Tutoring Physics, First weeks of second semester

We started looking at the electric field in a capacitor where the same amount of charge was distributed across different amounts of area.

Drawing a diagram can be a good first step in many problems.

Sometimes problems will have a balance of forces or other things canceling each other out. One example was the oil drops with charge where an electric field caused a force that exactly balanced gravity.

Using subscripts will become more important to keep track of things.

Went over some basics of derivatives and integrals for a bit.

Also talked about logarithms and natural logs.

Work and energy both have units of joules, they are used to describe different things, but do have the same units.

This class is a similar class to Physics 32 which I took with Phil Kesten at Santa Clara University. It is taught at CSUMB in Monterey though. Second term of calculus based physics.

I recommend the Giancoli book which we used, and actually Phil Kesten and Dr. Tauck wrote their own physics book as well which I got a copy of.

Inverse Trig Functions “Unlock” the Arguments of Trig Functions, Tutoring Geometry

We started by using SOHCAHTOA.

One mistake was not including an argument of the function. Sin, cos, tan do not mean anything without an argument. So sinx, cosx, tanx, etc.

Went over some ways to manipulate algebraic expressions.

Talked about how the inverse trig functions reverse the action of a trig function.

They are somewhat like a key which unlocks a box which holds the argument of a trig function and all that is left is the argument.

Reminded about using the degree symbol • and using units when applicable.

Used the Pythagorean Theorem a few times.

Briefly talked about the reciprocals of trig functions.

Tutoring Physics 2, focusing on components

We went over problems involving electric forces.

For one the horizontal components of the forces canceled out, using that information, we could calculate the sum of the vertical forces and figure out the charges by using Coulomb’s Law.

You can either use trig functions or the Pythagorean Theorem, or a combination, for some of these problems.

Labeling things with subscripts becomes more important once the problems get more complicated.

One problem involved early kinematic equations.

To figure out the units for something, like electric field, you can use the simplest equation you know for that thing and go from there.

Tutoring Precalculus, Log function tricks

We started with some problems involving logarithms and natural logarithms.

By acting the log function on something with an exponent, the exponent can be brought in front.

Continuous compound interest uses a function with the natural number e to the power of the rate multiplied by time.

Also went over test corrections.

One trick was divide both sides of an equation by two to simplify everything.

Vertex form of quadratic equation, Tutoring Precalculus

We started by looking at some problems with division of polynomials. Chose to use synthetic division, long division would be an option also.

To fully factor, for example, a fourth degree polynomial, if it comes out without a remainder then the combination of all factors is the fully factored form. Like you could factor 12 as 2*3*2.

Another problem seemed to call for the vertex form of a quadratic equation y = a(x-h)^2 + k where (h, k) is the vertex. Then you could convert that to standard form.

Talked a bit more about imaginary numbers.

Knowing a little can help a lot (to look things up) in math

Tutoring precalculus,

We started by looking at problems with sets of equations.

The first were sets of linear equations.

Others had quadratics. For one problem, if you recognized part of the equation was a perfect square, it made the problem easier.

Other times a single mistake like reversing a negative sign could cause problems later on, especially if done early on that means redoing a lot of work.

After the winter break he was a little rusty with a few things.

It’s important to be able to recognize the equations of parabolas, ellipses, and hyperbolas. If you know what something is, you can look up equations for it more easily.

“What do nanometers measure?”

You can measure anything with length in nanometers.

The real question maybe should be ‘what do nanometers measure conveniently?’

For example, you could say your height in meters. You could also say it in nanometers.

But to say it in meters, probably gives you a number somewhere under 10. Which is a convenient number to think about.

You can measure something much longer in meters, say the distance from California to NY. But at that point, you’re talking about a very large number and it becomes more convenient to use kilometers since then you can use a smaller number of that unit.

With nanometers, it’s convenient for things like the wavelength of visible light and things at the atomic level.

“How many Planck lengths are in an inch?”

You can ask Google questions like this if you know how to format them correctly.

“How do I find the angle of a right triangle knowing the ratios of the sides?”

You can use an inverse trig function of the ratio using SOHCAHTOA. Then you can subtract the two known angles from 180° to get the remaining angle.

Why is “the” square root always positive

Since there are often multiple answers to quadratic equations. That is the important thing to remember in mathematics, I think.

‘The square root” implies something singular and refers to the positive, principal square root. That is more semantic and personally, I think less important practically.

There are situations in which negative numbers do not make sense so it does make sense to only think of a positive value for the square root. But not all situations.

Personally, I would rather see two different symbols for square roots in general and for only finding the principal square root. I would like to see a new symbol for the square root finding the principal and have the usual symbol imply multiple solutions.

But I don’t get to decide these things.

Maybe the idea of a different symbol could catch on though.