2) The area problem- integral calculus Uses rectangles to approximate the area under a curve.
Limits: Yes – finally some calculus! Objective: To understand the definition of a limit and to graphically determine the left and right limits and to algebraically determine the value of a limit. If the function f(x) becomes arbitrarily close to a single number L (a y-value) as x approaches c from either side, then lim x c f(x) = L. *A limit is looking for the height of a curve at some x = c. *L must be a fixed, finite number. One-Sided Limits: lim x c+ f(x) =L 1 Height of the curve approach x = c from the right lim x c- f(x) =L 2 Height of the curve approach x = c from the left
Definition of Limit: If lim x c+ f(x) = lim x c- f(x) = L then, lim x c f(x)=L (Again, L must be a fixed, finite number.) f(2) = f(4) = Examples:
Basic Limits (for the book part) lim x 4 2x – 5 = lim x -3 x 2 = lim x cos x = lim x 1 sin =
Important things to note: The limit of a function at x = c does not depend on the value of f(c). The limit only exists when the limit from the right equals the limit from the left and the value is a FIXED, FINITE #! A common limit you need to memorize: (see proof page 65 ) Limits fail to exist: (ask for pictures) 1. Unbounded behavior – not finite 2. Oscillating behavior – not fixed 3. - fails def of limit Do you understand how to graphically find a limit? HW: 1.2 1-31(odd) Use your calculator!