I have a math test coming up on Friday (it is currently Monday). I don't feel prepared but I don't know where to even start studying. What do I do?
First thing is first, put away every piece of technology. No exceptions. Keep a calculator only if you are allowed one on the exam. Studying math with background classical music can be helpful, but if there wont be music during the exam, it is best to practice without it. On your first day studying, create a master list of all of the topics you will be tested on. If you missed a class or a homework assignment, put emphasis on that topic. Generally for math, teachers will provide practice problems for each section. Divide the problems up for Monday, Tuesday and Wednesday, sit down and work through them all without looking at an answer key or a solution. Once you have struggled through them, grade them. Redo them, even the ones you got right. This way you will have higher confidence on the problems you already know how to do. That confidence will ease your nerves on an exam. Try to avoid doing any of these problems for the first time on Thursday, use Thursday to ask questions and go over the ones you still don't understand. Look to your notes as well for hints as to what your teachers will test you on. Finally, when test day comes around, do the easiest problems first, to boost confidence. Then tackle the difficult ones. If you get stuck, move on to the medium level questions so you don't run out of time. Show every single thing that goes through your mind as you solve a problem and keep your work organized. Teachers can give you more partial credit if they can see your thought process was on the right track. Above all, choose three things you are thankful for before the exam and remind yourself that even though studying and test-taking can be rough, those three things will still be there later.
What is optimization? And why does the derivative help with optimization problems?
Optimization is a very practical application of calculus. We use it to design running tracks with the least amount of material that still meet the correct criteria. We use it to fit the full 12 ounces of soda in a can using the least amount of aluminum possible. Optimization is essentially making the most out of the bare minimum criteria. We learned previously that derivatives give the instantaneous slope of a graph. If the derivative is zero, we know the graph has leveled off and is likely changing directions. Previously we learned how to check if the point where the derivative is zero is a local maximum or minimum. Finding this max or min by taking the derivative of provided/derived mathematical models allows us to find dimensions or data regarding the best way to approach a situation.
I am given two equations and I need to determine if the two lines will intersect at any given point. If they intersect, I need to determine where they will cross paths. Where do I start? How can I check my work?
To determine if two lines cross, you need to begin by finding their slopes. The slope of a line is the "m" in the point-slope equation: y = mx + b. If you are given an equation that fits the point-slope form, then you can determine the slope that way. For instance, in the equation, y = 2x + 3, the slope is 2. You may have to do some rearranging, for instance, if you have 10y = 20x + 3, you will have to divide everything by 10 in order to isolate y. If you are not given an equation, and you are simply given two points, you can calculate the slope. First decide which point is plotted further to the left. This will be your x1 and y1 values. The point furthest to the right will be your y2 and y1 values. The slope is rise (y-values; rise and y kind of rhyme, if that helps) over run (x-values). Calculate by doing (y2-y1)/(x2-x1). Once you determine the slopes, it is time to figure out if they are the same. If the slopes are the same numerical value, the lines are parallel and will never intersect. There is an exception. If the lines are exactly the same, they intersect at every point. If they are different, then set both equations equal to each other. For instance, y = 2x +3 and y = 4x + 9 can be rewritten as 4x + 9 = 2x + 3, and x = -3. This is the x-value where they intersect. Take x=-3 and plug it into either equation to figure out what the y-component of the intersection is.