the plane in plain English |
|
|
74 | (3) |
|
Parabolic and absolute value functions -- even-steven |
|
|
77 | (1) |
|
A couple of oddball functions |
|
|
78 | (1) |
|
|
|
78 | (1) |
|
|
|
79 | (2) |
|
|
|
81 | (2) |
|
Shifts, Reflections, Stretches, and Shrinks |
|
|
83 | (4) |
|
Horizontal transformations |
|
|
83 | (2) |
|
|
|
85 | (2) |
|
Practice Questions Answers and Explanations |
|
|
87 | (4) |
|
Whaddya Know?
Chapter 5 Quiz |
|
|
91 | (1) |
|
|
|
92 | (3) |
|
|
|
95 | (22) |
|
Starting off with SohCahToa |
|
|
95 | (1) |
|
Two Important Trig Triangles |
|
|
96 | (1) |
|
Circling the Enemy with the Unit Circle |
|
|
97 | (9) |
|
Angles in the unit circle |
|
|
98 | (1) |
|
Measuring angles with radians |
|
|
98 | (1) |
|
Honey, I shrunk the hypotenuse |
|
|
99 | (1) |
|
|
|
100 | (6) |
|
Graphing Sine, Cosine, and Tangent |
|
|
106 | (1) |
|
|
|
107 | (1) |
|
Identifying with Trig Identities |
|
|
108 | (1) |
|
Practice Questions Answers and Explanations |
|
|
109 | (4) |
|
Whaddya Know?
Chapter 6 Quiz |
|
|
113 | (1) |
|
|
|
114 | (3) |
|
|
|
117 | (64) |
|
Chapter 7 Limits and Continuity |
|
|
119 | (22) |
|
Take It to the Limit -- NOT |
|
|
119 | (10) |
|
Using three functions to illustrate the same limit |
|
|
120 | (1) |
|
Sidling up to one-sided limits |
|
|
121 | (1) |
|
The formal definition of a limit -- just what you've been waiting for |
|
|
122 | (1) |
|
Limits and vertical asymptotes |
|
|
123 | (1) |
|
Limits and horizontal asymptotes |
|
|
123 | (3) |
|
Calculating instantaneous speed with limits |
|
|
126 | (3) |
|
Linking Limits and Continuity |
|
|
129 | (2) |
|
Continuity and limits usually go hand in hand |
|
|
130 | (1) |
|
The hole exception tells the whole story |
|
|
130 | (1) |
|
Sorting out the mathematical mumbo jumbo of continuity |
|
|
131 | (1) |
|
|
|
131 | (4) |
|
Practice Questions Answers and Explanations |
|
|
135 | (2) |
|
Whaddya Know?
Chapter 7 Quiz |
|
|
137 | (2) |
|
|
|
139 | (2) |
|
Chapter 8 Evaluating Limits |
|
|
141 | (40) |
|
Easy Does It -- Easy Limits |
|
|
141 | (2) |
|
|
|
142 | (1) |
|
|
|
142 | (1) |
|
The "Real Deal" Limit Problems |
|
|
143 | (14) |
|
Figuring a limit with your calculator |
|
|
143 | (4) |
|
Solving limit problems with algebra |
|
|
147 | (6) |
|
Take a break and make yourself a limit sandwich |
|
|
153 | (4) |
|
Evaluating Limits at ±lnfinity |
|
|
157 | (8) |
|
Limits of rational functions at linfinity |
|
|
158 | (2) |
|
Solving limits at ±infinity with a calculator |
|
|
160 | (1) |
|
Solving limits at ±infinity with algebra |
|
|
161 | (4) |
|
Practice Questions Answers and Explanations |
|
|
165 | (10) |
|
Whaddya Know?
Chapter 8 Quiz |
|
|
175 | (1) |
|
|
|
176 | (5) |
|
|
|
181 | (216) |
|
Chapter 9 Differentiation Orientation |
|
|
183 | (32) |
|
Differentiating: It's Just Finding the Slope |
|
|
184 | (4) |
|
|
|
186 | (2) |
|
|
|
188 | (1) |
|
The Derivative: It's Just a Rate |
|
|
188 | (3) |
|
Calculus on the playground |
|
|
188 | (2) |
|
Speed -- the most familiar rate |
|
|
190 | (1) |
|
The rate-slope connection |
|
|
190 | (1) |
|
The Derivative of a Curve |
|
|
191 | (4) |
|
|
|
195 | (6) |
|
Average Rate and Instantaneous Rate |
|
|
201 | (1) |
|
To Be or Not to Be? Three Cases Where the Derivative Does Not Exist |
|
|
202 | (3) |
|
Practice Questions Answers and Explanations |
|
|
205 | (4) |
|
Whaddya Know?
Chapter 9 Quiz |
|
|
209 | (2) |
|
|
|
211 | (4) |
|
Chapter 10 Differentiation Rules -- Yeah, Man, It Rules |
|
|
215 | (44) |
|
Basic Differentiation Rules |
|
|
216 | (6) |
|
|
|
216 | (1) |
|
|
|
216 | (1) |
|
The constant multiple rule |
|
|
217 | (1) |
|
The sum rule -- hey, that's some rule you got there |
|
|
218 | (1) |
|
The difference rule -- it makes no difference |
|
|
218 | (2) |
|
Differentiating trig functions |
|
|
220 | (1) |
|
Differentiating exponential and logarithmic functions |
|
|
220 | (2) |
|
Differentiation Rules for Experts -- Oh, Yeah, I'm a Calculus Wonk |
|
|
222 | (9) |
|
|
|
222 | (1) |
|
|
|
222 | (3) |
|
Linking up with the chain rule |
|
|
225 | (6) |
|
Differentiating Implicitly |
|
|
231 | (3) |
|
Differentiating Inverse Functions |
|
|
234 | (3) |
|
Scaling the Heights of Higher-Order Derivatives |
|
|
237 | (3) |
|
Practice Questions Answers and Explanations |
|
|
240 | (13) |
|
Whaddya Know?
Chapter 10 Quiz |
|
|
253 | (1) |
|
Answers to
Chapter 10 Quiz |
|
|
254 | (5) |
|
Chapter 11 Differentiation and the Shape of Curves |
|
|
259 | (62) |
|
Taking a Calculus Road Trip |
|
|
259 | (3) |
|
Climb every mountain, ford every stream: Positive and negative slopes |
|
|
260 | (1) |
|
I can't think of a travel metaphor for this section: Concavity and inflection points |
|
|
260 | (1) |
|
This vale of tears: A local minimum |
|
|
261 | (1) |
|
A scenic overlook: The absolute maximum |
|
|
261 | (1) |
|
Car trouble: Teetering on the corner |
|
|
261 | (1) |
|
It's all downhill from here |
|
|
261 | (1) |
|
|
|
262 | (1) |
|
Finding Local Extrema -- My Ma, She's Like, Totally Extreme |
|
|
262 | (11) |
|
Cranking out the critical numbers |
|
|
263 | (1) |
|
The first derivative test |
|
|
264 | (4) |
|
The second derivative test -- no, no, anything but another test! |
|
|
268 | (5) |
|
Finding Absolute Extrema on a Closed Interval |
|
|
273 | (5) |
|
Finding Absolute Extrema over a Function's Entire Domain |
|
|
278 | (3) |
|
Locating Concavity and Inflection Points |
|
|
281 | (4) |
|
Looking at Graphs of Derivatives Till They Derive You Crazy |
|
|
285 | (4) |
|
The Mean Value Theorem -- Go Ahead, Make My Day |
|
|
289 | (3) |
|
Practice Questions Answers and Explanations |
|
|
292 | (18) |
|
Whaddya Know?
Chapter 11 Quiz |
|
|
310 | (1) |
|
Answers to
Chapter 11 Quiz |
|
|
311 | (10) |
|
Chapter 12 Your Problems Are Solved: Differentiation to the Rescue! |
|
|
321 | (46) |
|
Getting the Most (or Least) Out of Life: Optimization Problems |
|
|
321 | (5) |
|
The maximum volume of a box |
|
|
322 | (1) |
|
The maximum area of a corral -- yeehaw! |
|
|
323 | (3) |
|
Yo-Yo a Go-Go: Position, Velocity, and Acceleration |
|
|
326 | (10) |
|
Velocity, speed, and acceleration |
|
|
328 | (1) |
|
Maximum and minimum height |
|
|
329 | (1) |
|
Velocity and displacement |
|
|
330 | (1) |
|
Speed and distance traveled |
|
|
331 | (2) |
|
Burning some rubber with acceleration |
|
|
333 | (1) |
|
|
|
333 | (3) |
|
Related Rates -- They Rate, Relatively |
|
|
336 | (10) |
|
|
|
336 | (3) |
|
|
|
339 | (2) |
|
Fasten your seat belt: You're approaching a calculus crossroads |
|
|
341 | (2) |
|
Try this at your own risk |
|
|
343 | (3) |
|
Practice Questions Answers and Explanations |
|
|
346 | (16) |
|
Whaddya Know?
