AP Calculus Unit 2 Review

What is the definition of the derivative?

• The derivative is the limit of the average rate of change of a function as the interval approaches zero.

• The derivative is the slope of a line at any point on a graph.

• The derivative measures the total distance traveled by a function.

• The derivative is the integral of a function over a given interval.

Mathematical Equivalence

ON

What is the quotient rule formula for derivatives?

• (u/v)' = (u'v + uv') / v^2

• (u/v)' = (u'v - uv') / v

• (u/v)' = u' / v'

• (u/v)' = (u'v - uv') / v^2

Explain the concept of continuity in relation to functions.

• A function is continuous if it is defined at a point, the limit exists at that point, and the limit equals the function's value.

• A function is continuous if it has a maximum value.

• A function is continuous if it is always increasing.

• A function is continuous if it has no asymptotes.

What are the conditions for a function to be differentiable at a point?

• A function is differentiable if it has a maximum or minimum at that point.

• A function is differentiable if it is always increasing.

• A function is differentiable at a point if it is continuous there and the limit of the difference quotient exists.

• A function is differentiable if it is defined at that point.

If f(x) is continuous at x = a, does it imply that f(x) is differentiable at x = a? Why or why not?

• No, continuity does not imply differentiability.

• If f(x) is continuous, it must have a derivative at that point.

• Continuity and differentiability are the same concept.

• Yes, differentiability is guaranteed by continuity.