# Math 41 — Calculus ACE

For information common across all discussion sections, consult the main Math 41 course website for Autumn 2010.

This page (maintained by the ACE TA, Jonathan Lee) provides information specific to the ACE section of Math 41; note that some of it may differ from the non-ACE sections!

This section has a CourseWork page; it is used for checking your grades and little else.

## Who, what, when, where

### Discussion section

• Tuesdays — 1:15–3:05pm
• Thursdays — 1:15–3:05pm
The primary difference from the mainstream Math 41 sections is that the ACE sections meet for two hours a day, instead of one. These are held in room 203 in the Lane History corner.

### Lecture section

There is no special Math 41ACE lecture session. Students attending the 41ACE discussion section are expected to attend one of the mainstream lectures taught by Mark Lucianovic (without signing up for it!):
• 10:00–10:50 MWF, room 380-380C (section 11)
• 11:00–11:50 MWF, room 200-203 (section 01)
See the Math 41 section page for details.

### Office hours (tentative)

• Tuesdays — 9:30–10:50am
• Thursdays — 9:30–10:50am
These are held in room 380-T, in the basement of the math building. Please note that you are not limited to my office hours; you're most welcome to attend those held by any Math 41 instructor. See the complete list of office hours.

### Assignment submission

This quarter, Math 41 assignments are due at 3:00pm on Tuesdays. As the Math 41ACE section leader, I will be collecting them at the end of section.

Pending...

Pending...

## What happened?

Daily summaries will be posted here. In order for the math symbols to display properly, you'll need a fairly recent web browser with JavaScript enabled, such as one of those listed here.
• December 2 — review handout
• November 30
1. integration practice from previous finals!
• November 16 — handout
• November 11 — handout
1. mean value theorem
2. Riemann sums — know how to rearrange sums of squares and cubes as sum of integers starting from $$1$$
3. know how to derive the formulas for area approximation using the left-hand, midpoint and right-hand rules
• November 9 — handout
1. physics problems
2. Newton's method
3. mean value theorem — think of terms of "average velocity"!
• November 4
1. more review!
• November 2
1. midterm review! related rates and optimization problems
• October 21 — handout
1. more on related rates — it can sometimes easier to break up a related rates problem into two smaller related rates problem; the formulas become simpler
2. familar geometry formulae, such as $$c = ab \sin \theta$$, are useful!
3. l'Hôpital's rule
• October 19 — handout
1. linearizations ($$y = mx + b$$) and approximation via differentials ($$\Delta y \approx \frac{dy}{dx} \cdot \Delta x$$)
2. related rates — check the units of your final answer as an easy safeguard!
3. take-up of first midterm; common problems
• October 14 — handout
1. the chain rule!
2. differentiating $$\log u$$ — there are two ways:
• using the formula $$\frac{d}{dx} \log_b (u) = \left(\frac{du}{dx}\right) / (\log(b) \cdot u)$$
• equivalently, using the formula $$\frac{d}{dx} \log_b x = 1 / (\log(b) \cdot x)$$ together with the chain rule
use whichever one you're comfortable with, but know why they're the same
• October 12 — more review for the midterm!
• October 7 — handout
1. lots of practise computing derivatives!
• October 5 — handout
1. intermediate value theorem: why the conditions are necessary, using it to find roots of equations, etc.
2. showing continuity/differentiability of piecewise-defined functions: every piece must be continuous/differentiable on its own interval, and for the points where they match up, the definition has to be used
3. intuition behind the derivative and its definition — describing things visually
4. working backwards from the definition to express a limit as a derivative — how to do variable substitutions $$h = x - c$$ to make this doable
5. finding the tangent line to a function $$f(x)$$ at a point $$(a,b)$$ — the derivative $$f'(a)$$ gives the slope $$m$$; the point $$(a,b)$$ then allows to solve for $$b$$ in the equation $$y = mx + b$$
6. using the definition of a derivative to compute limits; some more algebraic tricks
• September 30 — continuation of handout
1. using the squeeze theorem — as long as $$f(x) \leq h(x) \leq g(x)$$, $$h(x)$$ can be completely random as long as $$f(x)$$ and $$g(x)$$ are nice
2. computing limits — if taking the limit of $$f(x)$$ as $$x \to a$$, can assume for algebraic manipulations that $$x$$ is in some small interval around $$a$$ by truncating $$f$$ — obtain a new function with a smaller domain that still contains $$a$$
3. Etch-a-Sketch! — you can think of the $$\varepsilon$$-$$\delta$$-definition of continuity using this fun toy! the $$y$$-knob becomes $$\varepsilon$$, and the $$x$$-knob becomes $$\delta$$
4. how to think of a composition of functions $$f \circ g$$ and to "determine its domain" via a sequence of allowed values
• September 28 — handout
1. graphing functions; making the distinction between left-, right- and total limits
• September 23
1. odd and even functions — what this means visually (reflecting around axes/rotating around the origin), what the algebraic definition is and how to verify a function is odd/even
2. the vertical line test checks if a graph represents a function $$f$$, while the horizontal line test checks if a function $$f$$ is one-to-one (and so whether an inverse $$f^{-1}$$ exists)
3. the horizontal line test is really just the vertical line test applied to the "inverse graph" (that is, reflected around the diagonal line $$y = x$$ )
4. computing the inverse of a function, and why variable names are switched — think of names and ages and a function translating between the two
5. question — which functions are equal to their own inverses?
• September 21
1. introductions and greetings!
2. describing lines using $$y = mx + b$$  — to determine the equation of a line given two points, you can use the rise-run formula ($$m = \frac{\Delta y}{\Delta x}$$) or you can substitute known values for $$x$$ and $$f(x)$$ and then subtract equations to isolate your variables
3. scaling/translating graphs of functions horizontally/vertically (given $$f(x)$$, can get $$c_3 \cdot f(c_1 \cdot x + c_2) + c_4$$ ); how to interpret these with physical analogies (height as a function of time/age, translating/scaling corresponds to different growth rates/start times, circus acts with stilts, etc.)
4. graphing functions — given graphs $$f(x)$$ and $$g(x)$$, what it means to graph $$f(x) + g(x)$$

## Breaking news!

• October 25 — the score distribution for this section on the first midterm has been computed
• September 23 — a new room is coming soon! Stay tuned.

http://math.stanford.edu/~jlee/math41/
Jonathan Lee