Math 125: Calculus I
USC Fall 2022


Time and location.


This course is an introduction to calculus. Our main topics will be:

  1. functions, limits, and continuity
  2. derivatives and applications
  3. integrals and the fundamental theorem of calculus
  4. exponential and logarithmic functions.
We will follow sections 1.1-5.5 (inclusive) in the textbook by Stewart rather closely, with some exceptions.

I highly encourage (and expect) you to follow along concurrently with the relevant sections in the textbook. There may be some important material in the textbook that we do not cover in lectures (and vice versa). I recommend periodically re-reading the previous textbook sections in order to strengthen your understanding.

Mailing list.

If for some reason you aren't currently registered for this course on Blackboard please email Kyler to make sure you receive class emails.


Math 108 or placement exam.


The primary textbook is Essential Calculus (2nd edition) by James Stewart.

Problem sets.

In addition to the lectures, there will be weekly problem sets. These will be listed on this website (see the schedule below) and handed in via Gradescope. Late homeworks will not be accepted. However, we will drop the two lowest problem set scores.

Much of the course material will be developed in the problem sets. It is very important to do all of the problem sets to the best of your ability and to challenge yourself to solve the problems on your own, as this is the most effective way to absorb the material. We expect students to devote a significant amount of time to the problem sets.

While working on the problem sets, you are allowed to consult or collaborate with your peers, as well as textbooks and the internet (apart from cheating websites such as Chegg or Cramster). However, you must write down attributions for any peer, textbook, website etc from which you took any significant ideas. Moreover, you must attempt all problems on your own and your submitted solutions must be written out originally and individually. Submissions which are copied or suspiciously similar are subject to being rejected and potential disciplinary action.


There will be quizzes every week (with some exceptions) assigned during the discussion sections.


There will be two midterms (most likely during regular class time) and one final exam (scheduled by the registrar). All exams are graded on a curve, so please don't stress too much about the raw score. Note that the final exam will be shared with the other sections of Math 125.

Grading scheme.

Problem sets: 15%, quizzes: 10%, midterm exams: 40%, final exam: 35%. We will drop the two lowest problem set grades and the two lowest quiz grade.

Office hours.

Additional resources.

Tentative schedule.

