Intermediate Value Theorem
Flashcard
•
Mathematics
•
12th Grade
•
Practice Problem
•
Hard
+2
Standards-aligned
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1.
FLASHCARD QUESTION
Front
What is the Intermediate Value Theorem?
Back
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different values at the endpoints (f(a) ≠ f(b)), then it must take on every value between f(a) and f(b) at least once within that interval.
2.
FLASHCARD QUESTION
Front
What does it mean for a function to be continuous?
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. A function is continuous on an interval if it is continuous at every point in that interval.
3.
FLASHCARD QUESTION
Front
What is a zero of a function?
Back
A zero of a function f(x) is a value x = c such that f(c) = 0. It represents the x-intercept of the function on a graph.
Tags
CCSS.HSF-IF.C.7D
4.
FLASHCARD QUESTION
Front
How can you determine if a zero exists between two points using the Intermediate Value Theorem?
Back
To determine if a zero exists between two points a and b, check if f(a) and f(b) have opposite signs (f(a) * f(b) < 0). If they do, then by the Intermediate Value Theorem, there is at least one zero in the interval (a, b).
5.
FLASHCARD QUESTION
Front
What is a discontinuity in a function?
Back
A discontinuity occurs at a point in a function where it is not continuous. This can happen due to a jump, infinite behavior, or a removable discontinuity (like a hole in the graph).
Tags
CCSS.HSF-IF.C.7D
6.
FLASHCARD QUESTION
Front
Back
At x = 3, the function f(x) has a discontinuity because the denominator becomes zero, making f(3) undefined.
Tags
CCSS.HSF-IF.C.7D
7.
FLASHCARD QUESTION
Front
Back
This function models the cooling of an object over time, where 90 is the initial temperature, 70 is the ambient temperature, and the exponential term represents the rate of cooling.
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