Limits and Continuity
Limit of f(x)= 2x+3 as x approaches 2.
What is 7?
A hole in the graph at x=2 represents this kind of discontinuity.
What is a removable discontinuity?
The derivative of x2
What is 2x?
The derivative of a constant.
What is 0?
What are corners, cusps, vertical asymptopes, or discontinuities?
These three conditions must be true for a function to be continuous at a point.
What are: the limit exists, the function is defined, and the limit equals the function value?
When the left and right limits at a point are not equal, this type of discontinuity occurs.
What is a jump discontinuity?
This limit expression defines the derivative.
What is f'(x) = limh-->0 [f(x+h) -f(x)]/h?
The derivative of sin(x).
The derivative of 3x3-5x2+2x-1.
What is 9x2-10x+2?
This is the value of limx-->3 (x2-9)/(x-3).
What is 6?
A vertical asymptote on a graph indicates this type of discontinuity.
What is an infinite discontinuity?
The derivative represents these two main ideas about a function.
What are the instantaneous rate of change and the slope of the tangent line?
What is ex?
The line that just touches a curve at one point and has the same slope as the curve there.
What is the tangent line?
This type of discontinuity occurs in (x2-1)/(x-1) at x=1.
What is a removable discontinuity?
This phrase describes what it means when limx-->a f(x)=∞.
What is "the function grows without bound near x=a"?
If f'(a) > 0, the function is doing this at x = a.
What is increasing?
Using the product rule, this is the is the derivative of x2ex?
What is 2xex+x2ex?
The formula y-y1 = m(x-x1) is used to find this equation in calculus problems involving derivatives.
What is the equation of the tangent line?
The limit of (sin(5x))/x as x--> 0.
What is 5?
This theorem guarantees that a continuous function on [a,b] takes every value between f(a) and f(b).
What is the Intermediate Value Theorem?
If f'(2) =0, this means that the tangent line at x=2 is doing this?
What is a horizontal line (slope of 0)?
Using the quotient rule, this is the derivative of (x2)/(x+1).
What is [x(x+2)]/(x+1)2?
This rule is used to differentiate a function inside another function.
What is the Chain Rule?