Notes for Introductory Calculus (Math 120)

Calculus Volume 1, Section 2.1¹, In particular Example 1, as usual the Checkpoint immediately following that (also 1 in this case), and Exercises 4, 5, 6, 16 and 17.

Calculus Volume 1, Section 2.2², In particular Examples 4, 5, 7, 8, 9, 10 and 11, as usual the Checkpoints immediately following those (4, 6, 7, 8, 9 and 10 in this case), and Exercises 30, 31, 35, 36, 37, 46–49, 77 and 79.

Calculus Volume 1, Section 2.3³, All Examples and Checkpoints are worth studying, and Exercises 83, 85, 89, 91, 93, 97, 107, 111, 119, 121, 127, and 128.

Calculus Volume 1, Section 2.4⁴, All Examples and Checkpoints are worth studying, and Exercises 133, 137, 141, 147, 150, 151, 154, 157, 163, and 165.

Calculus Volume 1, Section 2.5⁵, In particular Examples 39, 41, 43 and 44; the Checkpoints immediately following those (28 and 30), and Exercises 177, 184, 185, 187, and 191.

Calculus Volume 1, Section 3.1⁶; in particular Examples 1,2, 3, 5, 6 and 9, Checkpoint items 1, 3 and 4, and Exercises 1, 7, 11, 13, 15, 25, 37, 39, 41 and 51.

Calculus Volume 1, Section 3.2⁷: all Examples and Checkpoint items are worth reviewing, and Exercises 55, 57, 65, 67, 79, 80 and 96.

Calculus Volume 1, Section 3.3⁸: all Examples, Checkpoint items 12 to 19 and Exercises 107, 109, 111, 119, 122, 127, 129, 130, 131, 133, 142, 143 and 147.

Calculus Volume 1, Section 3.4⁹: Examples 34 to 36, Checkpoint item 22, and Exercises 151, 159 and 165.

Calculus Volume 1, Section 3.5¹⁰; in particular Examples 39–44, Checkpoint items 25–30, and Exercises 175, 178, 181, 182, 191, 197 and 206.

Calculus Volume 1, Section 3.6¹¹ Examples 48, 48, 50, 52 and 53, all Checkpoint items, and Exercises 215, 217, 219, 221, 224, 229, 233, 235, 245, 251 and 257.

Calculus Volume 1, Section 3.7¹²; Examples 61–67, Checkpoint items 43–46, and Exercises 265, 267, 269, 271, 279, and 291.

Hint for Exercise 279. One approach is to use the "equation solving" strategy of making the inverse function disappear: solve for \(\sin(y)=x^2\) and then differentiate each side of that equation.

Calculus Volume 1, Section 3.8¹³ Examples 68, 69, 71 and 72, both Checkpoint items, and Exercises 301, 303, 305, 307, 311, 316, 325, and 329.

Calculus Volume 1, Section 3.9¹⁴ Examples 74, 75, 77, 78, 81 and 82, Checkpoint 54, and Exercises 333, 339, 347, 351 and 353.

We in particular emphasize the last topic of Logarithmic Differentiation, using the strategy of simplifying functions of the form \(\log(\dots)\) using the laws of logarithms like \(\log(a b) = \log(a) + \log(b)\text{.}\)

Calculus Volume 1, Section 4.1¹⁵: all Examples and Checkpoints and Exercises 1, 3, 5, 7, 9, 17, and 25.

Calculus Volume 1, Section 4.2¹⁶ all Examples and Checkpoints and a few Exercises from each of the ranges 50–55, 62–67, 68–71, 72–77, 78–83, 84–86; for example, Exercises 49, 51, 52, 57, 69, 73, 79 and 84.

Calculus Volume 1, Section 4.3¹⁷: in particular the Problem Solving Strategy, both Examples and Checkpoints, and a few Exercises from each of the ranges 91–98, 100–103, 104–107, 108–117, 118–128 and 129–134. (Some suggested selections are Exercises 91, 93, 97, 101, 107, 109, 119 and 129.)

Calculus Volume 1, Section 4.4¹⁸. Pay particular attention the Corollaries of the Mean Value Theorem in the second half: Theorems 6, 7 and 8: these will be extremely useful for applications later in this chapter.

Study Examples 14 and 15, Checkpoint 14, and a selection from Exercises 148–150, 152–156, 161–166, 167–169, 182–184, and 190–193.

