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.

- Tuesdays — 1:15–3:05pm
- Thursdays — 1:15–3:05pm

- Tuesdays — 9:30–10:50am
- Thursdays — 9:30–10:50am

Pending...

Pending...

- December 2 — review handout
- November 30
- integration practice from previous finals!

- November 16 — handout
- November 11 — handout
- mean value theorem
- Riemann sums — know how to rearrange sums of squares and cubes as sum of integers starting from \( 1 \)
- know how to derive the formulas for area approximation using the left-hand, midpoint and right-hand rules

- November 9 — handout
- physics problems
- Newton's method
- mean value theorem — think of terms of "average velocity"!

- November 4
- more review!

- November 2
- midterm review! related rates and optimization problems

- October 21 — handout
- 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
- familar geometry formulae, such as \( c = ab \sin \theta \), are useful!
- l'Hôpital's rule

- October 19 — handout
- linearizations (\( y = mx + b \)) and approximation via differentials (\( \Delta y \approx \frac{dy}{dx} \cdot \Delta x \))
- related rates — check the units of your final answer as an easy safeguard!
- take-up of first midterm; common problems

- October 14 — handout
- the chain rule!
- 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

- October 12 — more review for the midterm!
- October 7 — handout
- lots of practise computing derivatives!

- October 5 — handout
- intermediate value theorem: why the conditions are necessary, using it to find roots of equations, etc.
- 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
- intuition behind the derivative and its definition — describing things visually
- working backwards from the definition to express a limit as a derivative — how to do variable substitutions \( h = x - c \) to make this doable
- 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 \( y = mx + b \)
- using the definition of a derivative to compute limits; some more algebraic tricks

- September 30 — continuation of handout
- 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
- 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 \( a \)
- 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 \( \delta \)
- 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
- graphing functions; making the distinction between left-, right- and total limits

- September 23
- 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
- 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)
- the horizontal line test is really just the vertical line test applied to the "inverse graph" (that is, reflected around the diagonal line \( y = x \) )
- computing the inverse of a function, and why variable names are switched — think of names and ages and a function translating between the two
- question — which functions are equal to their own inverses?

- September 21
- introductions and greetings!
- 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
- 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.)
- graphing functions — given graphs \( f(x) \) and \( g(x) \), what it means to graph \( f(x) + g(x) \)

- 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