Summary of MAT122 material

Summing up... here's a glimpse of what we've done these past eight weeks:

We started with Newton's method:
- it uses the linearization (aka linear approximation, tangent line) to find better approximations to roots.
- any equation can be written as a root problem.
We approximated areas using rectangle rules:
- Right rectangle rule
- Left rectangle rule
- Midpoint rectangle rule
- Trapezoidal rule is the average of left and right: whenever you have two estimates, you've got a third estimate (some sort of average).
- Turns out that the midpoint and trapezoidal methods can be averaged to give an even better method (Simpson's)
The definite integral
- More formally, each of these rules above can be written as Riemann sums: the idea is to divy up the interval of interest, and, on each small subinterval, find a representative function value to serve to define a rectangular approximation to the area of that little subinterval.
- If we take the limit as the subinterval widths go to zero, and get a finite answer, the integral is defined as the limit.
- Some comparison theorems, mean value theorems, etc.
The fundamental theorem of calculus (linking derivatives with integration)
- Two parts: solving integrals using anti-differentiation (staring at them until a solution comes to you)
- Using integrals to define anti-derivatives
u-substitution
- u-substitution is the chain-rule backwards
- when you don't recognize an anti-derivative for an integrand, you should try u-substitution if possible, to clean up your integral (and perhaps your thinking).
Areas between curves (pretty easy)
Setting up integrals (harder!)
- Volumes via cross-sectional areas
- flow problems (radial symmetry)
- average value of a function
- mass density problems (bars, plates)
Volumes of revolution
- disks and shells
- watch out for the axis of rotation, and the radius -- it's often not x
Work problems
- W=Fd
- Use your units!
- Watch out for the difference between pounds and kilograms -- pounds are units of force, kilograms units of mass
Exponential functions
- positive
- positive bases, but all can ultimately be expressed using base e.
- The advantage of e is that it's the base for which the exponential is own derivative (and this is the only "interesting" function with this property (well, actually f(x)=0 is pretty darned interesting; and constants times e^x are also interesting, but don't add much new).
Inverse functions
- General properties of inverses
- obtaining them graphically
- obtaining their formulas by solving for x as a function of y (then swapping variables)
- Derivatives in terms of the original function
logarithms
- inverses of exponential functions
- we only need the natural log, ln
- properties are reflections of exponential properties
- antiderivative of 1/x, the "missing power" -- mysterious link between powers and logs.
- logs grow, but more slowly than any function you know.
Applications of exponentials and logs
- exponentials satisfy a very important differential equation,
  
  which makes them amazingly important.
- The "rule of 70" replaced by a "rule of logarithms"
- doubling times (population growth, compound interest)
- half-lives (radio-carbon dating, forensic body time-of-death)
L'Hopital's rule
- Useful for evaluating indeterminate forms (of the form 0/0, or )
- Useful for demonstrating that families of functions grow more rapidly than others.
- May not always work.
- Using logs, exponentials (or other transformations) to
  - Handle indeterminate forms, or
  - transform ugly functions into more easily handled functions
Inverse trig functions
- What can I expect for the final?
  - Just focus on arcsin and arctan.
  - Graphs of these functions (by reflection!)
  - Domains and ranges (by restricting sine and tangent to be one-to-one)
  - Know their derivatives.
  - You might be called on do some integrals (e.g. substitutions) using them).

Website maintained by Andy Long. Comments appreciated.