Limits and Continuity
What is a limit?
The limit of a function is the y value on a graph as we get infinitely close to a given x value
What is the tangent line of a function?
The tangent line is a line at a point x=a, which passes through the function at (a, f(a)), and which has the same slope as the function's slope at that point.
What is the derivative of a function?
The rate of change of one variable with the respect to another variable.
What is the formula for finding a limit
limit x-->a f(a) = L
What are the three major types of discontinuity?
Removable Discontinuity, Jump Discontinuity, Infinite Discontinuity
Find the slope of the tangent line from the equation on the board.
mt = 2
Solve for the derivative using the equation written on the board.
4x - 3
Define the conditions and conclusions of Intermediate Value Theorem
If f(x) is:
Continuous on [a,b]
y0 is between f(a) and f(b)
then there exists a c in (a,b) such that f(c) = y0
note: c may not be unique
Solve the equation written on the board
-1/9
Find the slope of the tangent line from the equation on the board.
1/4
Solve for the derivative using the equation written on the board.
1/sqrt(2x-1)
What is the formula for the slope of the tangent line?
m = limit x-->a (f(x)-f(a))/(x-a)
On the graph shown on the board, identify all discontinuities.
x is discontinuous at:
3, 4, -1, -2
Find the equation of the tangent line from the equation on the board.
y = 3x - 3
Solve for the derivative using the equation written on the board.
2/(5-x)^2
What conditions make a function differentiable?
f is continuous
f is smooth
f does not have a vertical tangent line
Solve the limit equation written on the board
5
Find the equation of the tangent line.
y = -1/4x - 3/4
Solve for the derivative using the equation written on the board.
3/(2-x)^2
What is the limit definition of the derivative?
f'(x) = limit h-->0 (f(x+h)-f(x))/h