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.

Finding X and Y Coordinates on a Graph

Slope Intercept Form and Finding the slope

Tutoring Algebra, slope and intercepts

We looked at slope intercept form and how m is the slope and b is the y- intercept.

To get the y-intercept, you set x = 0
to get the x-intercept, you set y = 0

To get the slope, you use two points (some are easier to work with than others)
x1, y1
x2, y2

Then take (y2 – y1)/(x2 – x1)

Subtracting a negative two is like like adding two

x – – 2 = x +2

If you have the slope and a point, you can plug in the x and y to get the value for b and then complete the equation.

Parabola focus and directrix, p value, tutoring precalculus

We reviewed for the test.

Started with parabolas. (h, k) stands for the vertex. The x coordinate being h and the y coordinate being k. In alphabetical order, just like x, y.

The focus is a distance p from the vertex as is the directrix, in opposite directions. The focus is inside the parabola.

If rays of light hit a parabola, they go to the focus.

Also looked at ellipses and hyperbolas.

Power, current, resistance Substituting Ohm’s Law, Tutoring physics with electricity

Amps and current were first thought of with Ben Franklin, so the convention is of positive charge moving.

Ohm’s law is useful when it’s applicable (which is likely throughout this class). V = IR

Power can be a variation of P = I V and power itself is energy divided by time. You can substitute Ohm’s law into that equation for variations.

Opposite charges attract, like charges repel.

Sometimes you will use things like kinematic equations with newer material.

Reviewing Some Ideas From Calculus I and leading up to it

Tutored calculus for a couple of hours the other day, just before the student started Calculus II. Here are some things we went over.

We talked about the purpose of a derivative and how you can optimize things early on and how integrals can find the areas under various curves and on into three dimensions, etc.

Went over how logarithms can be useful, what the natural number e is, and the behavior of the natural logarithm.

How algebra can simplify calculus before doing derivatives or integrals.

How to deal with fractional exponents.

How the order of approach for fractional exponents can simplify things.

Thinking about the behavior of functions for limits.

How exponents like 1/4 and 1/6 work.

How the ln function increases and decreases very slowly.

How to divide by fractions by multiplying by the reciprocal.

Using common denominators.

Going over integration by parts.

Doing some derivatives and keeping track of the sign.

How squared trig functions look.

Factoring out algebraic terms.

Use the product rule when differentiating a product.

Picking u and v for integration by parts won’t always work out, if it seems to not be working, you may want to choose something else.

Doing an integral is taking an anti-derivative and finding something that when you take the derivative of it will give you what you started with between the integral sign and the differential.

Tutoring Precalculus, Permutations and Combinations

Started with sequences. Arithmetic and Geometric. Looked at the relevant formulas then used them.

He needed to figure out whether series were arithmetic or geometric.

And looked at convergence for geometric series.

Also looked at permutations and combinations. Permutations are like a telephone number, the order matters. With combinations, they are more like pizza toppings- mushrooms and olives is like olives and mushrooms, the order does not matter.

Many calculators can compute these.

Tutoring precalculus, asymptotes, holes and end behavior

We mostly looked at functions with polynomials that had asymptotes and holes.

One polynomial had the form of a sum of two cubes, where knowing a related formula is helpful.

To get the coordinates of holes, you plug the x-coordinate that gets a factor of zero in the denominator into the reduced equation.

Finding the behavior close to the asymptote is important, it generally goes up very high or down very low.

To find end behavior, I plug in high magnitude positive and negative numbers. The terms with the highest exponents become more important then.

Tutoring Precalculus, Hyperbolas

We mostly looked at hyperbolas.

For the equations, one term (x or y squared) is positive and the other is negative. The graph will be either vertical for positive y or horizontal for positive x.

h,k are the center. The asymptotes will go through the center.

Sometimes associating something with an image can help. I thought of b/a corresponding with the letter b and being horizontal.

For factoring cubic equations, if you have one zero, you can divide using that information and then get to a quadratic function.