Karteikarten in diesem Deck (20)

• What does it mean for a function to be continuous?

  A function is continuous if you can draw its graph without lifting your pencil from the page.
  calculus continuity
• What is the first criterion for continuity?

  The function has a limit as \(x \to a\).
  calculus continuity
• What does the first criterion for continuity imply?

  The function must approach the same point from both sides as we reach \(x = a\).
  calculus limits
• What is the second criterion for continuity?

  The function is defined at \(x = a\).
  calculus continuity
• What does the second criterion for continuity imply?

  The actual point needs to be defined at \(x = a\).
  calculus functions
• What is the limit definition of continuity at x = a?

  The limit as x approaches 'a' must equal the function's value at 'a'.
  calculus limits continuity
• What condition must a function meet to be differentiable?

  The derivative must exist at x = a, defined by the limit: \(\(\lim_{x \to a} \frac{f(x) - f(a)}{x - a}\)\).
  calculus differentiability
• When is a function differentiable?

  A function is differentiable at x = a if the limit \(\(\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)\) exists.
  calculus differentiability
• What happens if a function is not continuous at x = a?

  If a function is not continuous at x = a, it cannot be differentiable there.
  calculus continuity differentiability
• What is a prerequisite for differentiability at a point?

  Non-existence

  Discontinuity

  Limit

  Continuity
  calculus properties
• What is illustrated by the image regarding limits?

  The image shows that the limit as x approaches 'a' equals the function's value at 'a'.
  calculus limits
• Can a function be continuous but not differentiable?

  Yes, a function can be continuous at a point but not differentiable there.
  calculus functions continuity
• What is an example of a point where a function can be continuous but not differentiable?

  Linear Function

  Smooth Curve

  Cusp

  Corner

  Vertical Tangent Line
  calculus differentiability
• What does the existence of a corner signal in a function?

  A corner indicates a point where the function is continuous but has different slopes.
  calculus geometry
• What is characterized by a cusp in a continuous function?

  A cusp represents a sharp pointed peak where the function is continuous but not differentiable.
  calculus geometry
• What happens at a point with a vertical tangent line?

  A vertical tangent line indicates an infinite slope, making the function non-differentiable at that point.
  calculus tangents
• What x-values make the function not continuous?

  • x = -5 → Jump Discontinuity
  • x = 0 → Infinite Discontinuity
  math continuity
• What x-values make the function not differentiable?

  • x = -4 → Corner
  • x = -5 → Discontinuous
  • x = 0 → Discontinuous
  • x = 5 → Cusp
  math differentiability
• At which x-value does the function have a jump discontinuity?

  x = 5

  x = -5

  x = 0

  x = -4
  math discontinuity
• At which x-value does the function have an infinite discontinuity?

  x = 0

  x = -4

  x = -5

  x = 5
  math discontinuity