- Today is "review day". First of all, some logistics:
- Exam is Monday, December 8th, from 8:00-10:00 a.m.
- You may have an 8.5x11 sheet of paper, front and back, with anything on it that will make you happy.
- You may have a calculator, but you are to use it only as a scientific calculator -- no graphing, integrals, derivatives, sums, etc.
- The exam will include only a little of the vector stuff. Perhaps I'll have you graph a few points, draw a few vectors, and, if appropriate, compute a dot or cross product.
- The final is comprehensive. The only other new material is
Taylor series.
- Lauren has scheduled a review for tomorrow, 1-3 (MEP300):
your last two homework sets will be back and on my door
by then.
- A reminder that your final exam score will also replace
your worst semester exam grade (provided it's better, of
course!). So it will pay to do well on the final.
- I'm willing to do a Sunday night review: 7-9?
- I wish that you could have been with me in Washington, talking
about how
to "tinker" with the climate(!)
- I'll have your homework back by tomorrow.
- Chapter 6:
- Inverse functions, especially logs. Invertible functions pass both the horizontal and vertical line tests.
- logs turn products into sums; exponentials turn sums into products
- Why e?
is that magic function whose derivative is equal to itself. Looking at the slope of its graph tells you the function value; knowing the function value at x tells you the slope of the tangent line at x.
- Exponential functions -- easy to invert, and extremely
important for modeling purposes. They solve a very
important differential equation:
- We only need base e exponentials and logs (ln) -- we can convert any other base to these using identities such as
- It is easy to invert a function given by a graph -- just reflect it across the line y=x.
- Inverse trigonometric functions -- even though we can't invert them! We simply restrict domains....
- Hyperbolic trig functions (cleverly concealed exponential functions -- the catenary) and their inverses
- L'Hopital's rule -- allowing us to find indeterminate limits through studying derivatives.
- Chapter 7: Integration:
- Integration by parts (the product rule backwards)
- Trigonometric integrals; using trig identities, of which
the most important is the Pythagorean Theorem in trig form
- Approximating areas with rectangles
- Riemann sums (partitions, subintervals, limits, ...)
- LRR, RRR, Midpoint
- Trapezoidal (when you have two estimates, you can get a third by taking a suitable average). In this case, it's LRR and RRR
- The Midpoint method is the best of that lot in general, but usually makes an error half the size of the trapezoidal method, and in the opposite direction. What does that suggest?
- Simpson's Rule: take a suitable average of midpoint (2) and trapezoidal (1).
- Improper integrals, which come in two flavors:
- function undefined on the endpoints, and
- endpoints going to infinity.
- Are integrals all about areas? No: the most important
equation to remember is this:
It's about the differentials. This is why it's so important not to drop them.
- Chapter 8:
- Arc length, illustrating how easy it is to create an integral for a quantity of interest -- arc length -- using only elementary pieces of mathematics (e.g. Pythagorean theorem)
- Chapter 10:
- Parametric equations and curves (in the plane): where we again encountered uniform circular motion; discovered the cycloid, as well as the new idea of Bezier curves.
- Introduction to the cycloid (solution to the brachristochrone and tautochrone problems)
- Motions as functions
- Arc length for parametrics
- Polar coordinates
- Polar curves
- Chapter 11:
- sequences (e.g. Zeno, factorials, Fibonacci sequence, powers)
- series (adding up an infinite number of things)
- Convergence: limits as the index tends to infinity.
-
as a series
- the geometric series
- the harmonic series
- Divergence test
- Comparison tests:
- comparison test -- bound a series term by term to a known series, to see if you can force the unknown series to a limit or divergence.
- limit comparison
- integral test
- Alternating series, and the corresponding test
- Associated error estimates, based on the integral test and the AST.
- ratio and root tests
- Absolute versus relative convergence
- Power series (infinite polynomials) -- because your calculator can only do sums, difference, products, and quotients.
- Taylor series (and MacLaurin series). These are generalizations of tangent lines. With higher degree (taking a partial sum of a Taylor series with more powers), we can capture not just the slope, but the inflection, and so on.
- Radius and interval of convergence -- when is "="
equality? For example, for what values of x can we write
- Integration, differentiation, composition to create new series from old. The radius doesn't change.
- Chapter 12: Vectors
- 3-D coordinates systems (understanding and drawing 2-D projections of physically realizable systems).
- Vectors: direction and length
- Position vectors: pointing from the origin to a point
- Adding vectors (butt to tip), creating parallelograms
- Norm -- measuring distances.
- The dot product (detecting orthogonal vectors -- valid in any dimension). The dot product is a scalar.
- By the way, don't forget my summaries of each section -- they've got my take on each section, my emphasis.