Intermediate Value Theorem (+ some continuity and Limits)

Front

What is the Intermediate Value Theorem?

Back

If a function f is continuous on the interval [a, b] and f(a) ≠ f(b), then for every value L between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = L.

Front

Define continuity in the context of functions.

Back

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

Front

What does it mean for a function to be discontinuous?

Back

A function is discontinuous if there is a break, jump, or hole in its graph.

Tags

CCSS.HSF-IF.C.7D

Front

What is a limit in calculus?

Back

A limit is the value that a function approaches as the input approaches a certain point.

Front

Back

The limit is -2.

Front

If f is continuous on [-1,1], f(-1)=4 and f(1)= -2, what can be said about the existence of a zero?

Back

There is a zero between -1 and 1 due to the Intermediate Value Theorem.

Front

Given f(x) = x^3 + x - 3, find an interval that contains a zero.

Back

The interval [1, 2] contains a zero.

Front

What is the definition of a continuous function?

Back

A function is continuous if it is uninterrupted and has no breaks in its graph.

Front

What is the significance of the Intermediate Value Theorem in finding roots?

Back

It guarantees that if a function changes signs over an interval, there is at least one root in that interval.

Front

What does DNE stand for in limits?

Back

DNE stands for 'Does Not Exist', indicating that a limit does not approach a finite value.