Methods of Mathematics: Calculus, Statistics and Combinatorics

**Professor: **
Bernd Sturmfels
(office 925 Evans, e-mail `bernd@math.berkeley.edu`)

**Office Hours:** Monday 11:30am-12:30pm and Tuesday 8:00-9:00am.

- Natth Bejraburnin, office hours Tuesday 4-5pm and Friday 11-12am in 787 Evans.
- Ralph Morrison, office hours Monday 1:30-2:30pm and Thursday 1:30-2:30pm in 941 Evans.
- Ngoc Tran, office hours Monday 1-2pm and Wednesday 1-2pm in 391 Evans.
- Markus Vasquez, office hours Wednesday 2-3pm and Thursday 2-3pm in 1056 Evans.

**For extra help:** check
out the
Student Learning Center.
Their 1-unit course meets
meets TuTh 11-12:30 in 113 Chavez.

Wednesday, Nov 28, 1-2pm: Ngoc in 334 Evans

Thursday, Nov 29, 1-2pm: Ngoc in 334 Evans

Friday, Nov 30, 1-2pm: Ngoc in 340 Evans

Monday, Dec 3, 4-6pm: Natth in 35 Evans (Review)

Wednesday, Dec 5, 8-10am: Bernd in 105 Stanley (Review)

Wednesday, Dec 5, 1:30-3:30pm: Ralph in 941 Evans

Wednesday, Dec 5, 4:00-5:00pm: Markus in 31 Evans (Review)

Thursday, Dec 6, 2:00-4:00pm: Markus in 1056 Evans

Thursday, Dec 6, 5:30-6:30pm: Ralph in 941 Evans

Friday, Dec 7, 1-3pm: Natth in 762 Evans

Monday, Dec 10, 9:00-10:30am: Bernd in 939 Evans

This is the first semester of an introductory college-level mathematics course. It is primarily intended for majors in the life sciences. Math 10a covers: Introduction to differential and integral calculus of functions of one variable. Representation of data, elementary probability theory, statistical models and testing. The syllabus appears below. The continuation in the spring (Math 10b) will cover: Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, differential equations.

- Friday, August 24: Sets and Functions, Polynomials, Hardy-Weinberg Law, Composition of Functions
- Monday, August 27: Exponential Function, Tumor Growth, Inverse Function, Logarithm Function, Log-Log Plots
- Wednesday, August 29: Statistics Basics, Data Types, Histograms, Step Functions
- Friday, August 31: Coin Tosses, Simulated Data, Gaussian Function, Normal Distribution
- Wednesday, September 5: Axioms for Area, Integral of Step Function, Properties of Integral, Limits
- Friday, September 7: Area under a Curve, Riemann Integral, Left and Right Endpoint Approximations, Midpoint Rule, Trapezoid Rule
- Monday, September 10: Random Variables, Outcomes, Events, Discrete Probability, Binomial Distribution
- Wednesday, September 12: Cumulative Distributions, Continuous Probability Distributions, Normal Distribution Revisited
- Friday, September 15: Slope of a Line, Slope of a Curve, Derivative of a Function
- Monday, September 17: Power Rule, Constant Multiple Rule, Sum Rule, Difference Rule, Product Rule, Chain Rule, Quotient Rule
- Wednesday, September 19: Implicit Differentiation, Higher Derivatives
- Friday, September 21: Graphing Functions, Extrema, Inflection Points, Asymptotes
- Monday, September 24: Local Extrema, Critical Points,
Convex/Concave, Second Derivative Test, Global Maxima and Minima

Wednesday, September 26: Review for First Midterm

Friday September 28: First Midterm Exam - Monday, October 1: Approximation of Functions, Taylor Polynomials
- Wednesday, October 3: Taylor Polynomials and Numerical Optimization
- Friday, October 5: Newton's Method
- Monday, October 8: Infinite Sums and Series
- Wednesday, October 10: More General Series, Test for Divergence, Ratio Test, Root Test, Power Series, Taylor Series
- Friday, October 12: Poisson Distribution, Sequences and Assembly of DNA sequences
- Monday, October 15: Box Models, Long Run Behavior, Expected Value, Standard Error, Sampling Distribution, Central Limit Theorem
- Wednesday, October 17: Parameters and Statistics, Sampling, Bias and Variability
- Friday, October 19: Statistical Models
- Monday, October 22: Estimators, Likelihood Function, Maximum Likelihood Estimation
- Wednesday, October 24: Multiple Random Variables, Joint Density Function, Independence
- Friday, October 26: Multinomial Distribution,
Hardy-Weinberg Model,
Genetics Example,
Body Temperature Example

Monday, October 29: Review for Second Midterm

Wednesday, October 31: Second Midterm Exam - Friday, November 2: Antidifferentiation
- Monday, November 5: Existence of Antiderivatives, Indefinite Integrals
- Wednesday, November 7: Fundamental Theorem of Calculus, Area Function
- Friday, November 9: Chain Rule Revisited, Substitutions in Definite Integrals, Exploiting Symmetry
- Wednesday, November 14: Integration by Parts
- Friday, November 16: Hypothesis testing, Type 1 and 2 errors, Test statistic, Rejection regions, P-value
- Monday, November 19: Z-tests

Wednesday, November 21: Statistics Review - Monday, November 26: Student's t curve, T-tests

Wednesday, November 28: Calculus Review

Friday, November 30: Ralph's Review

The slides for each lecture and the homework problems have been assembled into a reader. A hard copy of the Math 10A reader can be purchased for $29.25 at Copy Central. The reader is also available for free to registered students as a pdf file on BSpace. We also recommend that you purchase the custom edition of Stewart's Calculus. For supplementary material on Statistics please use the free online notes and lectures posted by Professor Philip Stark. A custom edition of Rosen's Discrete Mathematics will be made available for Math 10B.

Writing is an essential part of studying mathematics and other quantitative subjects. Please use this course as an opportunity to practise your writing skills. You are expected to use complete sentences in all your written work. We will take this into consideration when grading the homework and the exams.

There will be a quiz in your GSI section on every Friday, starting on August 31.

There will be two in-class midterms, on Friday, September 28, and on Wednesday, October 31. A review session will be held in the lecture preceding the midterm. *No books, notes, electronic devices, scratch paper or collaboration are permitted at any exam*. The final exam will be on Monday, December 10, 7-10pm, in 220 Hearst Gymnasium. Please plan your holiday travels accordingly. Your student photo ID is required at all exams. No make-up exams will be given.

Homework 10%, Quizzes 10 %, Midterms 20% each, Final 40%. If a student does not take midterm #1, then midterm #2 will count for 40% of the grade. If a student takes midterm #1 but not midterm #2, the final exam will count for 60%. Students who take neither midterm will fail the course. There will be no make-up exames. Incomplete grades are very rarely given, and only for a documented serious medical problem or genuine emergency, provided you have a C average on previous coursework.

- Week 1: Finite sets, functions, balls and boxes (the 12-fold way)
- Week 2: Counting, permutations, binomial coeficients
- Week 3: Discrete probability theory, conditional probability, Bayes' rule.
- Week 4: The Poisson approximation to the binomial distribution and Poisson random variables
- Week 5: Difference equations, recursion
- Week 6: Algorithms and dynamic programming
- Week 7: Differential equations I
- Week 8: Differential equations II
- Week 9: Inference for discrete distributions
- Week 10: Least squares and correlation
- Week 11: Linear equations
- Week 12: Matrix algebra
- Week 13: Relations and partially ordered sets
- Week 14: Graphs, trees and phylogenetics