Here I group the exercises in ranges by "question type", so start by trying one or two from each of the six ranges. For example, some suggested selections are Exercises 149, 153, 161, 169, 182 and 192.

Calculus Volume 1, Section 4.5¹⁹: the Problem Solving Strategy, the First Derivative Test, the Second Derivative Test all Examples and Checkpoints, and a selection from Exercises 194–200, 201–205, 206–210, 211–215, 216–220, 221–223 and 224–230.

Some suggested selections are Exercises 199, 201, 203, 213, 215, 217, 223, 225, 229.

Calculus Volume 1, Section 4.6²⁰ Examples 21–26, 28, 29 and 31, Checkpoints 20, 23–25, 27, 28 and 30, (We omit oblique asymptotes, so skip Example 30 and Checkpoint 29) and a selection from Exercises 251–255, 256–260, 261–270, 271–281, 285–288 and 294–298.

Here the exercises are grouped in ranges by "question type", so start by trying one or two from each of the seven ranges; some suggested selections are Exercises 251, 256, 257, 259, 261, 263, 265, 267, 271, 279, 281, 285, 306 and 307.

Calculus Volume 1, Section 4.7²¹ Examples 33–35 and 37, Checkpoints 31–34 and 36, and a selection from Exercises 311–314, 315–318, 319–321, 322–326, 335–336 and 351–355.

Here the exercises are grouped in ranges by "question type", so start by trying one or two from each of the seven ranges; some suggested selections are Exercises 311, 316, 320, 322, 335 and 353.

Calculus Volume 1, Section 4.8²² Examples 38–41, 43 and 44, Checkpoints 37–40, 42 and 43, and a selection from Exercises 356–361, 362–366, 367–385, 387–389 and 391–395.

Here the exercises are grouped in ranges by "question type", so start by trying one or two from each of the ranges; some suggested selections are Exercises 357, 359, 363, 367, 371, 377, 379, 387, and 393.

Calculus Volume 1, Section 4.10²³ all the Examples and Checkpoints and a selection from Exercises 4465–469, 470–473, 474–489, 490–498, 499–503 and 504–508.

Here the exercises are grouped in ranges by "question type", so start by trying several from each of the ranges; some suggested selections are Exercises 465, 467, 469, 471, 477, 487, 491, 493, 499, 501 and 505.

Hint: It often helps to simplify the function first, and then use the list of derivatives and indefinite integrals in the online test.

Calculus Volume 1, Section 5.1²⁴ If you are unfamiliar with the \(\Sigma\) notation for sums, the first part of that section should help. Study Example 4, Checkpoint 4, and Exercises 15, 19, 23, 27, 29, and 43.

Calculus Volume 1, Section 5.2²⁵ Examples 8–13, Checkpoints–12, and Exercises 61, 65, 73, 75, 79, 81, 89, 91, 93, 99, 101 and 107.

Calculus Volume 1, Section 5.3²⁶ Theorems 4 and 5, Examples 17, 18, 20 and 21; Checkpoints 16, 17 and 19; and Exercises 147, 149, 153, 155, 157, 161, 171, 177, 179, 183, 190, 191 and 195.

Calculus Volume 1, Section 5.4²⁷ Theorem 6, Examples 23–26, 28 and 29, Checkpoints 21, 22 and 24, and Exercises 207, 209, 211 and 223.

Calculus Volume 1, Section 5.5²⁸: Theorem 7, the Problem Solving Strategy that follows it, Examples 30–33 (and maybe 34 and 35), Checkpoints 25–28, (and maybe 29 and 30), and one or several exercises from each of the following ranges: 256–260, 261–270, 271–287 and 292–297; Some suggested selections are Exercises 257, 261, 265, 271, 275, 281, 293, 297.

Note that, for definite integrals one can either do it as described there (Theorem 8, Examples 34 and 35, Checkpoints 29 and 30) or (a) first get the indefinite integral \(\int f(x) dx = F(x) + C\) using substitution and then (b) use FTC: \(\int_a^b f(x) dx = F(b)-F(a)\text{.}\)

Calculus Volume 1, Section 5.6²⁹ Examples 37, 38, 39, 41, 44, 45, 47, 48, the Checkpoints that immediately follow each of those, and a few Exercises from each of the ranges 320–325, 328–339 and 355–357.

Calculus Volume 1, Section 5.7³⁰ All Examples and Checkpoint items are worth looking at; and study a few Exercises from each of the ranges 391–394, 397–400, and 411–414.