Tutoring, Reviewing for Geometry/Algebra Final

We started by graphing the overlap of two inequalities. If the inequalities are less than or greater than you use a dotted line since they do not include the values on the line.

At that point I sketch horizontal lines either above or below one line and vertical lines above or below the second line and then you can see the overlap easily.

We graphed some parabolas, using the equation of for the vertex and showing the axis of symmetry.

Did a few problems with point slope and slope intercept form.

A few problems with long division and synthetic division.

And ended by looking at some volumes of 3D shapes. Talked for a minute about how you can see if an equation describes a volume or an area. Basically looking for whether something is 3D or 2D.

Tutoring Precalculus, Trigonometric Functions and Properties

We started by looking at a problem with triangles. Two smaller triangles made up a larger triangle. The larger triangle was a right triangle. Basically, we started finding different angles, and use the law of sines at one point. Then we could use a trigonometric identity.

For another problem, it helped to create a new variable.

We spent a little time on graphs of trigonometric identities and also understanding the domain and range. That included looking at the amplitude and expansion and compression of functions.

Circle Circumference and Area Formulas

circle_circumference_area_equations_formulas

Formulas for area and circumference of a circle.

Synthetic Division of Polynomials

We started by looking  at long division and synthetic division of polynomials.

Long division was familiar, synthetic division was new.

Synthetic division can be faster and less written work, but is more straightforward with certain conditions. You can use one method to check the other.

There is a decent explanation here, but with a mistake in the final example.

http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/synthetic/synthetic_division.html

The ‘test zero’ goes on the outside. If you are dividing by (x-2), the test zero is 2, if you are dividing by x+6, the test zero is -6, etc. The test zero is the number that makes the denominator zero when you plug it in.

Synthetic division is straightforward when the coefficient for the variable in the denominator is of the form x + a and the leading coefficient of the numerator’s first term is 1.

Wikipedia describes how you can use it when things get more complicated, but I would probably just use long division if that is the case.

We then looked at some graphing of equations and inequalities. To find the x-coordinate of the vertex of a parabola you use -b/2a.

Also simplified some expressions.

“Is it possible to construct a triangle with the side lengths of 7, 8, and 15?”

In the usual sense of things (Euclidean Geometry), no.

You could have a ‘degenerate triangle‘, but that’s not exactly a triangle in the way we usually think about a triangle.

8 + 7 = 15.

So basically you want the sum of the two smaller sides to be larger (even slightly larger) than the largest side to get a triangle that isn’t degenerate, a triangle in the way we usually think about triangles.

triangle_8_7_15

Fractional Exponents, Logarithms, Quartic and Cubic Equations

We started by looking at some current homework which involved factoring. For most of the problems they started with a quartic or cubic equation and then factored into a quadratic and one or two other terms.

Cubic equations can have three solutions, quartic equations can have four.

Talked a little about one application of imaginary numbers since we saw complex numbers in some solutions. And went into more detail about how to do calculations with them.

Talked about fractional exponents for a bit and how to work with them. Also about logarithms. The base being 10 if no other base is shown.

Fractional exponents are almost always easier to work with than ‘cube roots’, ‘fourth roots’ etc. Square roots aren’t too bad but still can be harder to manipulate in some cases.

Last problem we worked on was a three dimensional volume of a cone with the top cut off. Looked at the application of similar triangles and the proportionality of the sides.

What is csc(225°)?

csc(225°)

The cosecant function is the reciprocal of the sine function.

So if you think about the ratios of a triangle for sine, you have opposite/hypotenuse.

For cosecant, it will be hypotenuse/opposite.

Placing the angle on the xy axes

225° is 45° past 180°.

csc_225

The hypotenuse for a unit circle is positive 1. And the opposite side is negative and a value of -1/√2

So you want to calculate 1/(-1/√2) = -√2

Tutoring Precalculus, Why Inverse Trigonometric Functions have Restricted Domains

The first thing we saw was some points taken off for answers on a test because they were not in the right range.

It would seem like there should be multiple answers for something like arccos(1/2), but in order for the inverse trigonometric functions to actually be functions (which has a few requirements) the domain is restricted.

sin(x) looks like this, it keeps going in both the positive and negative directions along the x-axis

sinx

arcsin(x) looks like this,

arcsinx

If it kept going farther either above the top or below the bottom, it would fail the vertical line test and not be a function. I think that may be the simplest way to think about why the domain is restricted.

The main change to his approach should be to put the triangles for trigonometric functions on a set of axes properly. Should make things easier to figure out and more accurate. Often he correctly used a 30 60 90 triangle for example, but sometimes not in the correct orientation which caused problems.

Knowing what the oscillating functions look like on a graph would be good too as well as having a few more of the values memorized, but also understanding them in several ways.

Circuits Complex Analysis Equation Simplification

You want a common denominator to simplify things.

I replaced jw x 10^-5 with the variable X, so I didn’t have to write it as many times.

You can factor out a term later on that cancels.

10^5 * 10^-5 = 1
complex_simplification

Tutoring Physics, some basics about circuits

We looked at circuits with resistors. Saw current, voltage, and resistance. The circuits included branches.

Current is the flow of electrons. Electrons have a negative charge.

Used Ohm’s law, V = IR and moved the variables around.

Current flows in an optimal combination when it comes to a branch. Electrons tend to flow where there is less resistance but the presence of more electrons creates more resistance.

Current flow is much like water flow through pipes. Though different in a few ways.

Used 1/R1 + 1/R2 = 1/R for two resistors in parallel.