Secant/Tangent/Normal Lines

It is important for you to discern the difference between the aforementioned lines.

A secant line is a line that is drawn through two distinct points on a curve. To find the slope of the secant line you just find the change in y and divide by the change in x. Then all we need to do is plug it into point-slope form. y - yk =m (x - xk)

A tangent line is a line that is drawn through only one point on a curve. Right now, we find the slope of the tangent line by finding the limit as h goes to zero of the difference quotient (the change in y divided by the change in x) between our point x=a and a point immediately to the right, x=a+h. Once we do some algebra manipulation we will get a single number as the slope(m). Then all we need to do is plug it into point-slope form. y - yk =m (x - xk) This will be the equation for the tangent line through that single point (a, f(a))

A normal line is a line that is drawn through the same point on the curve as the tangent line was. It is perpendicular to the tangent, therefore its slope is -1/m. This means that only the slope needs to change from your tangent line equation when it is written in point-slope form.
y - yk =(-1/m)* (x - xk) This will be the equation for the normal line through that single point (a, f(a))

Example

Problem 13b from Friday

Above is the solution to problem 13b on page 88. I realized my mistake soon after class. I was in too much of a hurry to think what the question was actually posing. There are two and only two solutions for the standard absolute value equation (abs(x)). If x is positive, abs(x) has the equation y = x. The slope of the tangent line for any positive x on abs(x) is 1. If x is negative, abs(x) has the equation y = -x. The slope of the tangent line for any negative x on abs(x) is -1. This is shown above when x = -3. The relationship mentioned previously is the reason for the negative sign outside the parentheses in the second step. The rest of the steps are sequential and should be easy to follow. I apologize for the mistake and I will apologize in advance for any future ones. Obviously, I am human and that is a general flaw for all of humanity. I will try to minimize it as much as I can as far as calculus goes.

New to Blogging

I thought this would be a great way to communicate the day's lessons to you guys. You would be able to log in daily and check what it is that I taught, pick up any missing pieces, and catch up quickly if you miss class. It may also be a good way for you guys to post some questions about a concept. I am just hoping that I won't get too much spam and random comments, but I will never know unless I try. Posts will be updated period 3 (as long as the school does not block this website) and I will monitor the responses from home.