# Math 142a Winter 2014

### Instructors

#### Professor:

Name Office E-mail Phone Office Hours Lecture Time Lecture Place
Prof. Daniel Rogalski AP&M 5131 drogalsk@math.ucsd.edu 534-4421 T 2-3pm, W 11am-12pm MWF 1-1:50pm PCYNH 122

#### Teaching Assistants:

Name Office E-mail Office Hours Section Times Section Place
Rob Won AP&M 6321 rwon@ucsd.edu M 2:30-3:30pm, Th 10-11am A01, W 6-6:50pm; A02, W 7-7:50pm HSS 2150
Jay Cummings AP&M 6321 jjcummings@ucsd.edu Th 1-2pm A03, W 5-5:50pm AP&M 5402

### Calendar

#### We will cover primarily the following sections of the text Advanced Calculus" by Patrick M. Fitzpatrick: Preliminaries, 1.1-1.3, 2.1-2.4, 3.1-3.7, 4.1-4.4. Below we list which sections of the text we expect to cover when, but this is only an approximation. As each lecture occurs we will write brief summaries of what was covered in each lecture. The homework assignments will also be posted in the calendar below. Homework is due on Fridays in the homework boxes in the basement of AP&M by 5pm. Please submit your homework to the box with the name of the TA of your section.

(1/6/14) Preliminaries, Section 1.1: The axioms of the real numbers: Field axioms, positivity axioms, the completeness axiom. Definition of upper bound and the supremum (least upper bound). Some consequences of the field and positivity axioms. Notation for intervals.

(1/8/14) Section 1.1: More discussion of the completeness axiom. The number 3 has no square root in the rationals. The number 3 does have a square root in the reals, namely the supremum of the set of nonnegative real numbers x such that x^2 < 3. (Please review proof by induction, which is covered in math 109, in the text).

(1/10/14) Section 1.2. Lower bounds and the infimum (greatest lower bound). The Archimedean principle: every positive real c is smaller than some natural number n. Lemmas about the position of the integers and rational numbers in the reals including (1) If n is an integer, there is no integer in the interval (n, n+1); (2) for a real number c, the interval [c, c+1) contains a unique integer; and (3) the rational numbers are dense in the reals, that is for any real numbers a < b the interval (a, b) contains a rational number.

Homework 1 due 1/10/14: Exercises 1.1 #2, 5, 15, 18, 19

(1/13/14) Section 1.3: Properties of the absolute value, including the triangle inequality. Formula for the difference of nth powers. Formula for the geometric series. The binomial formula.

(1/15/14) Section 2.1: Intuitive idea of a limit of a sequence and examples of sequences we will want and will not want to be convergent. Definition of the convergence of sequence. Using this definition to carefully prove convergence of {1/n}, {1/n^3 - 1/n + 1}, {(2^n -1)/(2^{n-1})}. Proof that the sequence 1, -1, 1, -1, ... does not converge.

(1/17/14) Section 2.1: Properties of convergent sequences. If {a_n} converges to a and {b_n} converges to b, then (i) {a_n + b_n} converges to a + b; (ii) {a_n b_n} converges to ab (different proof from text); (iii) {1/b_n} converges to 1/b (provided all b_n and b are not zero); and (iv) {a_n/b_n} converges to a/b (provided all b_n and b are not zero).

Homework 2 due 1/17/14: Exercises 1.2 #1, 6, 7; Exercises 1.3 #8, 10, 18, 21, 22, 27

(1/20/14) MLK Day (NO CLASS)

(1/22/14) Section 2.2 Bounded subsets of the real numbers. If {a_n} is a convergent sequence, the set {a_n | n \geq 1 } is bounded. Conversely, a bounded sequence need not converge. Another point of view on density: a set S of real numbers is dense if and only if every real number c is the limit of a some sequence of numbers in S. Closed sets. The interval [a,b] is closed, with proof. Also, (-infty, b] and [a, infty) are closed. On the other other hand, open intervals (a, b) or half open intervals (a, b], [a, b) are not closed.

(1/24/14) Section 2.3: Monotonically increasing and monotonically decreasing sequences. The monotone converge theorem. Examples using series: note that if { a_n} is a sequence with nonnegative terms, the series s_n = \sum_{k=1}^n a_n is automatically monotonically increasing. Thus it coverges if and only if it is bounded above. The harmonic series \sum_{k=1}^n 1/n is not bounded above, so it is not convergent.

