Questions and Answers
What does it mean for f to be continuous at x = a?
lim x→a- (f(x))=lim x→a+ (f(x)) = L and lim x→a (f(x))=(f(a))=L
What does it mean for f to be differentiable at x = a?
f is differentiable at x = a and lim x→a- (f'(x))=lim x→a+ (f'(x))
What does it mean for f to be increasing on the interval (a,b)?
f' > 0 on the interval (a,b)
What does it mean for f to be decreasing on the interval (a,b)?
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What indicates that f has a critical point at x = a?
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What does it mean for f to have a relative minimum at x = a?
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What does it mean for f to have a relative maximum at x = a?
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What indicates that f is concave up on the interval (a,b)?
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What indicates that f is concave down on the interval (a,b)?
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What indicates that f has an inflection point at x = a?
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What does it mean for f to have an absolute minimum at x = a?
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What does it mean for f to have an absolute maximum at x = a?
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What is the Intermediate Value Theorem regarding f(x) = k for some x on the interval [a,b]?
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What is the Mean Value Theorem regarding f'(x) = k for some x on the interval (a,b)?
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What indicates that a particle is at rest at t = k?
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What indicates that a particle changes directions at t = k?
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What indicates that a particle is speeding up or slowing down at t = k?
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Study Notes
Continuity and Limits
- A function ( f ) is continuous at ( x = a ) if the limits from both sides and the function value at that point equal ( L ):
- ( \lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x) = f(a) = L )
Differentiability
- A function ( f ) is differentiable at ( x = a ) if derivatives from both sides at that point are equal:
- ( \lim_{{x \to a^-}} f'(x) = \lim_{{x \to a^+}} f'(x) )
Increasing and Decreasing Functions
- A function ( f ) is increasing on the interval ( (a, b) ) if its derivative ( f' > 0 ) throughout that interval.
- A function ( f ) is decreasing on the interval ( (a, b) ) if its derivative ( f' < 0 ) in that interval.
Critical Points
- A function ( f ) has a critical point at ( x = a ) if ( f'(a) = 0 ) or ( f' ) is undefined.
Relative Extrema
- A function ( f ) has a relative minimum at ( x = a ) if the derivative changes from negative to positive at that point.
- A function ( f ) has a relative maximum at ( x = a ) if the derivative changes from positive to negative at that point.
Concavity
- A function ( f ) is concave up on the interval ( (a, b) ) if the second derivative ( f'' > 0 ) throughout that interval.
- A function ( f ) is concave down on the interval ( (a, b) ) if the second derivative ( f'' < 0 ) in that interval.
Inflection Points
- A function ( f ) has an inflection point at ( x = a ) if ( f''(a) = 0 ) or undefined, and the second derivative changes signs.
Absolute Extrema
- A function ( f ) has an absolute minimum at ( x = a ) if it is a critical point with the lowest value compared to all critical points and endpoints.
- A function ( f ) has an absolute maximum at ( x = a ) if it is a critical point with the highest value compared to all critical points and endpoints.
Theorems
- Intermediate Value Theorem: If ( f ) is continuous on ( [a, b] ) and ( f(a) \leq k \leq f(b) ), there exists at least one ( x ) in ( [a, b] ) such that ( f(x) = k ).
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Mean Value Theorem: If ( f ) is continuous on ( [a, b] ) and differentiable on ( (a, b) ), then there exists at least one ( c ) in ( (a, b) ) such that the slope of the tangent equals the average rate of change:
- ( \frac{f(b) - f(a)}{b - a} = k )
Particle Motion
- A particle is at rest at time ( t = k ) if the velocity ( v(k) = 0 ).
- A particle changes direction at time ( t = k ) when the velocity changes signs.
- A particle is speeding up or slowing down at ( t = k ) depending on whether velocity and acceleration have the same or opposite signs, respectively.
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Explore the essential concepts of continuity and differentiability in calculus with these flashcards. Each card provides key definitions and limits related to functions at specific points. Perfect for students looking to reinforce their understanding of foundational calculus principles.
