Today we're going to begin knowledge area 1 of the FE review exam, mathematics. And here is the outline from the reference handbook. These are the topics covered. Analytic geometry, calculus, roots of equations, vector analysis. But this doesn't cover very well the actual topics that I think are most likely going to be important, so this is the outline that I will follow. Analytic geometry and trigonometry, algebra, vectors, series, calculus, and then differential equations. So here again is the outline, and in this segment we'll start looking at analytic geometry and trigonometry. In particular, we first of all do an overview and then look at simple equations of straight lines, then polynomials and conics, and trigonometry. And in this sector, we'll just look at an overview first of all and then equations and properties of straight lines. Now the FE Reference Handbook contains many topics related to mathematics and here is just some of them here. It first starts talking about discreet math and symbols and then directed graphs, digraphs, goes on into finite state machines and then characteristics, adjectives, subjectives, bijectives, et cetera. But generally although these things are listed, I don't think they are going to be important topics and I very much doubt that they're going to be covered in the exam, so I won't cover them here. Instead, let's start with some very simple things; the equations of a straight line. So, here we have a straight line in two-dimensional space in xy passing through two points Points x1, y1 and x2, y2 with a slope of m. So here is the relevant section from the reference handbook. They give general forms of this equation, standard forms and relationships to formal line passing through two points. And reproducing them here, first of all the general form of the equation, of a straight line is Ax + By + C where A, B, and C are constant coefficients. The next form which is commonly known used is called the standard form y is equal to mx plus b where in this equation m is the slope of the line, in other words, the raise m in horizontal distance 1. And b is the y intercept or intersect the point where the straight line crosses the y axis. So in this case, of course, m is negative. The slope of the line is negative for this example. Another form is the point slope form, which gives the equation of the line of given slope m, passing through a point with coordinates x1 and y1, as shown here. And in addition, the slope of the line between those two points is simply equal to the height difference, which is y2 minus y1, divided by the horizontal distance, x2 minus x1. And some other important points to remember are that the slopes of parallel lines are equal. And the product of the slopes of two perpendicular lines, in other words which are rectangles to each other, is equal to minus 1. So, let's do an example on that. Question is, the slope of a line that connects two points with coordinates minus 1 and 2, and, 3 and minus 1, is most nearly which of these alternatives. So, we can compute the slope from this equation that we had on the previous slide, y2 minus y1 divided by x2 minus x1. So that is therefore equal to y2 is minus 1, y1 is 2, and in the denominator x2 is 3 and x1 is minus 1. So the answer, the slope is negative, or minus, three quarters. And the answer is B. The second problem relating to the same two points is that we now want to compute the slope of the line which is perpendicular to the line which is connecting those two points. Which of these alternatives is it? So, in this case we already know the slope of the line of line 1, we've already calculated it here. I'll call that m 1, is equal to minus three-quarters. And the line which is perpendicular to that, m2, is minus m1 over m1, because the product of those two slopes is equal to minus 1 if they're perpendicular to each other. Therefore the slope of the line which is perpendicular is equal to four thirds and the answer is B. Next example, the slope of a line whose equations is given by y plus x equals 2, is which of these? So first it's easiest to just write that equation. In the standard form y equals mx plus b. In other words, in this case, rewriting the equation we have y is equal to minus x plus 2, from which we see that m here is equal to minus 1. And B is equal to 2. So the answer is the slope of the line is minus 1. Next question. The standard form of the equation of a straight line which has a slope of 2, that intersects the y-axis at 3 is which of these alternatives? So in this case, we start from the standard form here, y is equal to mx plus b, because we're given the slope and the intersect. So we see that the slope is 2. In other words, m is equal to 2 and the intercept is 3. So B is equal to 3. So therefore, the standard form is y is equal to 2x plus 3 and the answer is A. Last example, the general form of the equation for a line that intersects the x axis at 2, and the y axis at 6, is which of these? So what we're looking for here is the general form of the equation, which is like this. Ax plus By plus C is equal to 0. But in this case, it's also easy, convenient to write it in standard form, y is equal to mx plus B. And we can compute the slope. So here is the picture. The line intersects the x axis at 2, in other words a point which has coordinates of 2 and 0, and the y axis at 6, in other words a point which has coordinates of 0 and 6. So, we can take those two points as our points 1 and 2. And the slope of the line between those two points is again equal to y2 minus y1 over x2 minus x1. So, y2 is 0. y1 is 6. x2 is 2. x1 is 0. So, the slope of the line is minus 3. And we also already know the intersect or the intercept is at B is equal to 6. So therefore the equation of the line is y equals minus 3x plus 6 and rewriting that in standard form we have y plus 3x minus 6 is equal to 0. So the answer is C. And this concludes our discussion of straight lines.
