Applications of Derivatives Review

Front

What is the derivative of a function?

Back

The derivative of a function measures the rate at which the function's value changes as its input changes. It represents the slope of the tangent line to the function's graph at a given point.

Front

What is the equation of the tangent line to a function at a point?

Back

The equation of the tangent line at a point (a, f(a)) is given by y - f(a) = f'(a)(x - a), where f'(a) is the derivative of the function at x = a.

Front

How do you find the derivative of f(x) = ax^n?

Back

The derivative is found using the power rule: f'(x) = n * ax^(n-1).

Front

What does it mean for a function to be continuous?

Back

A function is continuous if there are no breaks, jumps, or holes in its graph. Formally, f(x) is continuous at x = a if lim (x -> a) f(x) = f(a).

Front

What does it mean for a function to be differentiable?

Back

A function is differentiable at a point if it has a defined derivative at that point. This implies that the function is continuous at that point.

Front

What is the relationship between continuity and differentiability?

Back

If a function is differentiable at a point, it is also continuous at that point. However, a function can be continuous at a point but not differentiable there.

Front

What is the second derivative of a function?

Back

The second derivative is the derivative of the derivative. It provides information about the concavity of the function and can indicate points of inflection.

Front

How do you determine if a function has a local maximum or minimum using derivatives?

Back

To find local maxima or minima, use the first derivative test: find critical points where f'(x) = 0 or is undefined, then analyze the sign of f'(x) around these points.

Front

What is the Mean Value Theorem?

Back

The Mean Value Theorem states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What is a critical point?

Back

A critical point of a function is a point where the derivative is zero or undefined. These points are potential locations for local maxima, minima, or points of inflection.