Homework 3 due 1/24/14: Exercises 2.1 #1, 2(a), 6, 10, 14, 18; Exercises 2.2 #1, 2, 4

(1/27/14) Section 2.3-2.4: Sequences of the form a_n = c^n converge if |c| < 1 and diverge if |c| > 1 (if c = 1 it converges, c = -1 it diverges). Example: lim n/3^n = 0 (compare with (2/3)^n by showing n < 2^n for all n.) Definition of a subsequence of a sequence. If a sequence converges, any subsequence also converges to the same value.

(1/29/14) Review for Midterm

(1/31/14) Midterm Exam 1

Practice exam 1 with solutions (written by Prof. Rogalski----our exam will have similar format)
Practice exam 2 with solutions (This is an exam from Prof. Egger's Math 142a class in Winter 2008. Thanks to Prof. Eggers for sharing it. Problem 4 had an error, so I would concentrate on problems 1-3.)

### This quarter's midterm exam 1 with solutions

(2/3/14) Section 2.4: Subsequences. If {a_n} is a convergent sequence, then every subsequence of {a_n} converges to the same limit. Every sequence has a monotone subsequence (either monotonically increasing or monotonically decreasing. In particular, every bounded sequence has a monotone subsequence which is therefore convergent by the Monotone convergence theorem. A set S of real numbers is sequentially compact if every sequence of elements in S has a convergent subsequence whose limit is in S. Any closed, bounded set is sequentially compact (in particular, the interval [a, b] is sequentially compact).

(2/5/14) Section 3.1: Real valued functions. Continuity. Definition of continuity at a value x_0 using sequences. Examples of continuous and discontinuous functions. Properties of continuous functions: the sum and product of functions which are continuous at x_0 are also continuous at x_0. If f, g: D to R are continuous at x_0 and if g(x) is nonzero at x_0, then the quotient function f/g is continuous at x_0 also. The composition of continuous functions is continuous.

(2/7/14) Section 3.2: If f: D to R is a function on a domain D, then f(D) is called the image of f. If f(D) has a maximum (that is, it is bounded above and contains its supremum), we call it a maximum of the function f (on the domain D). Similarly, if f(D) has a minimum (equivalently, it is bounded below and contains its infimum), we call it a minimum of the function f. Examples of functions which do and do not have a maximum and minimum. Lemma: A continuous function f: [a, b] to R has bounded image. Theorem (Extreme value theorem): A continuous function f: [a, b] to R has both a maximum and minimum.

Homework 4 due 2/7/14: Exercises 2.3 #1, 2, 4, 5, 6, 8; Exercises 2.4 #1, 2, 7, 8.
(Note: I think 2.3 #6 is just as easy without using the comparison lemma, which we didn't explicitly cover; but feel free to use the comparison lemma or not as you wish.)

(2/10/14) Section 3.3 The intermediate value theorem. Theorem: if f: [a, b] to R is continuous, and f(a) < c < f(b) or f(b) < c < f(a), then c = f(x) for some x in [a, b]. Proof using the bisection method. This also depends on the the nested interval theorem (theorem 2.29), which we didn't cover before but covered in this lecture. Applications: showing a function must have a root, or showing a function must have a fixed point.

(2/12/14) Section 3.4 Uniform continuity: a function f: D to R is uniformly continuous if whenever {u_n}, {v_n} are sequences in D such that lim (u_n-v_n) = 0, then lim (f(u_n) - f(v_n)) = 0. Examples: f: R to R with f(x) = x^2 is not uniformly continuous. f: (0,1) to R with f(x) = 1/x is not uniformly continuous. Intuition: the fact that these functions have places where the tangent lines get arbitrarily close to vertical suggests they won't be uniformly continuous. Theorem: a continuous function f: [a, b] \to R is uniformly continuous.

(2/14/14) Section 3.5: The epsilon-delta criterion at a point: If f: D to R is a function then f satsifies the epsilon-delta criterion at x_0 if given epsilon > 0, there exists delta > 0 such that for x in D, |x - x_0| < delta implies |f(x) - f(x_0)| < epsilon. Example: Proving the epsilon-delta criterion for the function f(x) = x^2, at an arbitrary point x_0. Theorem: a function satisfies the epsilon-delta criterion at a point if and only if it is continuous at that point. A function f: D to R satisfies the epsilon-delta criterion on the domain D if given epsilon > 0, there exists delta > 0 such that whenever |u-v| < delta for u, v in D, then |f(u) - f(v)| < epsilon. This turns out to be a way to reformulate uniform continuity: A function f: D \to R satisfies the epsilon-delta criterion on the domain D if f is uniformly continuous.

Homework 5 due 2/14/14: Exercises 3.1 #1, 3, 6, 9; Exercises 3.2 #1, 4, 5; Exercises 3.3 #4, 6, 9.

