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.

Graphing Periodic Functions – Sine, Cosine, Amplitude and Period

We looked at periodic functions.

Started most problems by finding the amplitude and the period.

The amplitude is basically the absolute value of the coefficient multiplying the function. If there is no number, the amplitude of a sine or cosine function is 1.

The amplitude can be found by 2π/# the number in front of the x, as in cos(2x) the # is 2.

After that, you should know the basic shape of the sine and cosine functions. You can graph two periods (for this set of problems) and fill in the zeros.

Graphing Hyperbolas, Directrix, Focus

We worked on some upcoming material in the class with periodic functions.

Drew an angle on the unit circle and found the tangent of that angle. You draw the angle and a right triangle then find the ratio using SOHCAHTOA.

Graphed a parabola and looked at the focus and directrix. Also talked about some of the practical purposes of the shape of the parabola. If you forget the equations for these, you can google them if you know what to call them.

Did a problem with a hyperbola. And saw how the sign in front of the variables affects the orientation of the shape.

Looked at a little physics as well. For sound and light with higher wavelengths, the wavelength is shorter.

Tutoring Physics, Light, Indices of Refraction

We went over some problems with lenses, converging and diverging.

Then spent some time on problems with different indices of refraction. I would usually think of air to water and the behavior if you get mixed up.

Used Snell’s Law and another equation with the velocity of light, c, and the index of refraction.

n= c/v

The index of refraction is about 1 in air and is 1 in a vacuum. It doesn’t (in the context of this class) go below 1.