Tutoring math at CSUMB, systems of equations and quadratic equations

We started by looking at systems of two equations with two variables and using substitution to solve.

Then did a few problems with three equations and three variables.

Also used u-substitution to solve a few quadratic equations. If they can be factored, that can be easier. Sometimes quadratic equations cannot be easily factored, in which case it can be useful to use the Quadratic Formula.

Tutoring Precal for RLS, Double angle formulas and graphing trig functions

We started by looking at homework which used the double angle formula.

The basic approach was to do two things, could start in either order.

1. Draw the angle with a right triangle on the xy axes.
2. Use a double angle formula which is convenient

You can figure out the ratios for the trig functions using
SOHCAHTOA
CHOSHACAO

Then looked at a previous test with graphing trig functions. Should spend more time on that. But one thing you can do most of the time is plot points by plugging in x values and getting y values. You use again,

SOHCAHTOA
CHOSHACAO

It does help quite a bit to know the basic shape of the functions.

Tutoring RLS Precalculus, trig functions and Friday sheet

We looked at some problems with trig functions, especially sum and difference formulas.

For many of them, the angle does not need to be known if the sides of the triangle are known.

Used the half angle formula a bit. Reviewed the quadrants and which trig functions are positive and negative in different positions.

Then got into the Friday sheet which dealt with logarithms a bit.

Newtons and Distance in an equation

Newtons and meters are sometimes related to each other and you can see the relationships within equations.

One such equation is that work is equal to the dot product of force and distance. Force has units of Newtons, distance has units of meters.

There are other equations with both units as well.

Tutoring Algebra at CSUMB, vertical line test and horizontal line test

We looked at functions and inverses of function.

One quick way to see whether something is a function is to do the vertical line test. If a vertical line can ever pass through two points, it’s not a function.

An inverse function switches the x and y values. You can figure out if a function has an inverse that is a function by doing the horizontal line test.

Looked at telling the shape from a few types of equations, especially lines and parabolas.

Tutoring Precalculus, Series and Sigma Notation

We mostly looked at problems with sigma notation. At the bottom of the capital (Greek) letter Sigma, there is a variable with the starting point and at the top a finishing point if it is finite or infinity.

Then the sequence of numbers goes into an equation that are added together.

For a geometric series, there can be a finite sum or it can diverge.

Exponents can be used, and series can also alternate between positive and negative. You can describe the same series in ways that look different.

Looked a bit at combinations and permutations, including one that required a somewhat seldom used formula at least in high school math classes.

“Is the square root of -1 equal to just 1?”

There’s a way to check that. Or any similar problem with a square root.

Take what you think is the answer, 1 and square it. If you get -1, then yes, it is. If not, it isn’t.

1(1) = 1

Not -1, so no it is not.

The solution is an imaginary number. Using both real numbers and imaginary numbers is called ‘complex’.

The square root of -1 is called i. So i squared is -1.

Tutoring Math at RLS, Friday Sheet in April

We started by looking at the cosine functions of sums and differences.

Then used the Pythagorean Theorem on some triangles for some problems using the equations.

Fractional exponents like 1/2 and 1/3 are like the square root and cube root.

The ln function has e as a base.

If you take the ln or log of a function with an exponent, the exponent comes in front.

Used the formula for the log of a quotient once.

Some problems involved factoring and foiling.

Another problem had the intersection of an ellipse and a line. The answers ended up having whole numbers.

What does your calculator do when you press the sin/cos/tan button?

Apparently TI calculators use the CORIDC algorithm which involves rotation on a complex plane using complex numbers.

COordinate Rotation DIgital Computer

aka

Voldic’s algorithm

I would think that at least some calculators use (or used) the Taylor Series for the functions.

They would be something a calculator can do fairly easily, as opposed to the sine function itself. Taylor Series use polynomials.

That is more likely something you would see as a mathematics/physics student at the undergraduate level.

You would learn about the sine function being an ‘odd’ function and the cosine function being an ‘even’ function.

Each is an alternating series that starts with a positive term.

If you use more terms, you get more accuracy, but a calculator displays a limited number of terms. So a fairly small number of terms in the Taylor Series will give you a decent approximation for many things.

Also, these Taylor Series are more accurate with smaller values of x using less terms. If you use x = 0, they’re exactly right using only the first term.

How do we find the derivative of 1/(1+x) ?

I would recommend rewriting the expression first.

Sometimes if you change how something looks, it becomes easier to work with.

Now you can do the power rule and the chain rule.