- Your final homework is returned: graded 2 and 12 from the first
section, and 4 and 19 from the second. Lucky, both from the "front end"
of each assignment.
- #2, p. 665
- draw from endpoint to endpoint.
- don't forget direction of traversal.
- Use symmetry!
- 12, p. 665 -- this is an ellipse. Use Pythagoras in trig form to simplify.
- #4, p. 675 -- point/slope form
- 19, p. 675 -- Be careful! You can't always just analyze numerator and denominator separately: sometimes you get an indeterminate quotient.
- #2, p. 665
- A couple of students asked me about cosh and sinh....
-
(the catenary -- the hanging chain)
-
-
- Today we wrap up the course, with a bit of a summary, and your
questions.
- Logistics for the final:
- The final is Monday, 1:00-3:00.
- There's some new stuff and some old stuff. If your grade on the
old stuff is better than your worst hourly exam, that final grade will
replace your worst. Just a chance at some redemption!
- You are allowed an 8.5x11 cheat sheet of formulas or whatever.
- I'll be here Sunday night at 7:00-9:00 if anyone wants to come and
work problems or ask questions.
- Reminder: you didn't have a homework on the polar sections, but you might be expected to reproduce some of those we examined together (or examples from the sections) for the final exam.
- The final is Monday, 1:00-3:00.
- The Nephroid of Freeth:
- Length:
The easiest way to get the length of a polar curve is to use parametrics. Let's do it that way.
becomes
- Area:
For the area, let's start with a circle of radius r, centered at the origin, and see how we might adapt the "familiar formulas" to more exotic shapes, until we obtain
) is a linear function of another thing (
-- in the case of a circle).
Now, to some exercises:
- #5, p. 692
- 7
- 15
- 26
- 30
- 47
- 54
- The limit definition of the derivative (it is, afterall, the most
important definition in calculus):
- Integration
- Approximating areas with rectangles
- LRR, RRR, Midpoint; Trapezoidal (when you have two estimates, you have a third -- take a suitable average).
- The Midpoint method is the better than the trapezoidal method: it usually makes an error half the size of the trapezoidal method, and in the opposite direction. What does that suggest?
- Simpson's rule
- Are integrals all about areas? No: this is what it's all
about:
could be volume, mass, power, time, etc. (we usually try to choose a suitable name).
- Riemann sums (partitions, subintervals, limits, ...)
- Special summation formulas
- Definite and indefinite integrals
- The Fundamental Theorem of Calculus, Part I
- The Fundamental Theorem of Calculus, Part II
- Substitution (the chain rule backwards)
- Areas between curves
- Volumes
- Volumes of revolution
- Integration by parts (the product rule backwards)
- Arc length (this example illustrated 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))
- Inverse functions
- Why e?
- Exponential functions -- easy to invert, and extremely important for modeling purposes.
- Inverse functions, especially logs
- We only need base e exponentials and logs (ln)
- Easy to invert a function given by a graph
- Inverse trigonometric functions
- L'Hopital's rule
- Parametrics
- Parametric curves, e.g. the cycloid
- Graphing parametric curves or their orbits
- Lengths of parametric curves
- Speed of a particle and slope of the tangent line
- Polar coordinates -- distance and area
- Translating between polar and rectangular (Cartesian) coordinates