Questions and Answers
What is continuity at a point?
A function is defined, the limit as x approaches c of f(x) exists, and f(c) equals the limit as x approaches c of f(x).
What does the Intermediate Value Theorem state?
If f(x) is continuous on [a,b] and k is any number between f(a) and f(b), there exists at least 1 number c in [a,b] such that f(c)=k.
What are critical numbers?
Let f be defined at c; if f'(c)=0 or f'(c)=DNE, then c is a critical number.
What does the Mean Value Theorem state?
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What does Rolle's Theorem state?
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What are ways derivatives fail to exist?
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Study Notes
Continuity at a Point
- A function must be defined at a point for continuity.
- The limit of f(x) as x approaches c must exist.
- The value of the function at c must equal the limit as x approaches c: f(c) = lim (x→c) f(x).
Intermediate Value Theorem
- A function that is continuous on a closed interval [a, b] guarantees that any value k between f(a) and f(b) will be achieved.
- There exists at least one c in the interval [a, b] such that f(c) = k.
Critical Numbers
- Critical numbers occur at points where the derivative f'(c) is either zero or does not exist (DNE).
- These points are significant in identifying local extrema of the function.
Mean Value Theorem
- For a function continuous on [a, b] and differentiable on (a, b), there is at least one point c in [a, b] where the derivative equals the average rate of change:
- f '(c) = [f(b) - f(a)] / (b - a).
Rolle's Theorem
- When a function is continuous on [a, b], differentiable on (a, b), and the function values at the endpoints are equal (f(a) = f(b)), there exists at least one point c in [a, b] where the derivative is zero: f'(c) = 0.
Ways Derivatives Fail to Exist
- A derivative may not exist due to discontinuity in the function.
- It fails at corners, sharp turns, or cusps where the slope is undefined (e.g., f(x) = |x| at x=0).
- A vertical tangent line also indicates a point where the derivative does not exist.
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Description
Test your knowledge of key concepts in Calculus with these flashcards covering important definitions from Chapters 1 to 3. Topics include continuity, the Intermediate Value Theorem, and critical numbers. Enhance your understanding and retention of foundational calculus principles.
