*University of British Columbia*

This is webpage is currently under construction.

**Instructor:** Guillermo Martinez Dibene.

**Location:** Buchanan A 102.

**Time:** Mondays, Wednesdays and Fridays 11:00-11:50.

**Office hours:** Tuesdays from 15:30 to 17:30 at Mathematics Annex 1118 (MATHX1118). If you cannot make it at standard office hours, you can send me an email and I will reply to you with times where we can meet.

**Email:**dibene [at] math [dot] ubc [dot] ca

*All relevant information across all sections is in the main Web page for the course. *

The grading scheme and the textbook & problem book can be found there.

**There exists the Maths Learning Centre (MLC) in Leonards S. Klinck (LSK) building, room 301 & 302.** There you will find an army of TA (between three to six) willing to help students with their doubts. Please, try the exercises by your own before going there since the demand for TA time is high, the time supplied from TA is relatively short per students (about ten to fifteen minutes). The MLC is a space for undergraduate students to study math together, with friendly support from tutors, who are graduate and upper-year undergraduate students in the math department. The MLC is located at LSK301 and LSK302 and is open 5 days a week from 11am-5pm. Every undergraduate student studying Math is welcome here!

*Logbook for the section.*

Here is the course book we are following.

- Week 0. Chapter zero and Appendix A
**Sept 6th and 8th.**Birds-eye view of high-school topics.

- Week 1. Chapter one, section from 1.1 until 1.5
**Sept 11th.**Notion of limit \(\lim\limits_{x \to a} f(x) = L\). Definitions of Cost (C), Revenue (R), Price (p), Quantity (q) and Profit (P). Simple, compounded and continuously compounded interest.**Sept 13th.**Notions of the limit \(\lim\limits_{x \to \infty} f(x) = L\) and of \(\lim\limits_{x \to a} f(x) = \infty\) and of lateral (or sided) limits \(\lim\limits_{x \to a^+} f(x) = L\) and \(\lim\limits_{x \to a^-} f(x) = L\).*Notice that these can be merged and so obtain, for example, the notion of \(\lim\limits_{x \to a^+} f(x) = \infty\).*Several examples of limits. There was a wrong derivation in previous class, please read and work the proposed exercised in the correct derivation.**Sept 15th.**Definition of*Average rate of change.*The geometrical interpretation as a slope and examples.

- Week 2. Chapter one, section 1.6. Chapter two, sections 2.1 until 2.3
**Sept 18th.**Notion of continuity and different types of discontinuities that we shall encounter: removable, jump, infinite and essential. Some examples, see section 1.6 in the book.**Sept 20th.**A typical business problem, the intermediate value theorem (IVT) in a geometrical version and several examples about it.**Sept 22nd.**Average rate of change revisited. The notion of tangent line using average rate of change.

- Week 3. Chapter two, sections 2.4 until 2.7
**Sept 25th.**The rules of differentiation for addition (or sum), product (or multiplication) and quotient (or division). Also, a differentiable function at a point is necessarily continuous there.**Sept 27th.**Problems with tangent lines with some twist. Examples of these include finding common tangent line to two curves, finding a tangent line with a given slope, finding a tangent line passing through some point (not in the curve), finding two different tangent lines that are parallel to each other and many more.*I cannot stress out more that simple examples can quickly become very algebraically involved*(such as the one done today in class) and so, you need to practice.**Sept 29th.**Review with lots of practice problems. Solutions sketch.

- Week 4. Chapter two, sections 2.8 until 2.10
**Oct 2nd.**Chain rule: "the slope of the best linear approximation of the composition of two functions \(g \circ f\) is the slope of the composition of the corresponding best linear approximations, namely \((g \circ f)'(c) = g'(f(c)) f'(c)\)." Several examples.**Oct 4th.**First midterm.**Oct 6th.**Examples involving Chain rule. Feedback from student.

- Week 5. Chapter two, sections 2.11 until 2.14
**Oct 9th.**Happy thanksgiving.**Oct 11th.**A brief serious talk about the midterm results. Implicit differentiation.**Oct 13th.**The mean value theorem.

- Week 6. Chapter three, sections 3.1 until 3.3 & Notes on Price Elasticity of Demand and on Continuously Compounded Interest
**Oct 16th.**Price elasticity of demand. We didn't have time to revisit Continuously compounded interest, you should read the notes provided.**Oct 18th.**Exponential-vs-polynomial growth and a first look at Ordinary Differential Equations.**Oct 20th.**Optimisation I. Fundamental*Extreme-value theorem:*a continuous function defined on a closed interval \(f:[a, b] \to \mathbb{R} \) attains its maximum and minimum, that is to say, there are two points (at least) \(x_1^*\) and \(x_2^*\) in \([a, b]\) with \(f(x_1^*) \leq f(x) \leq f(x_2^*)\) for all \(x \in [a, b]\). We also saw that if a continuous function \(f:[a, b] \to \mathbb{R}\) is also differentiable on \((a, b)\) then any extreme point (or optimiser, as I called them) \(x^* \in (a, b)\) will satisfy \(f'(x^*)=0\).

