11-22-2011, 04:30 PM
Lesson 2: http://www.supportforums.net/showthread.php?tid=23421
The basic idea of a limit is that the graph approaches a point at some c. It should come at no surprise, then, that the definition of a limit is that the limit as x approaches c of f(x) equals L. This statement implies that the limit exists and that there is only one unique limit at this point. The graph does not have to exist at that point, or even be defined at it. The most important concept in differential calculus is how to find the slope of the tangent line. This is first done by finding the slope of the secent line, a line drawn between two different points on a graph, with the equation:
f(x + Δx) - f(x) over Δx
You may notice that this question is the difference quotient, where h equals Δx. As Δx approaches x, the slope of the secant line becomes closer and closer to the slope of the tangent line at that point. This is the purpose of finding the limit.
The conditions that would show a limit does not exist are:
It is important to remember these three truths.
Sum or Difference:
Calculus – With Daniel Faraday
What is calculus?
- Calculus is the study of change. Before calculus, the student focuses on things that are constant and defined at a location or coordinate, a number, a shape, or the area within a shape. Calculus focuses on the slope of a curve, the area under it, and things of that nature.
Why should I study calculus?
- Calculus is the gateway to higher and more complex branches of mathematics. Not only does it focus on drawing lines between previous concepts, it also focuses on building up to future ideas. It is a very complex form of math, but it holds very strong rewards for those who choose to go into it.
Lesson 1 - Limits
The basic idea of a limit is that the graph approaches a point at some c. It should come at no surprise, then, that the definition of a limit is that the limit as x approaches c of f(x) equals L. This statement implies that the limit exists and that there is only one unique limit at this point. The graph does not have to exist at that point, or even be defined at it. The most important concept in differential calculus is how to find the slope of the tangent line. This is first done by finding the slope of the secent line, a line drawn between two different points on a graph, with the equation:
f(x + Δx) - f(x) over Δx
You may notice that this question is the difference quotient, where h equals Δx. As Δx approaches x, the slope of the secant line becomes closer and closer to the slope of the tangent line at that point. This is the purpose of finding the limit.
The conditions that would show a limit does not exist are:
- The left hand and right hand side do not approach the same point.
- F(x) decreases or increases without bound as x approaches c.
- F(x) oscillates between two fixed values as x approaches c.
It is important to remember these three truths.
- The limit as x approaches c of b is b.
- The limit as x approaches c of x is c.
- The limit as x approaches c of x^n is c^n.
- Where b and c are real numbers and n is a positive integer.
Sum or Difference:
- lim[f(x) ± g(x)] = L ± K
- lim[f(x) * g(x)] = LK
- lim[f(x)/g(x)] = L/K (provided K ≠ 0)
- lim[f(x)]^n = L^n