ROC, Tangent Lines, and Derivative

Front

What is the first derivative of a function?

Back

The first derivative of a function, denoted as f'(x), y', or dy/dx, represents the rate of change of the function with respect to its variable.

Front

What does the limit definition of the first derivative state?

Back

The limit definition of the first derivative states that f'(x) = lim (h -> 0) [(f(x+h) - f(x))/h], which describes the slope of the tangent line to the curve at a point.

Front

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

Back

To find the equation of a tangent line, calculate the derivative at the point to find the slope, then use the point-slope form of a line: y - f(a) = f'(a)(x - a).

Front

What is the average rate of change of a function?

Back

The average rate of change of a function between two points is calculated as (f(b) - f(a)) / (b - a), representing the slope of the secant line connecting the two points.

Front

If a function's derivative is negative, what does this indicate about the function?

Back

If a function's derivative is negative, it indicates that the function is decreasing in that interval.

Front

What is the significance of a positive first derivative?

Back

A positive first derivative indicates that the function is increasing in that interval.

Front

What does the second derivative tell us about a function?

Back

The second derivative indicates the concavity of the function: if it is positive, the function is concave up; if negative, concave down.

Front

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

Back

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

Front

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

Back

A local maximum occurs where the first derivative changes from positive to negative, and a local minimum occurs where it changes from negative to positive.

Front

What is the derivative of f(x) = 4x^2 + 2x - 1?

Back

The derivative f'(x) = 8x + 2.