Note: this schedule is tentative and will be continuously updated to adapt to the pace of the course. Please check back regularly for updates and problem set assignments. z
$\#$ Date Material References Problem set
1 Monday 8/22/22 Course introduction. Basics on sets and examples (reals, rationals, intervals, etc). Unions of sets. Functions between sets. Different ways of representing functions (symbolic, verbal, graphical, etc). Vertical line test. §1.1
2 Wednesday 8/24/22 More on functions. Domain, codomain, and range. Catalogue of functions. §1.1,§1.2 Pset 1 (due Weds 8/31/22 by 11:59pm) Pset 1 (due Weds 8/31/22 by 11:59pm):
§1.1: 2,4,5,6,7,8,28,30,36,60,62, §1.2: 5,17,18,19,43,50
3 Friday 8/26/22 Catalogue of functions continued. More practice with domain and range. Polynomials and their typical graphs. Power functions. Rational functions. Sine and cosine. §1.2
4 Monday 8/29/22 Introduction to limits. Left and right sided limits. Various examples and computational techniques. Computer graphs. §1.3,§1.4,§1.5
5 Wednesday 8/31/22 More on limits. Continuity and limit laws. The precise $\epsilon,\delta$ definition. §1.3,§1.4,§1.5 Pset 2 (due Weds 9/7/22 by 11:59pm) Pset 2 (due Weds 9/7/22 by 11:59pm):
§1.3: 4,8,10,12,24,34,46, §1.4: 1,2,4,6,10,22,24,32,34,42,50,54, §1.5: 6,12,20,27,34
6 Friday 9/2/22 More on continuity. The intermediate value theorem. The squeeze theorem and examples. Limits involving infinity. §1.5,§1.6
Monday 9/5/22: Labor day - no class
7 Wednesday 9/7/22 More examples of limits involving infinity. Tangent lines and derivatives of functions. §2.1 Pset 3 (due Weds 9/14/22 by 11:59pm) Pset 3 (due Weds 9/14/22 by 11:59pm):
§1.6: 3,9,13,18,22,24,28,31,42, §2.1: 1,3,11,17,20, §2.2: 3,5,23,43,45
8 Friday 9/9/22 Computing derivatives from the definition. Various examples such as polynomials, square root function, the absolute value function, etc. Three situations where differentiability fails. Deriving the sum rule and constant multiple rule. The "naive product rule" fails. §2.2,§2.3
9 Monday 9/12/22 Rules for computing derivatives and examples. Derivatives of trigonometric functions. The power rule, product rule and quotient rule. §2.3,§2.4
10 Wednesday 9/14/22 More practice with derivative rules. The chain rule. Examples. §2.5 Pset 4 (due Weds 9/21/22 by 11:59pm) Pset 4 (due Weds 9/21/22 by 11:59pm):
§2.3: 10,16,20,24,27,30,35,41,45 (feel free to use Desmos or any other plotting software for graphing problems) §2.4: 3,6,9,28,36,41 §2.5: 1,7,13,30,35,42
11 Friday 9/16/22 Equations of tangent lines and normal lines. Examples involving velocity, acceleration, and maxima. §2.1,§2.2,§2.3
12 Monday 9/19/22 Implicit differentiation and related rates. Some examples. §2.6,§2.7
13 Wednesday 9/21/22 Proof that $\lim\limits_{x \rightarrow 0}\tfrac{\sin(x)}{x} = 1$ using squeeze theorem and some trigonometry. More related rates examples. §1.4 (see page 42), §2.7 Pset 5 (due Weds 9/28/22 by 11:59pm) Pset 5 (due Weds 9/28/22 by 11:59pm):
§2.6: 2,5,8,16,22,23,28,34,37, §2.7: 2,5,11,14,16,25,41, §2.8: 1,6,11,15
14 Friday 9/23/22 More related rates examples. Linear approximations. §2.7, §2.8
15 Monday 9/26/22 Another linear approximation example. Absolute and local extreme of fuctions. The extreme value theorem and Fermat's theorem. Critical numbers. The closed interval method. §2.8,§3.1
16 Wednesday 9/28/22 Midterm review
17 Friday 9/30/22 Midterm 1 during class. Solutions. Pset 6 (due Fri 10/7/22 by 11:59pm) Pset 6 (due Fri 10/7/22 by 11:59pm):
§3.1: 3,4,5,6,13,27,37,41,43,45, §3.2: 2,3,5,10,14,17
18 Monday 10/3/22 Examples of finding absolute maxima and minima. Proof of Fermat's theorem. Rolle's theorem and the mean value theorem. §3,1,§3.2
19 Wednesday 10/5/22 More on Rolle's theorem and the mean value theorem. Corollories and applications. §3.2,§3.3
20 Friday 10/7/22 More on the mean value theorem. The increasing / decreasing test for functions. The first derivative test. §3.2,§3.3 Pset 7 (due Mon 10/17/22 by 11:59pm) Pset 7 (due Mon 10/17/22 by 11:59pm):
§3.3: 3,4,6,10,11,13,14,15,22,23,37, §3.4: 1,4,10,18,34
