Applications of Derivatives - Rolle's & Mean Value Theorem, Concavity, Critical & Inflection Points,

Newton's Method is an iterative process used to approximate the zeros of a function. It involves making an initial guess, using the function and its derivative to refine the guess, and repeating the process until a satisfactory approximation is achieved.

The average rate of change is equal to the slope of the secant line and is calculated as the change in y divided by the change in x, where the change in y is f(b) - f(a) and the change in x is b - a.

A secant line is a line that passes through a curve at two points, while a tangent line touches the curve at only one point.

The slope of the tangent line is equal to the instantaneous rate of change, which is the first derivative evaluated at a specific point on the curve.

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point where the slope of the tangent line equals the slope of the secant line, which is the average rate of change.

Rolle's Theorem states that if a function is continuous on a closed interval, differentiable on the open interval, and has the same y-value at the endpoints, then there is at least one point within the interval where the first derivative is zero, indicating a horizontal tangent.

The absolute minimum is the lowest point of the function on the interval, while the absolute maximum is the highest point. These points occur at the endpoints of the interval or at critical points within the interval.

A critical point is a point on the curve where the first derivative is zero or undefined. It can indicate a local maximum, local minimum, or an inflection point.

The First Derivative Test uses the sign changes of the first derivative to identify local extrema. If the first derivative changes from positive to negative at a point, there is a local maximum. If it changes from negative to positive, there is a local minimum.

The Second Derivative Test uses the sign of the second derivative to identify local extrema. If the second derivative is positive, the function is concave up, indicating a local minimum. If the second derivative is negative, the function is concave down, indicating a local maximum.

An inflection point is a point on the curve where the concavity changes, from concave up to concave down or vice versa. At an inflection point, the second derivative is zero.

L'Hopital's Rule is used to evaluate limits involving the quotient of two functions that approach zero or infinity. It states that the limit of the quotient is equal to the limit of the quotient of their derivatives.

Newton's Method is an iterative process used to approximate the zeros of a function. It involves making an initial guess, using the function and its derivative to refine the guess, and repeating the process until a satisfactory approximation is achieved.

The average rate of change is equal to the slope of the secant line and is calculated as the change in y divided by the change in x, where the change in y is f(b) - f(a) and the change in x is b - a.