(2/17/14) President's Day (NO CLASS)

(2/19/14) Section 3.6 Monotone functions: Monotonically increasing/decreasing and strictly increasing/decreasing functions. Intervals. Convex sets. Theorem: if f: I to R is continuous and I is an interval, then f(I) is also an interval. Theorem: If f: D to R is a monotone function, and f(D) is an interval, then f is continuous.

(2/21/14) Section 3.6: One-to-one functions. A strictly increasing or strictly decreasing function is one-to-one. A one-to-one function f: D to R has an inverse f^{-1}: f(D) to R. If f is strictly increasing or strictly decreasing, then the inverse function f^{-1} has the same property. If f: D to R is strictly increasing or strictly decreasing and D is an interval, then f^{-1} is automatically continuous. Examples of inverse functions: ln x is the inverse of e^x. The nth root function of x is the inverse of x^n.

Homework 6 due 2/21/14: Exercises 3.4 # 5, 6, 10, 11 Exercises 3.5 #5, 7, 8
(Note: For any problem involving uniform continuity, even in Section 3.4, you can use the epsilon-delta formulation of uniform continuity given in Theorem 3.22 if you find it easier to use than the definition using sequences.)

(2/24/14) Section 3.7: Limits. Limit point x_0 of a set D: a real number such that there exist a sequence x_n of points in D, not equal to x_0, such that {x_n} converges to x_0. Given a function f: D to R and a limit point x_0 of D, we say the limit of f(x) as x approaches x_0 is equal to a number c if for all sequences {x_n} of elements in D (not equal to x_0) converging to x_0, the limit of {f(x_n)} is c. Definition of derivative. Some examples. Properties of limits: For sums, products, quotients, compositions of functions, limits behave as expected.

(2/26/14) Review for Midterm 2

(2/28/14) Midterm Exam 2

Sample exam for midterm 2 with solutions (our exam will have similar format)

Another sample exam (no solutions) (This is an exam from Prof. Egger's Math 142a class in Winter 2008. Problem 3 is about a topic from Section 3.7, which is not covered on our Midterm 2.)

### This quarter's midterm exam 2 with solutions

(3/3/14) Section 4.1 Definition of derivative. Interpretation as the slope of a tangent line. Calculating the derivative of f(x) = x^n from the definition. If a function is differentiable at a point, it is continuous at that point. An example of a continuous function which is not differentiable (the absolute value function f(x) = |x| at x_0 = 0).

(3/5/14) Section 4.1 Properties of derivatives. (f + g)'(x) = f'(x) + g'(x). The product and quotient rules, with proofs. Quotients of polynomials p(x)/q(x) are differentiable at all points where q(x) is not zero.

(3/7/14) Section 4.2. The Chain rule. The derivative of an inverse. As an example of the derivative of an inverse function, the function f^{-1}(y) = y^{1/n} (which is the inverse of f(x) = x^n). As another example, the inverse of f(x) = e^x is f^{-1}(y) = ln y, from which if one assumes that f'(x) is its own derivative, one gets that the derivative of ln y is 1/y.

Homework 7 due 3/7/14: Exercises 3.6 #1, 8, 13, 14; Exercises 3.7 #4; Exercises 4.1 #1, 4, 9, 12.

(3/10/14) Section 4.3 Local maxima and minima. If f is differentiable at a point x and has a local max or local min there, then f'(x) = 0. Rolle's theorem. The Mean value theorem. Application: if f is differentiable, the the number of zeroes of f is bounded by the number of zeroes of the derivative of f, plus 1.

(3/12/14) Section 4.3-4.4 A differentiable function f is constant if and only if f'(x) = 0 for all x. Two differentiable functions g and h have g'(x) = h'(x) for all x if and only if g(x) = h(x) + c for some constant c, for all x. If f is differentiable and f'(x)> 0 for all x on an interval I, then f is strictly increasing on I. Similarly, if f'(x) < 0 on an interval I then f is strictly decreasing on I. The 2nd derivative test. The Cauchy mean value theorem. (Theorem 4.24 is skipped).

(3/14/14) review

Homework 8 "due" 3/14/14: Exercises 4.2 #2, 4, 5 ; Exercises 4.3 #1, 10, 11, 15, 19, 21; Exercises 4.4 #3, 4
(Note: This homework is not to be handed in and does not count towards your grade. It is for you to do on your own in preparation for the final. There is likely to be a problem on the final similar to one of these problems.)

(3/21/14) Final Exam 11:30am-2:30pm

Sample final exam with solutions

Another sample final exam (no solutions) (This is an exam from Prof. Egger's Math 142a class in Winter 2008.)