MTH 312 -- Spring 2017 Calendar
- For Lectures refer to sections from the class text [F].
- Recitations are on Friday 4-5pm.
- Red colors are for due dates for HW.
- Orange colors are for quizzes.
- Green colors are for worksheets.
Lectures: Material Covered
- Lecture 1. 4/3: Introduction to Class. Start Chapter 6. Sec 6.1: Partitions, Lower and Upper sums and refinements.
HW # 1: Turn in neatly handwritten (or typed) solutions to problems
- Sec 6.1 # 3,7
- Sec 6.2 # 7,8.
in class on Monday 4/10.
- Lecture 2: 4/5: Sec 6.1, 6.2: Properties of refinements. Lower and upper integrals. Statement of Archimedes-Riemann Theorem.
- Lecture 3: 4/7 Sec 6.2: Proof of Archimedes-Riemann Theorem (if part).
- Recitation 1: 4/7 Sec 6.2: Examples to prove integrability of a bounded function using the Archimedes-Riemann Theorem. Worksheet # 1: Proving that Lipschitz functions on [a,b] are R-integrable using the Archimedes-Riemann Theorem.
Worksheet # 1: Turn in your worksheet solutions in class on Fri, April 4/14
- Lecture 4: 4/10 Sec 6.2: Proof of Archimedes-Riemann Theorem (only if part), and showing that the integral is the limit of lower or upper sums. Sec 6.4 A continuous function on a closed bounded interval is R-integrable (did part of the proof). Read Sec 6.3, 6.4 for next class.
Quiz #1 on Fri, 4/14 (Definition of R-integrability of a bounded function on a bounded domain, Archimedes-Riemann Theorem)
HW # 2: Turn in neatly handwritten (or typed) solutions to problems
- Sec 6.2 # 10
- Sec 6.3 # 6.
- Sec 6.4 # 4.
in class on Monday 4/17.
- Lecture 5: 4/12 Sec 6.3-6.4: R-i.ntegrability of a continuous function on a closed bounded interval, R-integrability of monotone increasing functions. Started R-integrability of a continuous function on an open interval (a,b) that is bounded on the closed interval [a,b].
- Lecture 6: 4/14 Sec 6.3-6.4: R-integrability of a continuous function on an open interval (a,b) that is bounded on the closed interval [a,b]. Properties of the R-integral: linearity, monotonicity and additivity. Next week: More properties of the integral and the fundamental theorems of Calculus.
- Recitation 2: 4/14 Worksheet # 2: Continuous Functions and Integrability. Work on this worksheet during the week and bring to recitation on Fri, April 4/21. Quiz # 1 in class.
- Lecture 7: 4/17 Sec 6.3, 6.5 properties of the integral: Additivity, Monotonicity, linearity and triangle inequality. Main ideas of proofs based on the Archimedes-Riemann Theorem. Statement of the first fundamental theorem of Calculus. Next class, Sec 6.5
HW # 3: Turn in neatly handwritten (or typed) solutions to problems
- Sec 6.5 # 2,3
- Sec 6.6 # 2ac, 7
in class on Monday 4/24.
- Lecture 8: 4/19 Sec 6.5 first fundamental theorem of Calculus (FTC) and proof. Statement of second FTC and some remarks. Next class, Sec 6.6: Mean value Theorem for integrals and proof of second FTC.
- Lecture 9: 4/21 Sec 6.6 Mean value theorem for integrals. The indefinite integral is continuous for integrable f.
- Recitation 3: 4/21 Worksheet # 3: FTC.
Worksheet # 2 corrections due on Monday 4/24.
- Lecture 10: 4/24 Sec 6.6 Proof of the second FTC. Examples.
HW # 4 : Turn in neatly handwritten (or typed) solutions to problems
- Sec 6.6: # 3,5,8
- Sec 8.1: # 3
in class on Monday 5/1
Worksheet # 3 due on Friday 4/28
Quiz #2 on Fri, 4/28 (Definition of antiderivative, indefinite integral, statements of first and second FTC, Statement of Mean value theorem for integrals and its proof.
- Lecture 11: 4/26 Sec 8.1 Taylor polynomials: contact of order n, existence, uniqueness. Review Cauchy Mean Value Theorem and its consequences on page 111.
Next, sec 8.2: Lagrange Remainder Theorem.
