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.

Preparing for an Algebra Test

We went through the review packet for the test.

One thing to watch for was to only combine like terms, so either numbers or variables with the same exponent.

We saw the difference of perfect cubes more than once, which has a corresponding formula.

You can always multiply by different forms of one to do things like getting a common denominator. Something divided by itself is one.

Need to be careful with parentheses and notation to make the work easy to follow and check.

If the denominators of two equal fractions are equal then the numerators will also be equal, that can save a few steps.

Usually quadratic functions have two solutions, sometimes they have one or none.

Matrix Addition, Subtraction, Multiplication – Tutoring Math

We worked on matrices.

Starting with matrix addition and subtraction, which were not a problem.

Then got into multiplication  by numbers as well as addition and subtraction. Sometimes factoring was helpful there.

For matrix multiplication, you multiply the row elements by the column elements and add them up to get the products elements. Sometimes if the dimensions of the matrices do not match up correctly, you cannot multiply them.

Tutoring Trigonometry, Use Trig functions on right triangles!

We went over problems with trig functions.

The functions only work with right triangles.

Looked at both the angle of elevation and the angle of depression in a few problems.

The final two problems were more complicated and involved two equations and two unknowns as well as calculating the tangent function of two angles and factoring as well as multiplying algebraic expressions.

Graphing a hyperbola “neatly”, tutoring Algebra II

We started by graphing a hyperbola “neatly” as the directions stated. That involved finding the vertices, foci and drawing the diagonal asymptotes.

Checked one point by plugging in an x value to see if the graph was accurate.

The sign in front of the variables is important to determine orientation of the conic sections.

The natural number is e and is the base for the natural logarithm. e is approximately 2.7