18.014 | Fall 2010 | Undergraduate

Calculus with Theory


Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Recitations: 2 sessions / week, 1 hour / session


Apostol, Tom M. Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra. Waltham, Mass: Blaisdell, 1967. ISBN: 9780471000051.

(Vol. 2 will be needed for those who wish to continue on to 18.024 Multivariable calculus with theory.)

Additional course notes by James Raymond Munkres, Professor of Mathematics, Emeritus, are also provided.


We will assume a working knowledge of Calculus I (18.01 Single Variable Calculus), but no familiarity with proofs or proof writing.


We will cover the same material as 18.01 but with an emphasis on proofs and conceptual understanding rather than computation. Topics include:

  • Axioms for the real numbers;
  • The Riemann integral;
  • Limits, theorems on continuous functions;
  • Derivatives of functions of one variable;
  • The fundamental theorems of calculus;
  • Taylor’s theorem;
  • Infinite series, power series, rigorous treatment of the elementary functions.

Recitation Assignments

At the beginning of each unit, students will receive a schedule that includes problems to complete in advance of the next recitation. Students should be prepared to present their proofs in clear detail. This will give them the opportunity to very deliberately improve their proof writing skills.

Problem Sets

Problem sets are assigned weekly, and due the following week. At the end of the course, the lowest two pset scores for each student will be replaced by his or her average score on all of the psets.


There will be three in-class, one hour exams, and one three-hour comprehensive final exam.


Problem sets 20%
Midterms 40%
Final exam 30%
Participation 10%


Real numbers
0 Proof writing and set theory  
1 Axioms for the real numbers  
2 Integers, induction, sigma notation  
3 Least upper bound, triangle inequality  
4 Functions, area axioms  
The integral
5 Definition of the integral Pset1 due
6 Properties of the integral, Riemann condition  
7 Proofs of integral properties  
8 Piecewise, monotonic functions Pset2 due
Limits and continuity
9 Limits and continuity defined  
10 Proofs of limit theorems, continuity Pset3 due
11 Hour exam I Covers lectures 1-9
12 Intermediate value theorem  
13 Inverse functions  
14 Extreme value theorem and uniform continuity Pset4 due
15 Definition of the derivative  
16 Composite and inverse functions  
17 Mean value theorem, curve sketching Pset 5 due
18 Fundamental theorem of calculus  
19 Trigonometric functions  
Elementary functions; integration techniques
20 Logs and exponentials Pset 6 due
21 IBP and substitution  
22 Inverse trig; trig substitution Pset 7 due
23 Hour exam II Covers lectures 10-20
24 Partial fractions  
Taylor’s formula and limits
25 Taylor’s Formula  
26 Proof of Taylor’s formula Pset 8 due
27 L’Hopital’s rule and infinite limits  
Infinite series
28 Sequences and series; geometric series Pset 9 due
29 Absolute convergence, integral test  
30 Tests: comparison, root, ratio Pset 10 due
31 Hour exam III Covers lectures 21-29
32 Alternating series; improper integrals  
Series of functions
33 Sequences of functions, convergence  
34 Power series  
35 Properties of power series Pset 11 due
36 Taylor series  
37 Fourier series  
38 Final exam  

Course Info

As Taught In
Fall 2010
Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes