Unit 2 Flashcard Review: Definition of a derivative

Front

What is the definition of a derivative?

Back

The derivative of a function at a point is the instantaneous rate of change of the function with respect to its variable at that point.

Front

What does f'(x) represent?

Back

f'(x) represents the derivative of the function f(x), indicating the slope of the tangent line to the graph of f at the point x.

Front

How do you find the equation of the tangent line to a function at a given point?

Back

The equation of the tangent line can be found using the point-slope form: y - f(a) = f'(a)(x - a), where a is the point of tangency.

Front

What is the significance of f'(2) = -1 in the context of a function?

Back

It indicates that at x = 2, the slope of the tangent line to the function f(x) is -1, meaning the function is decreasing at that point.

Front

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

Back

A function is differentiable at a point if it has a defined derivative at that point, which implies that the function is continuous and has no sharp corners or vertical tangents there.

Front

Where is the graph of a function not differentiable?

Back

The graph is not differentiable at points where there are sharp corners, vertical tangents, or discontinuities.

Front

What is the relationship between the derivative and the slope of a tangent line?

Back

The derivative at a point gives the slope of the tangent line to the graph of the function at that point.

Front

Back

This limit represents the definition of the derivative of f at the point a, as x approaches a.

Front

What is the instantaneous rate of change?

Back

The instantaneous rate of change of a function at a point is the value of the derivative at that point.

Front

How do you determine if a function is increasing or decreasing using its derivative?

Back

If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.