Chapter 12 Quiz |
|
|
362 | (1) |
|
Answers to
Chapter 12 Quiz |
|
|
363 | (4) |
|
Chapter 13 More Differentiation Problems: Going Off on a Tangent |
|
|
367 | (30) |
|
Tangents and Normals: Joined at the Hip |
|
|
368 | (6) |
|
|
|
368 | (2) |
|
|
|
370 | (4) |
|
Straight Shooting with Linear Approximations |
|
|
374 | (4) |
|
Business and Economics Problems |
|
|
378 | (7) |
|
Managing marginals in economics |
|
|
378 | (7) |
|
Practice Questions Answers and Explanations |
|
|
385 | (8) |
|
Whaddya Know?
Chapter 13 Quiz |
|
|
393 | (1) |
|
Answers to
Chapter 13 Quiz |
|
|
394 | (3) |
|
UNIT 5 INTEGRATION AND INFINITE SERIES |
|
|
397 | (226) |
|
Chapter 14 Intro to Integration and Approximating Area |
|
|
399 | (40) |
|
Integration: Just Fancy Addition |
|
|
400 | (1) |
|
Finding the Area Under a Curve |
|
|
401 | (2) |
|
|
|
403 | (8) |
|
Approximating area with left sums |
|
|
404 | (2) |
|
Approximating area with right sums |
|
|
406 | (2) |
|
Approximating area with midpoint sums |
|
|
408 | (3) |
|
Getting Fancy with Summation Notation |
|
|
411 | (6) |
|
|
|
411 | (3) |
|
Writing Riemann sums with sigma notation |
|
|
414 | (3) |
|
Finding Exact Area with the Definite Integral |
|
|
417 | (4) |
|
Approximating Area with the Trapezoid Rule and Simpson's Rule |
|
|
421 | (6) |
|
|
|
421 | (2) |
|
Simpson's rule -- that's Thomas (1710-1761), not Homer (1987-) |
|
|
423 | (4) |
|
Practice Questions Answers and Explanations |
|
|
427 | (8) |
|
Whaddya Know?
Chapter 14 Quiz |
|
|
435 | (1) |
|
Answers to
Chapter 14 Quiz |
|
|
436 | (3) |
|
Chapter 15 Integration: It's Backwards Differentiation |
|
|
439 | (40) |
|
|
|
439 | (2) |
|
Vocabulary, Voshmabulary: What Difference Does It Make? |
|
|
441 | (1) |
|
The Annoying Area Function |
|
|
441 | (5) |
|
The Power and the Glory of the Fundamental Theorem of Calculus |
|
|
446 | (3) |
|
The Fundamental Theorem of Calculus: Take Two |
|
|
449 | (11) |
|
Why the theorem works: Area functions explained |
|
|
454 | (2) |
|
Why the theorem works: The integration-differentiation connection |
|
|
456 | (2) |
|
Why the theorem works: A connection to -- egad! -- statistics |
|
|
458 | (2) |
|
Finding Antiderivatives: Three Basic Techniques |
|
|
460 | (9) |
|
Reverse rules for antiderivatives |
|
|
460 | (3) |
|
|
|
463 | (2) |
|
|
|
465 | (4) |
|
Finding Area with Substitution Problems |
|
|
469 | (2) |
|
Practice Questions Answers and Explanations |
|
|
471 | (5) |
|
Whaddya Know?
