We mostly looked at

Also finding the positions of angles and the names of equivalent angles.

Less than zero can be read as ‘negative’ and more than zero can be read as ‘positive’.

For these trig functions, the hypotenuse is always positive. And many times, using the unit circle is useful. Although, sometimes multiplying the ratios by a common factor can also be useful.

Compound interest and half lives – tutoring precalculus

We looked at problems with continuous interest in terms of banking and similar problems for elements with radioactive decay. The same equation can basically be used for both situations, you can change letters if it seems to make more sense that way.

The lnx and e^x functions can counteract each other, much like arcsinx and sinx.

There is another equation for half life specifically, but it’s easy enough to get to an equation with that idea using the original equation and therefore not memorizing more equations than necessary.

How do I solve 3^{2x}=81?

A few options. The option on the left assumes that the answer is a positive whole number.

The option on the left would be more useful for a not-whole number.


We started by looking at evaluating trig functions of angles. They involved 30 60 90 triangles and 45 45 90. So we began by looking at those triangles.

It’s easy enough to derive the ratios for the isosceles triangle.

Then we compared that to the other triangle.


Often, it helps to draw the angles on the xy axes starting from the 0° position (usually East). Then connect the angle to the x-axis to make a triangle.

Also, the functions are pronounced ‘sine’, ‘cosine’, ‘tangent’. If you’re going to say them out loud, say those, not “sin”, “cos”.

If you get to “csc”, how would you even say that out loud?

Tutoring Precalculus, Logarithms

We went over problems involving logarithms.

Log has a base of 10 if nothing is written and ln has a base of e. e is ~ 2.7, it’s a number with a lot of significance, kind of like pi.

log 10 = 1
log 100 = 2
You can think about it as a base, an exponent, and a solution.

e^x and lnx are inverse functions, meaning they can counteract each other, like sin (sin^-1(x))

Sometimes doing some algebra initially before jumping into the logarithms is helpful.

Looked a bit at the shape of lnx.

“How do I write the equation of a circle with center (5,-7) and radius 9 units?”

I would write this equation so it’s similar to how the equation of an ellipse would look. The denominators will be the same. And you could write this in different ways, but I think this way clearly shows the center and radius.

Having one on the right side of the equation, shows you the radius in the denominators and would show the major and minor axes if it was an ellipse.

Tutoring Physics, Dot Products and Cross Products Return

We looked at induced currents, loops, magnetic fields.

Spent some time on the right hand rule.

Also looked at the difference between dot products and cross products. They are important for the current material.

The result of a dot product is a scalar, the result of a cross product is a vector.

One has the sine function in it and the other has a cosine.

So if you took the dot product of two perpendicular vectors, you would get zero.

If you took the cross product of two parallel vectors, you would get zero.

Good not to mix them up. Neither one of them is simply ‘multiplication’, though sometimes they reduce to that depending on the scenario. The cross product will still have a direction in addition to a simple multiplication even with orthogonal vectors.

Algebra, Variables with Exponents Positive, Negative, and to the Zero

An explanation of why numbers to the zero power equal 1.

Dealing with exponents in various situations.

Tutoring Calculus, Series, Sequences, Improper Integrals and Limits

We focused on series and sequences and looked a bit at improper integrals.

All of these are connected to limits. If something diverges, which is easier to determine than convergence, you simply say it diverges. If a test is inconclusive, you try another test.

Geometric series might be the easiest to deal with. Sometimes a series doesn’t exactly look like a typical geometric series and needs to be manipulated a bit algebraically. If you have the ratio and the first term, there is a simple formula to determine the sum.

Many times for series, the variable with the highest exponent becomes more important.

An alternating series just needs to be decreasing to be convergent. To be absolutely convergent requires more tests.

The algebra before doing the calculus can be very important. Changing the look of an expression can make it easier to work with.

Knowing the trig functions is necessary for some of these problems.

U-substitution was a useful technique for some of the integration. Before calculus 3, it’s probably a good idea to review integration by parts.

e ~ 2.7

Remember the chain rule when taking the derivative of e^u (most of the time) unless taking the derivative with respect to u.

If you make a substitution in integration, you must change the bounds or label the bounds more carefully.

Two Kinematic Equations Illustrated

I drew this diagram of two equations in physics. I used different colors to show different things. The letter x stands for position, the letter v stands for velocity, the letter a stands for acceleration, and the letter t stands for time. The diagram of the person shows an example of someone starting at a certain position, moving with acceleration, and ending up at another position over an amount of time.

Dibuje este diagrama de dos ecuaciones en física. Use diferentes colores para mostrar diferentes cosas, la letra x expresa la posición, la letra v para expresar la velocidad, la letra a para expresar la aceleración y la laetra t a para expresar el tiempo. El diagrama muestra a un ejemplo de alguien que empieza en un determinado lugar se mueve con cierta aceleración, y termina en otra posición en un determinado tiempo.