Calculus Flashcards: Continuity and Differentiability

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 ).
• 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.