- It is time to evaluate the course. I'm sure that you've received
emails to that effect....
- Final exam: April 30th, 8:00-10:00 a.m.
- Today is a review day.
- For the final, you may have a one page sheet of notes (front and back). There will be a little new material (section 12.4 and 12.5, plus a little of chapter 10), but the main emphasis will be on old material.
- 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....
- L'Hopital's rule -- allows us to find indeterminate limits through studying derivatives.
- Chapter 7: Integration:
- 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).
- Integration by parts (the product rule backwards)
- Trigonometric integrals; using trig identities, of which
the most important is the Pythagorean Theorem in trig form
- 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)
- Surface area -- rotating an arc about an axis
- Chapter 9:
- Differential equations -- extremely important in the sciences -- relating rates of change to function values.
- F=ma -- mother of many physical differential equations
- Newton's law of cooling:
, where
- Chapter 11:
- sequences (e.g. Fibonacci sequence, factorials, powers; Zeno)
- series (adding up an infinite number of things)
- Convergence: limits as the index tends to infinity.
- geometric series
- 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
- 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.
- Equations of lines and planes using vectors;
- The cross product (detecting parallel vectors). Only valid in three dimensions. Computing them with determinants.
- The cross product is a vector, perpendicular to the two vectors involved in the product. Hence it allows us to create a right-hand system of vectors.
- Computing areas of parallelograms and volumes of parallelpipeds using cross products.
- Corresponding physical quantities, like forces, Torque, work
- 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
- By the way, don't forget my summaries of each section -- they've got my take on each section, my emphasis.