We started with some problems involving bearings. The direction can get mixed up pretty easily, but the problems were fairly straightforward.
Another problem asked for the exact value of cos 255 degrees without a calculator. It seemed like a half-angle formula might be a good approach.
One problem involved area which meant needing to know the area formula for a triangle 1/2 bh.
And using inverse trig functions was useful. Other problems in the review did require the law of cosines and the law of sines.
We started by looking at graphing ellipses.
Began by with the equation for a circle of radius = 1. Then saw how coefficients change the shape into an ellipse.
The ‘eccentricity’ in a way measures how different an ellipse is from a circle.
Saw how the center is found as well as the axes. X is connected to horizontal and y is connected to vertical.
Ended by looking at hyperbola and a Friday sheet that included Sigma (summation) notation.
We first looked at getting a bearing measurement between three cities.
The law of cosine was used. As more information is found, you can use the Law of Sine and properties of triangles.
Another problem had a unit vector in which the magnitude was 1.
And other problems involved adding vectors which meant sometimes multiplying them by coefficients and then lining them up head to tail.
We started with a problem involving gravitational potential energy, kinetic energy, and elastic potential energy with a mass and a spring.
Work and energy both have units of Joules.
Made an analogy to a trampoline for the spring. Used the concept of conservation of energy. Energy changed forms a few times.
Another problem involved friction and an incline with and without friction. The version with friction was a bit more complicated and required more algebra, factoring out common terms.
For a calculus problem, getting the volume in terms of one variable using a ratio of radius/height was helpful.
We started by looking at problems from the most recent test.
Getting the x and y intercepts just involves setting one variable or the other equal to zero since that is where the equation will intersect the axes.
A quadratic equation will generally have two solutions, it can be less.
To find horizontal asymptotes, I just plug in a large number for x and a negative large magnitude number. You can think of positive/negative infinity or even positive/negative 100.
It helps to know the general shapes of functions, lines, parabolas, cubic functions, etc.
The hardest problem might have been figuring out the equation of a cubic function based on the graph. Two zeros were given as well as another point. That means that one of the terms with a zero would be squared since there were only two zeros.
We looked at problems involving the Law of Sines and the Law of Cosines. You can use these equations with any triangle, they do not have to be right triangles. Depending on the information you have, you use one or the other.
You’ll have two sides and an angle generally or two angles and a side. And as you fill in more information you may want to switch to the Law of Sines since it tends to be less work.
And if you have two angles, you can just subtract them from 180 degrees to get the third angle. The law of cosines is really like a more general form of the Pythagorean Theorem, it has a similar function but does not require the triangle to be a right triangle.
Using memory in your calculator can be helpful for these problems as well since you often need to combine several numbers that are not whole numbers.
Why you should move beyond ‘undefined’ and think more in terms of infinity and negative infinity for some problems
Some other problems involved looked at shapes of graphs.
Division by zero should not longer be thought of as simply ‘undefined’. In some of these problems it indicated a vertical asymptote which can be graphed.
The values are closer related to positive and negative infinity.
Besides features like that though, you can simply plot points to get the shape of the equation. But you need to be careful to not cross the asymptotes, etc.
A few of the problems involved synthetic division.
They might start as third degree or fourth degree polynomials with one or more of the roots given. There was an equation that would give a second root if one was a complex number (if all the coefficients of the original polynomial are real).
The complex conjugate will also be a root.
For example, if 1 + i is a root, then 1- i is also a root. (given the right conditions)
If you have two roots and use synthetic division (or long division) you will have two roots and a quadratic equation, which will be much easier to factor.
We looked some at the previous test. But focused on material for the upcoming test.
One thing was the product rule, he had made a mistake before with it on another test.
Often there were small algebraic errors. The new material seems fairly comfortable though.
If you make a mistake early on it can continue to be in your work for the entirety of the problem and depending on the direction of where you go with the problem, it can get bigger. Let’s say you have four instead of two, then you square that number, suddenly the mistake got larger.
Calculus involves a lot of algebra, geometry, etc. Sometimes there can be many steps and mistakes add up and get bigger. Often times it is better to slow down a bit and be careful rather than hunt for mistakes later on.