- Lecture 12: 4/28 Sec 8.2: Cauchy Mean Value Theorem. Lagrange Remainder Theorem. Detecting maxima and minima using the remainder term.
- Recitation 4: 4/28 Worksheet # 4: A discrete approximation to the ln function. Please work on this over the week and bring to Recitation 5. There will be time to complete this worksheet in class during next recitation.
- Lecture 13: 5/1 Sec 8.3: Convergence of Taylor polynomials. Review of convergence of series of real numbers. Examples: Sec 8.6: A non-analytic infinitely differentiable function.
HW # 5: Turn in neatly handwritten (or typed) solutions to problems
- Sec 8.1: # 5, 6
- Sec 8.2: # 8, 11
- Sec 8.3: # 3, 5
in class on Friday 5/12
- Lecture 14: 5/3 Sec 8.3: Convergence of Taylor polynomials. Convergence Theorem. Example: cos(x)
- Lecture 15: 5/5 Review for Midterm (print review sheet on Canvas and bring to class)
- Recitation 5: 5/5 Worksheet 4: Spend time in class doing this.
Worksheet # 4 Will be due on Wed 5/10, along with WK# 3 corrections.
- Lecture 16: 5/8 MIDTERM
- Lecture 17: 5/10 Sec 8.5, Cauchy form of remainder term. Generalized Mean Value Theorem. Derivation of Lagrange form from Cauchy form.
- Lecture 18: 5/12 Sec 8.7 Weierstrauss's Approximation Theorem
HW # 6: Turn in neatly handwritten (or typed) solutions to problems
- Sec 8.5: # 4, 6
- Sec 9.1: # 1a, 2, 6
in class on Friday 5/19
Quiz #3 on Fri, 5/19 on Sec 8.5. Cauchy form of the Remainder term.
and generalized mean value theorem for integrals.
- Recitation 6: 5/12 Worksheet # 5: Convergence of Series: Comparison test, p-test and alternating series test. Please work on the sheet and bring to recitation on Friday 5/19.
Worksheet # 5 due on Friday 5/19 during recitation.
- Lecture 19: 5/15 Sec 9.1 Cauchy sequences of real numbers, Cauchy convergence criteria for a sequence of real numbers.
- Lecture 20: 5/17 Sec 9.1 Proof of Cauchy convergence criteria. Infinite series of real numbers.
- Lecture 21: 5/19 Sec 9.1 Tests for convergence of series (comparison, integral, p, alternating, absolute convergence)
- Recitation 7: 5/19 Worksheet # 5 solutions in class. Quiz # 3.
- Lecture 22: 5/22 Sec 9.2 Pointwise convergence of a sequence of functions
- Lecture 23: 5/24 Sec 9.3 Uniform convergence of a sequence of functions
HW # 7: Turn in neatly handwritten (or typed) solutions to problems
- Sec 9.2: # 4, 5
- Sec 9.3: # 2,3
in class on Wednesday 5/31
- Lecture 24: 5/26 Sec 9.3 Uniformly Cauchy sequences of functions. Weierstrass uniform convergence criteria of a sequence of functions.
- Recitation 8: 5/26 Worksheet # 6: Pointwise and Uniform convergence of a sequence of functions.
Worksheet # 6 and Worksheet # 5 corrections due on Friday 6/2 during recitation.
Quiz #4 on Fri, 6/2 on Sec 9.2 and 9.3. Study the definitions of pointwise convergence and uniform convergence of a sequence of functions, the definition of uniformly Cauchy for a sequence of functions and the Weierstrass uniform convergence criteria (Theorem 9.29 on page 247)
- Lecture 25: 5/29 Memorial Day holiday
- Lecture 26: 5/31 Sec 9.4 Uniform limit of a sequence of functions
HW # 8: Turn in neatly handwritten (or typed) solutions to problems
- Sec 9.4: # 1,2,3 (Hint: Use the first FTC)
- Sec 9.5: # 7 (Hint: use Theorem 9.40)
in class on Wednesday 6/7
- Lecture 27: 6/2 Sec 9.4 Uniform limit of a sequence of functions (see notes on Canvas)
- Recitation 9: 6/2 Quiz # 4.
- Lecture 28: 6/5 Sec 9.4 Uniform limit of a sequence of functions (see notes on Canvas)
- Lecture 29: 6/7 Fourier Series
- Lecture 30: 6/9 Review for final (print review sheet on Canvas and bring to class)
- Recitation 10: Review for final