- Week 7. Chapter three, section 3.5. I also wrote notes for the first order criteria.
**Oct 23rd.**Optimisation II. The idea of global (or absolute) optima (or extrema) vs global (or local) optima (or extrema). A first look at worded problems.**Oct 25th.**Optimisation III. The concept of critical point and singular point (the candidates, as I call them). More worded examples.**Oct 27th.**Optimisation IV. Second order criteria. The following example.

- Week 8. Chapter three, section 3.5 and section 3.2.
**Oct 30th.**Optimisation: final chapter. Birds-eye view on the possible ways to go tackle an optimisation problem. These two examples.**Nov 1st.**Guest lecturer. Examples on Related Rates.**Nov 3rd.**Review for midterm two. And solution to exercise three.

- Week 9. Chapter three, section 3.6.
**Nov 6th.**Introduction to the idea of convexity and concavity.**Nov 8th.**Second midterm and its solution.**Nov 10th.**Introduction to the idea of asymptote and tracing of a few examples.

- Week 10. Chapter three, section 3.6. Specifically, 3.6.5 and 3.6.6.
**Nov 13th.**Remembrance day long weekend.**Nov 15th.**Example (final 2006).**Nov 17th.**Example (final 2012) and some exercises.

- Week 11. Chapter three, section 3.4. Specifically, 3.4.1 and 3.4.2.
**Nov 20th.**Introduction of constant approximation and linear approximation (also known as -AKA- first order approximation or linearization). Introduction to error bounds.**Nov 22nd.**Error bound for the first derivative \(R_1(a; h) \leq M h^2\) where \(M > 0\) is any bound of \(f''(x)\) for \(a < x < a + h\) when \(h > 0\) or for \(a + h < x < a\) when \(h < 0\).**Nov 24th.**Examples and exercises regarding linear approximation.

- Week 12 and final. Chapter three, section 3.4. Specifically, 3.4.3 and 3.4.4.
**Nov 20th.**Examples of linear and quadratic approximation. Bounding worst-case error term.**Nov 22nd.**Taylor polynomials of higher degree.**Nov 24th.**Grand finale.

*Written Assignments (roughly biweekly).* All assignments are focused on *exposition of ideas* rather than on maths content (that is Webwork for). Short arguments should be given in the main steps of the solution. You don't need to make an essay out of the assignment, but *an assignment containing only formulae, even if they make sense, sall receive a mark of zero (0).*

Please **remember** to write your **name** or preferred name and **student number.**

**To pick up your assignments**, you will need to go to the MLC (see above for information of the MLC) with your **student ID** and two things: (1) under no circumstances you do not have the right to access the drawers where the assignments are guarded and (2) you cannot ask for other students' assignments, *doing so would violate UBC Privacy Information Policy.*

**First assignment.***Notice the hand-in date for Wed, Sept 20th.***Second assignment.***Notice that the hand-in date is for next Wednesday, Sept 27th and*This is because Oct 4th is the first midterm and you need to practice at least another written assignment.**not**for Oct 4th.**Third assignment.***Notice that the hand-in date is for Wed, Oct 18th.*Solution.**Fourth assignment.***Notice that the hand-in date is for Wed, Oct 25th until 17:00hrs.***Fifth assignment.***Notice that the hand-in date is for Wed, Nov 1st.***Sixth assignment.***Notice that the hand-in date is for Wed, Nov 15th.***Seventh assignment.***Notice that the hand-in date is for Wed, Nov 22th.***Eighth and final assignment.***Notice that the hand-in date is for Fri, Dec 1st.*

*Schematic relation of the maths ideas in the course.*

*Policy for the midterms.*

To be announced. The only disclosable information is that it will be in class time (50 minutes), so likely you won't have time to try out the exercises during the midterm and you will need to know how to solve them in advance. Study all the available material. Here it is the **official midterm one policy.**

*More material for you to practice.*

- The UBC wiki resources for MATHS 104 and 184.
- You can go to the UBC Library and look for the book "Problem Book For First Year Calculus" by Bluman, George (he is a Maths professor here in UBC) in
*electronic format.*You can download it with your CWL (your tuitions pay this service).*Problems 1 to 46 in pages 163 to 170 are good practice for this course.* - Notes on business terminology and a solved example. Also, there is a basic version.
- Business problems for
*week 1.* - Business problems for
*week 4.* - Elasticity of Demand problems and Continuously Compounded Interest problems for
*week 5.* **Extra problems for no credit.**- Continuity and Intermediate Value Theorem.
- Chain rule. Mark has also provided some related rates problems with solutions.
- Implicit Differentation.
- Mean Value theorem & Logarithmic differentiation.
- Exponential growth and a first look at ordinary differential equations. I have added a solution to exercise 3.
- Optimisation and its solution set. Mark has also provided some optimisation problems.

Return to Guillermo's main Web page

* Last update:* October 22, 2017