What makes a function continuous?
A limit exists for all f(x) approaching a and the lim at f(x) = f(a)
Are all polynomial functions continuous?
What are the local min and max?
Points where f prime evaluates to zero (ie No slope bc at bottom of pit)
What is the derivative of 1 - x?
What is the derivative of -5 + 11 - 6x?
What is the derivative of x/40?
What is the integral of 1/sqrt(x)?
How many feet in a mile?
What is the extreme value theorem?
if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once.
Can a function have no maximum and minimum values?
Yes, x^3 is an example
What is Fermat's Theorem?
If f has a local max or min at c, and if f'(c) exists, then f'(c) = 0.
What is a critical number?
A number c in the domain of f such that either f'(c) = 0 or f'(c) does not exist.
Is the local min or max always a critical number?
What is the closed interval method? (to find absolute max/min)
1. Find values of f at critical points. 2. Find values of f at end points. 3. Largest/smallest wins.
What is the Mean Value Theorem?
f(b) - f(a) = f'(c)(b-a) At a point(s) the tangent line is parallel to the secant line.
When is a function increasing?
f'(x) > 0
When is a function decreasing?
f'(x) < 0
What is the First Derivative test?
A. (f' + to - at c) = (f has loc max at c)
B. (f' - to + at c) = (f has loc min at c)
C. (f' stays the same at c) = (f does not have min/max at c)
If f' is increasing on I what does concavity look like?
Concave up :)
If f' is decreasing on I what does concavity look like?
Concave down :(
What is an inflection point?
Point where curve switches concavity
Limits and Continuity