21 Monday 10/10/22 More on the increasing / decreasing test and example. More on the first derivative test. Concave up and down functions. The concavity test and the second derivative test. Examples where the second derivative test is inconclusive. §3.3
22 Wednesday 10/12/22 Review of concavity test and second derivative test. Inflection points. Guidelines for curve sketching and examples. §3.3,§3.4
Friday 10/14/22: Fall recess - no class
23 Monday 10/17/22 More on curve sketching. Optimization problems. §3.4,§3.5 Pset 8 (due Mon 10/24/22 by 11:59pm) Pset 8 (due Mon 10/24/22 by 11:59pm):
§3.5: 4,6,8,12,17,24,36 §3.7: 1,6,9,15,19,21,42
24 Wednesday 10/19/22 More examples of optimization problems. §3.5
25 Friday 10/21/22 Antiderivatives. The "anti power rule" and other examples. §3.7
26 Monday 10/24/22 Approximating the area under a curve. Introduction to the integral. Computing sum of first $n$ numbers and first $n$ squares. §4.1,§4.2 Pset 9 (due Mon 10/31/22 by 11:59pm) Pset 9 (due Mon 10/31/22 by 11:59pm):
§4.1: 3,9,15,17 §4.2: 3,5,11,15,17,21,23 §4.3: 3,5,7,11,17
27 Wednesday 10/26/22 More on integrals. Using left endpoints, right endpoints, midpoints, random points. Integrals which can be computed using known area formulas. Integrals of functions which are both positive and negative. §4.1,§4.2, Appendix B
28 Friday 10/28/22 Examples of integral computations using Riemann sums. Formulas for sum of the first $n$ numbers, sum of the first $n$ squares, etc. Basic properties of integrals. The evaluation theorem (a.k.a. the second part of the Fundamental Theorem of Calculus). §4.2,§4.3, Appendix B
29 Monday 10/31/22 Definite versus indefinite integrals. Review of the evaluation theorem and other basic properties of integrals. Various examples. Proof of the evaluation theorem. §4.3 Pset 10 (due Mon 11/7/22 by 11:59pm) Pset 10 (due Mon 11/7/22 by 11:59pm):
§4.3: 24,29,31,40,48 §4.4: 1,5,9,10,13,15 §4.5: 2,3,4,5,10,17,23,30
30 Wednesday 11/2/22 Examples involving total displacement versus total distance traveled (integrate velocity versus integrate the absolute value of velocity). Integrals involving absolute values by breaking into a sum of integrals. Part I of the fundamental theorem of calculus. §4.3,§4.4
31 Friday 11/4/22 More on the fundamental theorem of calculus. Derivatives of integrals when the limits of integration are functions of $x$ (e.g. $\tfrac{d}{dx}\int_{x^2}^{x^3} \sin(t^3)dt$). The u-substitution rule and examples. §4.4,§4.5
32 Monday 11/7/22 Midterm review
33 Wednesday 11/9/22 Midterm 2 during class. Solutions.
Friday 11/11/22: Veterans day - no class
34 Monday 11/14/22 The average value of a function and the mean value theorem for integrals. Inverse functions. §4.4, §5.1 Pset 11 (due Mon 11/21/22 by 11:59pm) Pset 11 (due Mon 11/21/22 by 11:59pm):
§4.4: 20,21,31 §4.5: 38,40,43,47 §5.1: 9,13,22,25,28,35,38,43 §5.2: 1,2,4,13,15,23,32,39,59
35 Wednesday 11/16/22 Derivatives of inverse functions. The natural logarithm. §5.1, §5.2
36 Friday 11/18/22 Proof of the laws of logarithms. Logarithmic differentiation and integrals requiring logarithms (e.g. $\int \tan(x)dx$). Definition of $e$ and the natural exponential function. §5.2,§5.3
37 Monday 11/21/22 The natural exponential as the inverse of the natural logarithm. Differentiation and integral involving exponentials. Logarithms with any base. §5.4,§5.5
Wednesday 11/23/22: Thanksgiving break - no class Pset 12 (due Mon 12/5/22 by 11:59pm) Pset 12 (due Mon 12/5/22 by 11:59pm):
§5.3: 3,6,10,15,17,21,23,37,39,62,63 §5.4: 5,6,10,23,33,34,41,44 §5.5: 3,13
Friday 11/25/22: Thanksgiving break - no class
38 Monday 11/28/22 More on natural logarithms and exponentials. Derivatives and integrals of expressions like $a^{bx}$. Logarithms with base $b$ and change of base formula. Limit formula for $e$. §5.3,§5.4
39 Wednesday 11/30/22 Exponential growth. Examples such as population growth, Newton's law of cooling. §5.5
40 Friday 12/2/22 (last day of class) More on exponential growth. Radioactive decay. Continuously compounding interest. §5.5
Wednesday 12/7/22: 2-4pm final exam in ZHS 159 (check schedule)
Image sources: here.