Chapter 15 Quiz |
|
|
476 | (1) |
|
Answers to
Chapter 15 Quiz |
|
|
477 | (2) |
|
Chapter 16 Integration Techniques for Experts |
|
|
479 | (42) |
|
Integration by Parts: Divide and Conquer |
|
|
479 | (7) |
|
|
|
482 | (2) |
|
Integration by parts: Second time, same as the first |
|
|
484 | (2) |
|
|
|
486 | (5) |
|
Integrals containing sines and cosines |
|
|
486 | (3) |
|
Integrals containing secants and tangents (or cosecants and cotangents) |
|
|
489 | (2) |
|
Your Worst Nightmare: Trigonometric Substitution |
|
|
491 | (6) |
|
|
|
492 | (2) |
|
|
|
494 | (2) |
|
|
|
496 | (1) |
|
The A's, B's, and Cx's of Partial Fractions |
|
|
497 | (6) |
|
Case 1 The denominator contains only linear factors |
|
|
497 | (1) |
|
Case 2 The denominator contains irreducible quadratic factors |
|
|
498 | (2) |
|
Bonus: Equating coefficients of like terms |
|
|
500 | (1) |
|
Case 3 The denominator contains one or more factors raised to a power greater than 1 |
|
|
501 | (2) |
|
Practice Questions Answers and Explanations |
|
|
503 | (14) |
|
Whaddya Know?
Chapter 16 Quiz |
|
|
517 | (2) |
|
Answers to
Chapter 16 Quiz |
|
|
519 | (2) |
|
Chapter 17 Who Needs Freud? Using the Integral to Solve Your Problems |
|
|
521 | (36) |
|
The Mean Value Theorem for Integrals and Average Value |
|
|
522 | (3) |
|
The Area between Two Curves -- Double the Fun |
|
|
525 | (4) |
|
Volumes of Weird Solids: No, You're Never Going to Need This |
|
|
529 | (8) |
|
|
|
529 | (3) |
|
|
|
532 | (1) |
|
|
|
533 | (4) |
|
|
|
537 | (3) |
|
Surfaces of Revolution -- Pass the Bottle `Round' |
|
|
540 | (4) |
|
Practice Questions Answers and Explanations |
|
|
544 | (7) |
|
Whaddya Know?
Chapter 17 Quiz |
|
|
551 | (1) |
|
Answers to
Chapter 17 Quiz |
|
|
552 | (5) |
|
Chapter 18 Taming the Infinite with Improper Integrals |
|
|
557 | (24) |
|
L'Hopital's Rule: Calculus for the Sick |
|
|
558 | (5) |
|
Getting unacceptable forms into shape |
|
|
559 | (1) |
|
Looking at three more unacceptable forms |
|
|
560 | (3) |
|
Improper Integrals: Just Look at the Way That Integral Is Holding Its Fork! |
|
|
563 | (7) |
|
Improper integrals with vertical asymptotes |
|
|
563 | (2) |
|
Improper integrals with one or two infinite limits of integration |
|
|
565 | (3) |
|
|
|
568 | (2) |
|
Practice Questions Answers and Explanations |
|
|
570 | (5) |
|
Whaddya Know?
Chapter 18 Quiz |
|
|
575 | (1) |
|
Answers to
Chapter 18 Quiz |
|
|
576 | (5) |
|
Chapter 19 Infinite Series: Welcome to the Outer Limits |
|
|
581 | (42) |
|
Sequences and Series: What They're All About |
|
|
582 | (4) |
|
|
|
582 | (2) |
|
|
|
584 | (2) |
|
Convergence or Divergence? That Is the Question |
|
|
586 | (16) |
|
A no-brainer divergence test: The nth term test |
|
|
586 | (2) |
|
Three basic series and their convergence/divergence tests |
|
|
588 | (4) |
|
Three comparison tests for convergence/divergence |
|
|
592 | (6) |
|
The two "R" tests: Ratios and roots |
|
|
598 | (4) |
|
|
|
602 | (4) |
|
Finding absolute versus conditional convergence |
|
|
602 | (1) |
|
The alternating series test |
|
|
603 | (3) |
|
Keeping All the Tests Straight |
|
|
606 | (1) |
|
Practice Questions Answers and Explanations |
|
|
607 | (9) |
|
Whaddya Know?
Chapter 19 Quiz |
|
|
616 | (1) |
|
Answers to
Chapter 19 Quiz |
|
|
617 | (6) |
| Index |
|
623 | |
