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Differentiability quiz #1 Flashcards

Differentiability quiz #1
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  • What are the necessary conditions for a function to be differentiable at a point?
    A function is differentiable at a point if it is continuous at that point and has no sharp corners or turns there, meaning the left and right derivatives are equal.
  • Can a function be continuous at a point but not differentiable there? Explain with an example.
    Yes, a function can be continuous but not differentiable at a point if it has a sharp corner or cusp there. For example, the absolute value function f(x) = |x| is continuous at x = 0 but not differentiable at x = 0 because the left and right derivatives are not equal.
  • How do you determine if a piecewise function is differentiable at the boundary between its pieces?
    To determine if a piecewise function is differentiable at a boundary, first check that the function is continuous at that point by ensuring the left and right function values are equal. Then, check that the left and right derivatives at the boundary are also equal. If both conditions are met, the function is differentiable at the boundary.
  • What two main conditions must be met for a function to be differentiable at a point?
    The function must be continuous at that point and have no sharp corners or turns there, meaning the left and right derivatives are equal.
  • Can a function be continuous at a point but not differentiable there? Give an example.
    Yes, for example, f(x) = |x| is continuous at x = 0 but not differentiable there because the left and right derivatives are not equal.
  • What is the first step in checking if a piecewise function is differentiable at the boundary between its pieces?
    First, check that the function is continuous at the boundary by ensuring the left and right function values are equal.
  • After confirming continuity at a boundary in a piecewise function, what must you check next to determine differentiability?
    You must check that the left and right derivatives at the boundary are also equal.
  • Why are polynomials always differentiable everywhere?
    Polynomials are always differentiable because they are continuous and have no sharp corners or discontinuities.
  • What graphical feature indicates that a function is not differentiable at a point, even if it is continuous there?
    A sharp corner or cusp in the graph at that point indicates the function is not differentiable there.
  • If a function is not continuous at a point, what can you immediately conclude about its differentiability at that point?
    If a function is not continuous at a point, it is not differentiable at that point.