Derivatives and Continuity Flashcard

Front

Which of the following functions is not differentiable on its entire domain? Options: f(x) = x^2, f(x) = 1/x, f(x) = sin(x), f(x) = |x|

Back

f(x) = |x|

Front

The derivative of a constant function is always: Options: Zero, One, Infinity, Undefined

Back

Zero

Front

If a function is differentiable at a point, then it is also: Options: Continuous at that point, Discontinuous at that point, Undefined at that point, None of the above

Back

Continuous at that point

Front

A function is differentiable at a point if: Options: The function is continuous at that point, The function has a vertical asymptote at that point, The function has a removable discontinuity at that point, The derivative of the function exists at that point

Back

The derivative of the function exists at that point

Front

Which of the following statements is true about limits and continuity? Options: If a function is continuous at a point, then the limit of the function at that point exists, If the limit of a function at a point exists, then the function is continuous at that point, Limits and continuity are unrelated concepts, None of the above

Back

If a function is continuous at a point, then the limit of the function at that point exists.

Front

Define differentiability in the context of calculus.

Back

A function is differentiable at a point if it has a defined derivative at that point.

Front

What is the relationship between continuity and differentiability?

Back

If a function is differentiable at a point, it must be continuous at that point.

Front

What is the derivative of f(x) = x^3?

Back

f'(x) = 3x^2.

Front

What does it mean for a function to be continuous at a point?

Back

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point.

Front

What is the limit of f(x) = 1/x as x approaches 0?

Back

The limit does not exist as x approaches 0.