sin^2x = (sinx)^2 Squared Trig Function Notation

We went into verifying some equations with trigonometric. They involved angles in both degrees and radians.

Sometimes with equations that had notation of something like sin^2x, which means (sinx)^2 but is the more common notation.

No calculators used in the problems, so the angles all were variations of 45-45-90 or 30-60-90.

Moment of Inertia, Summation of torques

We started by talking about impulse. It is notated using the letter J, not the letter I. In the context of chapter 9, I was the momentum of inertia which has a similar role that mass does for linear motion for rotational motion.

Figuring out the situation of what is going on in a problem can be the most difficult thing. Many times a diagram will help. Thinking about the directions of things can also be useful in getting started. For example with a sliding block and a horizontal restoring force. Also, you can often do problems in several different ways. With the spring and friction, conservation of energy was problem the easiest route.

The other problem had a summation of torques and figuring out the moment of inertia. One mistake would be to not take into account the torque due to the ladder, which can be thought of as a point mass at the center.

Solving Inequalities and finding intervals, Tutoring Precalculus

We started by looking at the Friday Sheet.

With inequalities, the steps are basically

1. Use factoring/etc to find boundary points
2. Plot the boundary points with open or closed circles on a number line
3. Test a point in each of the three regions

Spent a bit of time plotting graphs. Even if you don’t really know the shape, you can plot points. It helps quite a bit to know the shape in advance though.

Reviewed the equation for the vertex of a parabola. x = -b/2a. And also reviewed where you see all of those letters in the general quadratic equation.

Elastic and Inelastic Collisions, Notes from Tutoring Physics

We went over a collision that was in two different parts assumed to be elastic and inelastic.

For an elastic collision, energy is conserved and momentum is conserved.

For an inelastic collision, only momentum is conserved.

So momentum is conserved in either situation.

You can derive equations by using those two ideas for an elastic collision (which takes a few steps). Or you can also use equations that are similar to the purpose of the ‘range equation’ seen previously  in this class.

The work for the inelastic collision ends up being simpler.

Getting Arc Length, Tutoring Precalculus

We started with a word problem trying to find arc length based on latitude and longitude.

Reviewed the equations for the circumference of a circle and the area of a circle.

You can get arc length by multiplying the angle (in radians) by the radians. Thinking of the equation for circumference can help for that since it’s the angle of an entire circle in radians multiplied by the radius. Or you can multiply the fraction of the entire circle by the circumference.

Looked at a few more complicated graphing problems of trig functions.

In which quadrant is the point (-5, 4)?

Saw this question, thought it could be best shown with a visual.

The first coordinate is the x-coordinate, the second is the y-coordinate.


How can DMX unintentionally help you with math? Distinguishing between domain and range

We started with some inequality problems and interval notation. The thing we focused on was square brackets vs curved brackets [ and (. One is inclusive, like a closed dot, and the other is exclusive like an open circle on a number line

When boundary points are found, three regions are divided. It’s good to pick convenient numbers in each region to test them.

We also looked at U notation (union) and the upside down version (intersection). Sometimes that is used. Same meanings as the words ‘or’ and ‘and’.

Domain refers to the horizontal (x-coordinates) a trick to remember that is DMX, if you know who DMX is.

Also plotted a few equations using points.

The sum of perfect cubes equation came into play again.

We looked at how the discriminant can determine whether a quadratic equation can be factored.

And u-substitution was used again.

Tutoring Chemistry, pH Ka, Acids & Bases

You can find the pH by taking the negative log of the concentration of H+ or H3O+ and the pOH by taking the negative log of the OH-

There are various ways to do calculations with Ka and the pH.

We checked out Bronsted-Lowery acids and bases and the theory with proton donors/acceptors.

Also, with chemistry, you want to be very careful with the names since changing them slightly can mean a different substance.

And rounding early is not a good plan since it can introduce inaccuracy.

Variations in plotting trig functions, tutoring Precalculus

We looked at graphing different trigonometric functions with variations.

Started with sin x and cos x. Looked at expansions/contractions, shifts vertically and horizontally.

Used the chocolate trick to remember the less common trig functions

Vertical asymptotes are also useful to plot when they appear.

Tutoring SAT Math Calculator Section

We focused on questions in the calculator section that she was having trouble with.

The first had a bag of marbles and was about finding probabilities. The basic idea is to find the number of specific option(s) you want and divide by the total number of options. There are variations on that.

For another problem, the vertex of a parabola was needed. You can use -b/2a to get the x-coordinate. That can also be derived using calculus by taking the derivative of ax^2 + bx + c and setting it equal to zero since the slope at the vertex is zero.

Sometimes using different forms of an equation, including what is originally listed, can be more useful for certain applications than a derived form.

Getting a common denominator is important for some problems.

The SAT often has answers that you would arrive at by making common mistakes.

Mean, median, and mode can show up. Most common would be mean (average).

sin^2x + cos^2x = 1, that showed up in practice.